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S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end

 /gnudict 256 dict def gnudict begin /Color false def /Solid false
def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift
-33 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def
/vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto}
bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul
def /Lshow { currentpoint stroke M   0 vshift R show } def /Rshow {
currentpoint stroke M   dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M   dup stringwidth pop -2 div vshift
R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor
Solid {pop []} if 0 setdash }  {pop pop pop Solid {pop []} if 0 setdash}
ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL
{ stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth
mul /userlinewidth exch def       10 mul /udl exch def } def /PL {
stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0
{ PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 {
PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl
1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL
} def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8
{ PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash    gsave 1 setlinecap M 0 0 V stroke grestore
} def /Dia { stroke [] 0 setdash 2 copy vpt add M   hpt neg vpt neg
V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke   Pnt }
def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V   currentpoint stroke
M   hpt neg vpt neg R hpt2 0 V stroke   } def /Box { stroke [] 0 setdash
2 copy exch hpt sub exch vpt add M   0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke   Pnt } def /Crs { stroke [] 0 setdash
exch hpt sub exch vpt add M   hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash
2 copy vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke   Pnt  } def /Star { 2 copy
Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add
M   0 vpt2 neg V  hpt2 0 V  0 vpt2 V   hpt2 neg 0 V  closepath fill
} def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt
-1.62 mul V   hpt 2 mul 0 V   hpt neg vpt 1.62 mul V closepath fill
} def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M   hpt neg
vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V closepath
stroke   Pnt  } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V   hpt 2 mul 0 V   hpt neg vpt -1.62 mul V
closepath fill} def /DiaF { stroke [] 0 setdash vpt add M   hpt neg
vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V closepath fill }
def /Pent { stroke [] 0 setdash 2 copy gsave   translate 0 hpt M 4
{72 rotate 0 hpt L} repeat   closepath stroke grestore Pnt } def /PentF
{ stroke [] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt
L} repeat   closepath fill grestore } def /Circle { stroke [] 0 setdash
2 copy   hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash
hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90
450  arc } bind def /C1 { BL [] 0 setdash 2 copy        moveto    
   2 copy  vpt 0 90 arc closepath fill                vpt 0 360 arc
closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto        2 copy
 vpt 90 180 arc closepath fill                vpt 0 360 arc closepath
} bind def /C3 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0
180 arc closepath fill                vpt 0 360 arc closepath } bind
def /C4 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 180 270
arc closepath fill                vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto        2 copy  vpt 0 90 arc   
    2 copy moveto        2 copy  vpt 180 270 arc closepath fill   
            vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill               vpt 0 360
arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto      
2 copy  vpt 0 270 arc closepath fill               vpt 0 360 arc closepath
} bind def /C8 { BL [] 0 setdash 2 copy moveto       2 copy vpt 270
360 arc closepath fill               vpt 0 360 arc closepath } bind
def /C9 { BL [] 0 setdash 2 copy moveto       2 copy  vpt 270 450 arc
closepath fill               vpt 0 360 arc closepath } bind def /C10
{ BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto        2 copy vpt 90 180 arc closepath fill  
             vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash
2 copy moveto        2 copy  vpt 0 180 arc closepath fill        2
copy moveto        2 copy  vpt 270 360 arc closepath fill         
      vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy
moveto        2 copy  vpt 180 360 arc closepath fill              
 vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash  2 copy
moveto        2 copy  vpt 0 90 arc closepath fill        2 copy moveto
       2 copy  vpt 180 360 arc closepath fill                vpt 0
360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 
      2 copy  vpt 90 360 arc closepath fill                vpt 0 360
arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath
fill                vpt 0 360 arc closepath } bind def /Rec   { newpath
4 2 roll moveto 1 index 0 rlineto 0 exch rlineto        neg 0 rlineto
closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub
exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy
moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2
copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy
exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0
setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill
Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec
fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch
vpt sub vpt vpt2 Rec fill        2 copy vpt Square fill        Bsquare
} bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare
} bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare
} bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2
copy exch vpt sub exch vpt Square fill        Bsquare } bind def /S11
{ BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub
exch vpt2 vpt Rec fill        Bsquare } bind def /S12 { BL [] 0 setdash
2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec
fill        2 copy vpt Square fill Bsquare } bind def /S14 { BL []
0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill      
 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 {
BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave
translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate
45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45
rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate
0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0
0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0
S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6
stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke
grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore
} bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore }
bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore }
bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore }
bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore }
bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore }
bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore }
bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore }
bind def /DiaE { stroke [] 0 setdash vpt add M   hpt neg vpt neg V
hpt vpt neg V   hpt vpt V hpt neg vpt V closepath stroke } def /BoxE
{ stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2 neg V hpt2
0 V 0 vpt2 V   hpt2 neg 0 V closepath stroke } def /TriUE { stroke
[] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V   hpt 2 mul
0 V   hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke
[] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt 2 mul
0 V   hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def /CircE { stroke [] 0 setdash   hpt
0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore
0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M   hpt
neg vpt neg V hpt vpt neg V   hpt vpt V hpt neg vpt V Opaque stroke
} def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M   0 vpt2
neg V hpt2 0 V 0 vpt2 V   hpt2 neg 0 V Opaque stroke } def /TriUW {
stroke [] 0 setdash vpt 1.12 mul add M   hpt neg vpt -1.62 mul V  
hpt 2 mul 0 V   hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW {
stroke [] 0 setdash vpt 1.12 mul sub M   hpt neg vpt 1.62 mul V   hpt
2 mul 0 V   hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke
[] 0 setdash gsave   translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def /CircW { stroke [] 0 setdash   hpt 0
360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore
} def end
 
@fedspecial end
%%BeginFont: PazoMath-Italic
%!PS-AdobeFont-1.0: PazoMath-Italic 001.001
%%CreationDate: Thu Sep 14 17:4:17 2000
%%VMusage: 120000 150000
11 dict begin
/FontInfo 14 dict dup begin
/version (001.001) readonly def
/Copyright ((c) Diego Puga, 2000.) readonly def
/Notice (Copyright (c) Diego Puga, 2000. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def
/FullName (Pazo Math Italic) readonly def
/FamilyName (PazoMath) readonly def
/ItalicAngle -9.50 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (Regular) readonly def
end readonly def
/FontName /PazoMath-Italic def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 97 /alpha put
dup 98 /beta put
dup 100 /delta put
dup 101 /epsilon1 put
dup 104 /eta put
dup 108 /lambda put
dup 109 /mu put
dup 112 /pi put
dup 115 /sigma put
dup 182 /partialdiff put
readonly def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.00100 0 0 0.00100 0 0] readonly def
/UniqueID 5116634 def
/FontBBox {-70 -277 902 733} readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: PazoMath
%!PS-AdobeFont-1.0: PazoMath 001.001
%%CreationDate: Thu Sep 14 17:4:17 2000
%%VMusage: 120000 150000
11 dict begin
/FontInfo 14 dict dup begin
/version (001.001) readonly def
/Copyright ((c) Diego Puga, 2000.) readonly def
/Notice (Copyright (c) Diego Puga, 2000. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def
/FullName (Pazo Math) readonly def
/FamilyName (PazoMath) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (Regular) readonly def
end readonly def
/FontName /PazoMath def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 68 /Delta put
dup 165 /infinity put
dup 181 /proportional put
dup 213 /product put
dup 229 /summation put
readonly def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.00100 0 0 0.00100 0 0] readonly def
/UniqueID 5116633 def
/FontBBox {-40 -283 878 946} readonly def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: PazoMathBlackboardBold
%!PS-AdobeFont-1.0: PazoMathBlackboardBold 001.001
%%CreationDate: Thu Sep 14 17:4:17 2000
%%VMusage: 120000 150000
11 dict begin
/FontInfo 14 dict dup begin
/version (001.001) readonly def
/Copyright ((c) Diego Puga, 2000.) readonly def
/Notice (Copyright (c) Diego Puga, 2000. Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def
/FullName (Pazo Math Blackboard Bold) readonly def
/FamilyName (PazoMathBlackboardBold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (BlackboardBold) readonly def
end readonly def
/FontName /PazoMathBlackboardBold def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 82 /R put
readonly def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.00100 0 0 0.00100 0 0] readonly def
/UniqueID 5116637 def
/FontBBox {15 -176 934 709} readonly def
currentdict end
currentfile eexec
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%%EndFont 
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 @endspecial 679 774 a(T)-5 b(itel:)845 922 y FY(Neuralt)20
b(netv\346rk)h(til)g(tegnkendelse)679 1163 y FT(Projektperiode:)845
1278 y FY(P2,)845 1392 y(4.)f(febr)o(uar)f(\226)i(27.)e(maj,)i(2002)679
1688 y FT(Projektgruppe:)845 1802 y FY(B333)679 2098
y FT(Gruppemedlemmer:)845 2213 y FY(Jasper)e(Kjersgaar)o(d)h(Juhl)845
2327 y(Anders)g(Rune)h(Jensen)845 2442 y(Lau)f(Bech)g(Lauritzen)845
2556 y(Rasmus)h(J\370r)o(gensen)845 2671 y(Martin)f(Qvist)845
2785 y(Ole)h(Laursen)679 3081 y FT(V)-9 b(ejleder:)845
3196 y FY(Thomas)21 b(D.)g(Nielsen)679 3491 y FT(Bivejleder:)845
3606 y FY(Claus)f(Monrad)g(Spliid)679 3938 y FT(Antal)f(kopier:)g
FY(10)679 4119 y FT(Rapport)h(\226)g(sideantal:)d FY(93)679
4300 y FT(Appendiks)h(\226)i(sideantal:)d FY(6)679 4481
y FT(T)-9 b(otalt)19 b(sideantal:)f FY(99)p 2431 856
1613 4 v 2431 4405 4 3550 v 2480 939 a FT(Synopsis:)2480
1044 y FU(Et)35 b(af)g(de)f(omr\345der)g(hvor)g(neurale)h(netv\346rk)f
(ud-)2480 1149 y(m\346rker)27 b(sig)h(er)f(tegngenkendelse)h(som)g(har)
h(fun-)2480 1254 y(det)39 b(anvendelse)g(p\345)g(h\345ndholdte)h
(computer)o(e.)e(I)2480 1359 y(rapporten)19 b(unders\370ges)e
(mulighederne)g(for)h(at)i(an-)2480 1464 y(vende)40 b(et)h(fr)o
(emadf\370r)o(ende)c(neuralt)j(netv\346rk)g(til)2480
1569 y(at)26 b(genkende)f(enkeltst\345ende)g(h\345ndskr)o(evne)g(tegn.)
2480 1674 y(Optr\346net)16 b(med)f(fejlfunktionen)h(sum)f(af)h
(kvadrater)2480 1779 y(vil)21 b(dette)f(netv\346rk)g(maksimer)o(e)f
(sandsynligheden)2480 1884 y(for)j(rigtig)e(klassi\002kation.)h
(Netv\346rket)h(implemen-)2480 1989 y(ter)o(es)16 b(i)g(C++,)g(og)g
(der)f(indsamles)h(et)g(datas\346t)h(til)f(test.)2480
2193 y(For)31 b(at)i(kunne)g(udf\370r)o(e)d(testene)j(unders\370ges)d
(den)2480 2298 y(empiriske)20 b(metode,)i(som)g(bygger)e(p\345)j(en)g
(positivi-)2480 2403 y(stisk)e(tankegang)g(om)g(at)h(kun)g(det)f(der)f
(kan)h(sanses,)2480 2508 y(kan)i(bidrage)f(med)g(viden.)g(V)-8
b(ed)21 b(hj\346lp)i(af)g(den)f(em-)2480 2614 y(piriske)17
b(metode)h(kan)h(man)h(udf\370r)o(e)d(eksperimenter)2480
2719 y(og)23 b(opn\345)i(p\345lidelige)c(konklusioner)h(der)h(dog)f
(ikke)2480 2824 y(er)c(garanter)o(ede)g(at)h(v\346r)o(e)f(korr)o(ekte)e
(gener)o(elt.)2480 3028 y(T)-7 b(o)25 b(former)f(for)g(forbehandling)h
(unders\370ges,)d(cen-)2480 3133 y(tr)o(ering)d(og)h(skalering.)e
(Anvendelse)i(af)h(disse)e(viser)2480 3238 y(sig)k(at)h(v\346r)o(e)e
(v\346sentlige.)g(T)l(il)h(at)h(optr\346ne)f(netv\346r)o(-)2480
3343 y(ket)32 b(unders\370ges)d(to)j(algoritmer)-6 b(,)31
b(en)h(simpel)f(gra-)2480 3448 y(dientnedstigning)e(og)f(konjuger)o(et)
g(gradient)g(som)2480 3553 y(er)23 b(en)h(mer)o(e)f(avancer)o(et)h
(algoritme)f(\226)h(det)f(viser)g(sig)2480 3658 y(dog)33
b(at)g(gradientnedstigning)g(er)f(hurtigst.)h(Efter)2480
3763 y(at)24 b(have)g(fundet)g(st\370rr)o(elsen)d(p\345)i(det)g
(optimale)g(net-)2480 3868 y(v\346rk,)c(testes)g(der)f(om)i(en)g
(eftertr\346ning)e(af)i(netv\346r)o(-)2480 3973 y(ket)25
b(p\345)h(en)g(bestemt)g(testperson)e(\370ger)g(genkendel-)2480
4078 y(sespr)o(ocenten.)d(Dette)i(viser)e(sig)g(at)i(v\346r)o(e)e
(tilf\346ldet;)2480 4183 y(92,3\045)26 b(uden)e(eftertr\346ning)g(og)g
(99,3\045)h(med)f(en)h(ef-)2480 4288 y(tertr\346ning.)p
4041 4405 V 2431 4408 1613 4 v eop
%%Page: 1 3
1 2 bop 463 893 a FZ(Forord)463 1340 y FY(Denne)33 b(rapport)f(er)h
(udarbejdet)e(som)j(P2-opgave)e(p\345)g(Aalbor)o(g)g(Universitets)h
(basisud-)463 1454 y(dannelse)28 b(og)g(er)f(r)o(esultatet)g(af)g(4)g
(m\345neders)g(arbejde)g(med)h(emnet)g(neurale)f(netv\346rk)h(og)463
1569 y(tegngenkendelse.)h(For)o(uds\346tningerne)h(for)f(at)f(l\346se)h
(den)g(er)f(kendskab)h(til)g(teorien)g(bag)463 1684 y(funktioner)e(af)f
(\003er)o(e)g(variable)g(og)h(der)o(es)e(partielle)h(a\003edede,)g
(samt)g(kendskab)h(til)g(line\346r)463 1798 y(algebra.)588
1913 y(I)20 b(rapporten)g(br)o(uges)h(f\370lgende)g(notation:)588
2094 y FS(\017)41 b FY(Stor)o(e)19 b(bogstaver)i(og)g(fed)f(skrift:)g
(en)h(matrix)h FT(M)q FY(.)588 2275 y FS(\017)41 b FY(Sm\345)20
b(bogstaver)g(og)h(fed)f(skrift:)h(en)g(vektor)g FT(v)q
FY(.)588 2456 y FS(\017)41 b FY(T)-7 b(ransponering)20
b(af)g(vektor:)i FT(v)1647 2426 y FO(T)1692 2456 y FY(.)588
2637 y FS(\017)41 b FY(Det)20 b(indr)o(e)g(pr)o(odukt)h(\(prikpr)o
(oduktet\))e(mellem)i(to)h(vektor)o(er:)e FT(u)2749 2607
y FO(T)2795 2637 y FT(v)q FY(.)588 2818 y FS(\017)41
b FY(Middelv\346r)o(dien)18 b(af)k FL(x)r FY(:)g FS(h)s
FL(x)s FS(i)q FY(.)588 2999 y(Rapporten)e(m\345)h(frit)f(distribuer)o
(es)g(i)h(en)g(hvilken)g(som)h(helst)f(form.)588 3312
y FL(God)f(forn\370jelse)i(med)e(rapporten,)p 719 3721
1182 6 v 2000 3721 V 719 3836 a(Lau)h(Bech)f(Lauritzen)625
b(Jasper)20 b(Kjersgaard)i(Juhl)p 719 4065 V 2000 4065
V 719 4179 a(Rasmus)f(J\370r)o(gensen)667 b(Anders)21
b(Rune)h(Jensen)p 719 4409 V 2000 4409 V 719 4523 a(Ole)f(Laursen)869
b(Martin)21 b(Qvist)1929 5599 y FY(1)p eop
%%Page: 2 4
2 3 bop 463 893 a FZ(Indhold)463 1423 y FT(1)83 b(Indledning)2376
b(5)588 1537 y FY(1.1)85 b(Pr)o(oblemformulering)73 b(.)41
b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)151 b(5)588 1652 y(1.2)85
b(Oversigt)22 b(over)e(rapporten)59 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)151
b(6)463 1850 y FT(2)83 b(Introduktion)19 b(til)g(neurale)g(netv\346rk)
1568 b(7)588 1964 y FY(2.1)85 b(Biologiske)22 b(neurale)f(netv\346rk)72
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)151 b(7)588 2079 y(2.2)85 b(Kunstige)22
b(neurale)e(netv\346rk)60 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)151
b(8)588 2193 y(2.3)85 b(T)-7 b(r\346ning)21 b(og)h(anvendelse)34
b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)151 b(9)588 2308
y(2.4)85 b(Historisk)22 b(baggr)o(und)43 b(.)f(.)f(.)g(.)g(.)h(.)f(.)g
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)
f(.)g(.)g(.)110 b(10)463 2505 y FT(3)83 b(H\345ndholdte)18
b(computere)1886 b(12)588 2620 y FY(3.1)85 b(Eksister)o(ende)20
b(gr\346nse\003ade)76 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(12)588 2735 y(3.2)85 b(Forslag)22 b(til)f(forbedr)o(et)e
(gr\346nse\003ade)64 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(13)588 2849
y(3.3)85 b(Ydelsesm\346ssige)22 b(krav)e(til)h(netv\346rket)52
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)110 b(14)463 3047 y FT(4)83 b(Konstruktion)18
b(af)j(tegngenkendelsessystem)1188 b(15)588 3161 y FY(4.1)85
b(Oversigt)22 b(over)e(pr)o(oces)64 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
(.)g(.)110 b(15)778 3276 y(4.1.1)98 b(Fra)20 b(tegning)i(til)f(billede)
i(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(15)778 3391 y(4.1.2)98
b(Fra)20 b(billede)h(til)g(vektor)65 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(16)778 3505 y(4.1.3)98 b(Fra)20 b(vektor)h(til)g(forbehandlede)f
(netv\346rksdata)66 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(16)778 3620 y(4.1.4)98 b(Fra)20 b(netv\346rksdata)g(til)h(r)o
(esulter)o(ende)e(vektor)83 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)110 b(16)778 3734 y(4.1.5)98 b(Fra)20 b(r)o(esulter)o(ende)f
(vektor)i(til)g(bogstav)46 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)110 b(17)588 3849 y(4.2)85 b(Forbehandling)36
b(.)42 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)
f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(17)778 3963 y(4.2.1)98 b(Skalering)21 b(af)e(billedet)77
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(18)778 4078 y(4.2.2)98
b(Centr)o(ering)20 b(af)g(bogstavet)72 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(18)778 4192 y(4.2.3)98 b(Praktiske)21 b(overvejelser)65
b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)110 b(20)588 4307 y(4.3)85 b(Netv\346rksarkitektur)
36 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(20)778
4422 y(4.3.1)98 b(Per)o(ceptr)o(onen)72 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)110 b(20)778 4536 y(4.3.2)98 b(Flerlagsmodeller)65
b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(21)778 4651 y(4.3.3)98
b(Aktiveringsfunktioner)48 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(24)778
4765 y(4.3.4)98 b(Startv\346r)o(dier)19 b(for)h(v\346gte)41
b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)110 b(26)588 4880 y(4.4)85 b(T)-7
b(r\346ningsmetoder)52 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
110 b(27)778 4994 y(4.4.1)98 b(Fejlfunktioner)43 b(.)e(.)g(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)g(.)110 b(27)778 5109 y(4.4.2)98 b(Ber)o(egning)21
b(af)f FS(r)s FL(E)j FY(med)d(tilbagef\370rselsalgoritmen)36
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(28)778 5224
y(4.4.3)98 b(Gradientnedstigning)45 b(.)c(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(30)778 5338 y(4.4.4)98 b(Konjuger)o(et)21 b(gradient)71
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(33)1929 5599 y(2)p eop
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3 4 bop 1802 -129 a FY(Indhold)p 463 -96 2975 4 v 778
192 a(4.4.5)98 b(Generalisering,)21 b(overtr\346ning)g(og)g
(termineringskriteriet)59 b(.)41 b(.)g(.)g(.)110 b(40)588
306 y(4.5)85 b(Maksimering)21 b(af)f(rigtige)h(klassi\002kationer)26
b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
110 b(45)463 504 y FT(5)83 b(Implementering)2145 b(49)588
618 y FY(5.1)85 b(Implementering)22 b(af)e(netv\346rket)61
b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)110 b(49)778 733 y(5.1.1)98 b(Krav)20
b(til)i(implementeringen)82 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(49)778 848
y(5.1.2)98 b(V)-8 b(alg)21 b(af)f(pr)o(ogrammeringsspr)o(og)80
b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
110 b(49)778 962 y(5.1.3)98 b(Design)22 b(af)e(netv\346rket)70
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(50)778 1077 y(5.1.4)98
b(Beskrivelse)21 b(af)e(netv\346rksalgoritmer)57 b(.)41
b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(52)588 1191 y(5.2)85 b(Indsamling)22 b(af)e(datas\346t)71
b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(54)588 1306
y(5.3)85 b(Beskrivelse)21 b(af)f(hj\346lpepr)o(ogrammer)25
b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)
f(.)g(.)g(.)110 b(55)778 1420 y(5.3.1)98 b(Opdeling)21
b(i)g(validerings-)g(og)g(tr\346ningss\346t)57 b(.)41
b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(55)778
1535 y(5.3.2)98 b(Centr)o(ering)20 b(af)g(bogstaverne)40
b(.)i(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)110 b(55)778 1650 y(5.3.3)98 b(Skalering)21
b(af)e(billederne)45 b(.)c(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(55)778 1764
y(5.3.4)98 b(Sammenl\346gning)22 b(af)e(billeder)46 b(.)41
b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g
(.)110 b(56)778 1879 y(5.3.5)98 b(Konvertering)22 b(til)f(billedformat)
55 b(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
(.)g(.)110 b(56)463 2076 y FT(6)83 b(Empirisk)19 b(metode)2112
b(57)588 2191 y FY(6.1)85 b(Positivisme)23 b(og)e(empiri)j(.)41
b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(57)588 2305
y(6.2)85 b(Empirismens)22 b(baggr)o(und)82 b(.)41 b(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)110 b(59)588 2420 y(6.3)85 b(Bevisf\370r)o(else)73
b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(59)778 2535 y(6.3.1)98 b(Deduktive)21 b(slutninger)k(.)41
b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)110 b(59)778 2649 y(6.3.2)98 b(Induktionspr)o
(oblemet)31 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(61)778 2764
y(6.3.3)98 b(Slutning)22 b(til)f(bedste)f(forklaring)64
b(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)110 b(62)588 2878 y(6.4)85 b(Empiriske)21 b(fors\370g)72
b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(63)588
2993 y(6.5)85 b(P\345lidelighed)21 b(og)g(anvendelighed)54
b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)g(.)110 b(64)778 3107 y(6.5.1)98 b(Anvendelse)20
b(i)h(dette)f(pr)o(ojekt)29 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(66)463
3305 y FT(7)83 b(T)-9 b(est)2613 b(67)588 3420 y FY(7.1)85
b(Empiriske)21 b(overvejelser)49 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)
110 b(67)588 3534 y(7.2)85 b(Praktiske)21 b(begr\346nsninger)37
b(.)k(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(68)588 3649 y(7.3)85
b(Opstilling,)23 b(udf\370r)o(else)c(og)i(fortolkning)h(af)e
(indledende)h(test)29 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(68)778 3763 y(7.3.1)98 b(Or)o(dforklaring)44 b(.)d(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)110 b(69)778 3878 y(7.3.2)98 b(Opstilling)22
b(af)e(netv\346rk)27 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(70)778
3992 y(7.3.3)98 b(Centr)o(ering)39 b(.)i(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)110 b(70)778 4107 y(7.3.4)98 b(Sigmoid)21
b(vs.)g(tanh)68 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(71)778
4222 y(7.3.5)98 b(Bestemmelse)21 b(af)g FN(h)j FY(og)f
FN(m)53 b FY(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(71)778 4336 y(7.3.6)98
b(Konjuger)o(et)21 b(gradient)g(vs.)f(gradientnedstigning)50
b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(73)778 4451
y(7.3.7)98 b(Antal)21 b(skjulte)g(neur)o(oner)47 b(.)41
b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)110 b(74)778 4565 y(7.3.8)98 b(Ekstern)21
b(skalering)46 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(75)778
4680 y(7.3.9)98 b(Intern)21 b(skalering)g(vs.)g(ekstern)62
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)110 b(76)778 4794 y(7.3.10)56 b(Regionsskalering)e(.)41
b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(76)778 4909 y(7.3.11)56
b(St\370rr)o(else)20 b(af)g(valideringss\346ttet)57 b(.)41
b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(77)588 5024 y(7.4)85 b(Opstilling)23 b(af)d(de)g(endelige)g(test)j(.)
41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)110 b(78)778 5138 y(7.4.1)98 b(Resultater)21
b(\226)f(forskellige)h(startv\346r)o(dier)54 b(.)41 b(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(78)778 5253
y(7.4.2)98 b(Resultater)21 b(\226)f(krydstest)76 b(.)41
b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)110 b(79)1929 5599 y(3)p eop
%%Page: 4 6
4 5 bop 1802 -129 a FY(Indhold)p 463 -96 2975 4 v 588
192 a(7.5)85 b(Eftertr\346ning)47 b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(79)588 306 y(7.6)85 b(Praktiske)21
b(overvejelser)82 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(81)588 421 y(7.7)85 b(Evaluering)21 b(af)f(metoden)49
b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(82)463 618
y FT(8)83 b(Konklusion)2312 b(84)588 733 y FY(8.1)85
b(Perspektivering)57 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)110 b(85)778 848 y(8.1.1)98 b(Anvendelighed)21 b(af)e(neurale)i
(netv\346rk)j(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
g(.)110 b(85)778 962 y(8.1.2)98 b(Implementering)22 b(p\345)e
(h\345ndholdt)71 b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)g(.)110 b(86)778 1077 y(8.1.3)98 b(L\370bende)20
b(eftertr\346ning)73 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(86)463
1274 y FT(Litteratur)2529 b(89)463 1472 y(A)60 b(Konjugeret)18
b(gradient)h(i)i(praksis)1651 b(91)588 1586 y FY(A.1)62
b(Bevis)21 b(for)g(at)f(konjuger)o(ede)g(vektor)o(er)g(er)h(line\346rt)
f(uafh\346ngige)j(.)41 b(.)g(.)h(.)f(.)g(.)g(.)110 b(91)588
1701 y(A.2)62 b(Skaler)o(et)20 b(konjuger)o(et)h(gradient)36
b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)g(.)110 b(92)463 1899 y FT(B)70 b(Datas\346ttet)
2357 b(94)588 2013 y FY(B.1)76 b(Centr)o(ering)56 b(.)41
b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110
b(94)588 2128 y(B.2)76 b(Skalering)44 b(.)d(.)g(.)g(.)h(.)f(.)g(.)h(.)f
(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)110 b(95)463 2325 y FT(C)65
b(Startv\346rdier)18 b(for)k(v\346gte)1912 b(96)1929
5599 y FY(4)p eop
%%Page: 5 7
5 6 bop 463 845 a FJ(Kapitel)44 b(1)463 1298 y FZ(Indledning)463
1745 y FY(Efter)o(h\345nden)26 b(som)j(du)f(l\346ser)g(denne)f
(rapport,)g(genkender)h(du)g(de)f(enkelte)h(elementer)g(i)463
1859 y(teksten)f(og)f(oms\346tter)h(dem)f(til)h(noget)g(der)e(kan)i(br)
o(uges)f(vider)o(e)f(i)i(informationsbehand-)463 1974
y(lingen)c(i)g(din)f(hjerne.)h(Du)f(t\346nker)h(m\345ske)f(ikke)h
(p\345)e(l\346sepr)o(ocessen)h(som)h(en)g(genkendelse)463
2088 y(for)o(di)17 b(den)h(forl\370ber)f(p\345)g(et)h(ubevidst)g(plan,)
f(men)i(ikke)f(desto)g(mindr)o(e)f(for)o(etager)g(din)h(hjerne)463
2203 y(en)j(genkendelse)g(af)f(bogstaverne)g(og)h(or)o(dene)f(i)h
(teksten.)588 2317 y(Mens)k(pr)o(ocessen)g(er)f(ganske)i(ubesv\346r)o
(et)e(for)h(dig,)g(er)g(den)g(et)g(meget)h(st\370rr)o(e)e(pr)o(oblem)
463 2432 y(for)18 b(computer)o(e.)f(Det)h(er)g(ikke)g(indlysende)h
(hvor)o(dan)e(et)h(pr)o(ogram)g(skal)g(b\346r)o(e)f(sig)h(af)g(med)f
(at)463 2547 y(genkende)h(et)g(bogstav)g(ud)g(fra)f(punkterne)h(i)g(et)
g(billede.)f(Samtidig)h(er)g(pr)o(oblemet)f(ikke)h(kun)463
2661 y(teor)o(etisk)j(\226)f(et)g(eksempel)h(p\345)f(en)h(praktisk)g
(anvendelse)f(er)g(ved)h(h\345ndholdte)g(computer)o(e.)588
2776 y(En)g FL(h\345ndholdt)e(computer)i FY(er)g(en)h(lille)g
(lommecomputer)g(til)g(at)f(holde)h(styr)g(p\345)f(aftaler)f(og)463
2890 y(lignende.)h(Br)o(uger)o(gr\346nse\003aden)f(best\345r)g(af)g(en)
g(trykf\370lsom)i(sk\346rm)f(og)h(et)e(peger)o(edskab.)588
3005 y(P\345)30 b(disse)i(sm\345)f(computer)o(e)g(er)g(der)f(ikke)i
(plads)f(til)g(et)g(tastatur)-6 b(,)31 b(og)g(derfor)g(m\345)g(man)463
3119 y(br)o(uge)19 b(andr)o(e)f(metoder)h(til)h(at)f(kommuniker)o(e)h
(med)f(computer)o(en.)f(En)i(m\345de)e(at)h(g\370r)o(e)g(dette)463
3234 y(p\345)24 b(er)g(at)h(skrive)f(til)h(computer)o(en,)g(dvs.)f
(skrive)h(p\345)f(den)g(trykf\370lsomme)i(sk\346rm)f(med)f(det)463
3349 y(medf\370lgende)j(peger)o(edskab,)f(men)i(dette)f(kr\346ver)g(at)
g(computer)o(en)h(kan)f(genkende)h(den)463 3463 y(tekst)21
b(der)f(bliver)g(skr)o(evet.)588 3578 y(En)j(metode)h(til)g(at)g
(genkende)f(denne)h(tekst)g(kan)g(\002ndes)g(ved)f(at)h(efterligne)f
(den)h(m\345de)463 3692 y(den)j(menneskelige)g(hjerne)g(pr)o(ocesser)o
(er)e(information)j(p\345.)e(Hjernen)h(best\345r)f(af)g(et)h(net)g(af)
463 3807 y(s\345kaldte)21 b(neur)o(oner)-6 b(,)22 b(et)g
FL(neuralt)h(netv\346rk)p FY(.)f(Hvad)f(dette)h(n\346rmer)o(e)g
(indeb\346r)o(er)e(gennemg\345s)463 3921 y(sener)o(e)e(i)i(rapporten;)e
(det)h(er)g(blot)h(v\346sentligt)g(at)e(vide)h(at)g(kunstige)h(neurale)
f(netv\346rk,)g(mo-)463 4036 y(deller)24 b(af)f(hjernens)i(netv\346rk,)
f(har)g(vist)g(sig)h(br)o(ugbar)o(e)e(til)h(at)g(l\370se)h(visse)f
(typer)g(pr)o(oblemer)463 4151 y(som)i(det)e(er)g(sv\346rt)h(at)f
(opstille)h(fuldst\346ndige)g(modeller)g(for)-6 b(.)24
b(Et)g(eksempel)h(p\345)f(et)h(s\345dant)463 4265 y(pr)o(oblem)20
b(er)h(netop)g(h\345ndskriftsgenkendelse.)463 4572 y
FI(1.1)119 b(Problemformulering)463 4777 y FY(\330nsket)34
b(er)f(ved)g(hj\346lp)i(af)e(et)g(kunstigt)i(neuralt)e(netv\346rk)h(at)
f(opn\345)h(den)f(bedst)g(mulige)463 4891 y(genkendelse)22
b(af)e(enkeltst\345ende)h(tegn)h(skr)o(evet)f(p\345)f(en)i
(h\345ndholdt)g(computers)g(skrivefelt.)463 5006 y(P\345)e(gr)o(und)i
(af)e(neurale)g(netv\346rks)h(beskaf)o(fenhed)e(for)o(uds\346tter)h
(dette)g(en)h(implementering)463 5120 y(som)h(kan)e(testes.)588
5235 y(En)d(r\346kke)h(sp\370r)o(gsm\345l)f(\370nskes)i(besvar)o(et)d
(for)h(at)h(give)f(indblik)i(i)f(de)f(metoder)g(og)h(teorier)463
5349 y(der)i(omgiver)h(pr)o(oblemet:)1929 5599 y(5)p
eop
%%Page: 6 8
6 7 bop 1558 -129 a FY(Kapitel)20 b(1:)25 b(Indledning)p
463 -96 2975 4 v 567 192 a(1.)41 b(Hvor)o(dan)19 b(og)j(hvorfor)f
(funger)o(er)e(et)i(kunstigt)h(neuralt)e(netv\346rk?)567
373 y(2.)41 b(Hvilke)18 b(pr)o(oblemstillinger)h(er)f(der)g(ved)g
(anvendelse)g(af)g(kunstige)h(neurale)f(netv\346rk)671
487 y(p\345)i(h\345ndholdte)h(computer)o(e?)f(Og)h(hvor)o(dan)f(kan)h
(disse)g(l\370ses?)567 668 y(3.)41 b(Hvor)o(dan)19 b(konstr)o(uer)o(es)
h(et)g(neuralt)g(netv\346rk)g(til)g(at)g(l\370se)g(tegngenkendelsespr)o
(oble-)671 783 y(met)g(med)h(en)g(h\370j)g(genkendelsespr)o(ocent?)567
964 y(4.)41 b(Hvor)o(dan)27 b(afg\370r)g(man)h(om)g(et)g(netv\346rk)g
(er)f(bedr)o(e)g(end)g(et)h(andet?)f(Kan)h(man)g(v\346r)o(e)671
1078 y(sikker)19 b(p\345)f(at)h(man)g(har)f(fundet)h(det)f(bedste)h
(netv\346rk?)g(Hvorfor)f(kan)h(man)g(ikke)h(det?)567
1259 y(5.)41 b(Hvor)o(dan)15 b(kan)i(neurale)f(netv\346rk)h(tilpasses)f
(individuelle)h(br)o(uger)o(e?)f(Kan)h(det)f(betale)671
1374 y(sig?)21 b(Kan)g(det)f(lade)g(sig)h(g\370r)o(e)f(p\345)g(en)h
(h\345ndholdt)g(computer?)463 1682 y FI(1.2)119 b(Oversigt)29
b(over)h(rapporten)463 1886 y FL(Intr)o(oduktion)e(til)i(neurale)h
(netv\346rk)f FY(beskriver)f(neurale)g(netv\346rks)h(historie.)h
(Desuden)f(in-)463 2001 y(tr)o(oducer)o(es)20 b(de)h(forskellige)g
(terminologier)i(der)e(br)o(uges)g(i)h(forbindelse)f(med)g(neurale)g
(net-)463 2115 y(v\346rk.)588 2230 y FL(H\345ndholdte)g(computer)o(e)h
FY(beskriver)h(h\345ndholdte)g(computer)o(es)g(gr\346nse\003ade)g(og)g
(speci-)463 2344 y(\002kationer)e(og)g(hvilken)h(ind\003ydelse)f(det)f
(f\345r)g(for)h(et)f(netv\346rk.)588 2459 y FL(Konstruktion)28
b(af)f(tegngenkendelsessystem)h FY(beskriver)f(og)h(analyser)o(er)e
(forskellige)i(me-)463 2574 y(toder)22 b(til)g(neurale)f(netv\346rk,)h
(her)o(under)f(forbehandlingsmetoder)h(og)h(tr\346ningsmetoder)-6
b(.)463 2688 y(Desuden)35 b(beskrives)f(tegngenkendelsespr)o(ocessen.)h
(Afsnittet)f(danner)g(det)g(teor)o(etiske)463 2803 y(gr)o(undlag)21
b(for)f(de)h(forskellige)g(test)g(vi)f(sener)o(e)g(udf\370r)o(er)-6
b(.)588 2917 y FL(Implementering)16 b FY(beskriver)h(netv\346rkspr)o
(ogrammet)h(og)g(de)f(forskellige)h(sm\345pr)o(ogram-)463
3032 y(mer)k(der)f(er)g(blevet)g(br)o(ugt)h(til)g(test.)g(Desuden)g(r)o
(etf\346r)o(digg\370r)o(es)d(valg)i(af)g(pr)o(ogrammerings-)463
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3376 y(r)o(etisk)d(set.)g(Analysen)h(skal)f(br)o(uges)h(til)g(at)f
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3490 y(af)i(testene.)588 3605 y FL(T)-9 b(estafsnittet)24
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2718 y(neuralt)i(netv\346rk.)g(Netv\346rket)g(skulle)h(da)e(kunne)i
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2948 y(skulle)28 b(v\346r)o(e)e(i)i(stand)f(til)h(at)g
FL(generaliser)o(e)g FY(ud)g(fra)e(disse)i(og)g(l\370se)f(det)h(givne)f
(pr)o(oblem)h(ved)463 3062 y(fr)o(emtidige,)20 b(ukendte)g(data.)588
3177 y(Overraskende)k(nok)i(er)g(dette)f(faktisk)g(muligt,)h(selv)g
(med)f(en)h(simpel)g(model)g(af)f(hjer)o(-)463 3291 y(nen.)c(P\345)g
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(hjerne.)g(Hjernen)f(best\345r)g(af)f(omkring)i(100)e(mia.)h
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(som)j(suppler)o(es)f(af)f(ti)i(gange)f(s\345)h(mange)f(ikke-neurale)g
(st\370t-)463 4376 y(teceller)30 b(\(f.eks.)f(til)h(ener)o
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4835 y(kan)21 b(v\346r)o(e)f(helt)h(op)g(over)g(200.000\))c(i)k
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(sig)g(fra)f(neur)o(on)h(til)h(neur)o(on)f(ved)f(hj\346lp)i(af)e(kemi-)
463 5064 y(ske)k(r)o(eaktioner)f([Harvey)-9 b(,)27 b(1994].)f
(P\345virkningen)k(fra)e(en)h(bestemt)f(neur)o(on)h(til)g(en)g(anden)
463 5178 y(kan)19 b(enten)g(v\346r)o(e)e(sv\346kkende)i(eller)f
(forst\346rkende)h(og)g(afh\346nger)f(af)g(styrken)h(og)g(fr)o(ekven-)
463 5293 y(sen)j(af)f(de)g(elektriske)h(impulser)g(som)h(den)f
(f\370rste)f(neur)o(on)h(udsender)-6 b(.)21 b(Hvis)h(den)g(samlede)1929
5599 y(7)p eop
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8 9 bop 1167 -129 a FY(Kapitel)21 b(2:)k(Intr)o(oduktion)c(til)h
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 @endspecial 534 1186 a FH(Figur)26 b(2.1:)i(Neuralt)e(netv\346rk)g
(fra)h(hjernebarken)f(\226)h(kun)f(5\045)i(af)f(de)g(faktiske)g(neur)o
(oner)e(i)h(omr\345det)i(er)534 1291 y(synlige.)463 1561
y FY(signalstyrke)k(fra)d(alle)i(de)f(forskellige)h(indkommende)h
(signaler)f(ved)f(en)h(bestemt)g(neu-)463 1675 y(r)o(on)26
b(overskrider)f(dens)g(t\346rskel,)h(\204fyr)o(er\223)d(neur)o(onen)j
(og)g(sender)g(et)f(signal)h(vider)o(e)f(ad)g(de)463
1790 y(udg\345ende)20 b(forbindelser)-6 b(.)588 1904
y(Ef)o(fektiviteten)24 b(af)i(kontakt\003aderne)g(mellem)g(neur)o
(onerne,)g FL(synapserne)p FY(,)h(kan)f(\346ndr)o(es)463
2019 y(med)19 b(tiden)g(af)f(det)h(signal)h(der)e(passer)o(er)f(gennem)
j(dem.)f(Denne)g(\346ndring)h(af)e(overf\370rings-)463
2134 y(ef)o(fektiviteten)e(formodes)i(at)f(v\346r)o(e)g(\345rsagen)g
(til)h(at)f(det)g(er)g(muligt)i(for)e(mennesker)h(at)f(huske)463
2248 y(og)k(at)g(l\346r)o(e)e([Br)o(unak)i(og)g(Lautr)o(up,)f(1988].)
588 2363 y(Et)j(sk\370nsm\346ssigt)i(overslag)e(p\345)g(antallet)f(af)h
(aritmetiske)g(ber)o(egninger)f(som)j(en)e(com-)463 2477
y(puter)k(skulle)i(for)o(etage)d(for)h(at)h(udf\370r)o(e)e(den)i(samme)
g(ber)o(egning)f(som)i(alle)e(neur)o(onerne)h(i)463 2592
y(hjernen)g(til)f(sammen)h(udf\370r)o(er)d(analogt,)i(er)g(af)f
(st\370rr)o(elsesor)o(denen)g(10)2792 2562 y FP(18)2886
2592 y FY(operationer)h(pr)-6 b(.)463 2706 y(sekund)33
b([Br)o(unak)e(og)i(Lautr)o(up,)f(1988].)d(T)-5 b(il)33
b(sammenligning)i(kan)d(den)g(hurtigste)h(com-)463 2821
y(puter)e(i)g(ver)o(den)g(per)f(november)i(2001)d(teor)o(etisk)i(klar)o
(e)f(1)q(,)q(23)20 b FS(\001)h FY(10)2650 2791 y FP(13)2748
2821 y FY(operationer)31 b(pr)-6 b(.)30 b(se-)463 2936
y(kund)j([T)-8 b(op500,)31 b(2001].)g(De)i(to)g(tal)f(kan)h
(naturligvis)g(ikke)h(dir)o(ekte)d(sammenlignes)j(idet)463
3050 y(hjernens)20 b(parallellitet)d(funger)o(er)h(gr)o(undl\346ggende)
h(anderledes)e(end)i(en)g(elektr)o(onisk)g(pr)o(o-)463
3165 y(cessor)-6 b(,)19 b(men)h(illustr)o(er)o(er)e(dog)h(at)g(hjernen)
h(besidder)e(en)h(meget)h(stor)f(ber)o(egningskapacitet.)463
3473 y FI(2.2)119 b(Kunstige)30 b(neurale)f(netv\346rk)463
3677 y FY(Den)16 b(simpleste)g(model)g(af)f(hjernens)i(komplicer)o(ede)
e(netv\346rk)h(med)f(forbindelser)h(p\345)f(kryds)463
3791 y(og)22 b(tv\346rs)f(er)g(et)g FL(fr)o(emadf\370r)o(ende)f
FY(netv\346rk,)h(se)g(\002gur)h(2.2)e(p\345)h(den)g(f\370lgende)g
(side.)g(En)g(neur)o(on)463 3906 y(r)o(epr\346senter)o(es)30
b(nu)j(blot)f(af)g(et)g(tal)g(under)f(ber)o(egningerne)h(i)g(nettet,)g
(og)h(de)f(illustr)o(er)o(ede)463 4021 y(forbindelser)e(mellem)h(neur)o
(onerne)f(stammer)g(fra)f(den)h(m\345de)g(v\346r)o(dier)f(f\370r)o(es)g
(fr)o(emad)g(i)463 4135 y(netv\346rket.)22 b(Hver)f(forbindelse)h(har)g
(en)g FL(v\346gt)f FY(i)h(form)g(af)g(et)g(tal)f(associer)o(et)h(med)g
(forbindel-)463 4250 y(sen.)588 4364 y(Dataene)c(inds\346ttes)h(som)h
(tal)f(i)g(den)g(ene)g(ende)f(af)g(netv\346rket)h(\(ved)f(i'erne)h
(p\345)f(\002gur)i(2.2\))463 4479 y(og)g(sendes)g(fr)o(emad)e(lag)h
(for)h(lag)f(ved)g(at)g(den)h(v\346gtede)f(sum)h(af)f(de)g
(tilg\345ende)g(forbindelser)463 4593 y(til)f(en)g(neur)o(on)h
(inds\346ttes)f(i)g(en)g(funktion)h(hvis)g(r)o(esultatet)e(er)g(v\346r)
o(dien)g(for)h(neur)o(onen.)g(V\346r)o(-)463 4708 y(dien)24
b(for)f(en)h(tilg\345ende)g(forbindelse)f(ber)o(egnes)g(som)i(pr)o
(oduktet)e(mellem)h(v\346r)o(dien)h FL(x)3285 4724 y
FO(i)3331 4708 y FY(for)463 4823 y(den)17 b(neur)o(on)g(den)g(udg\345r)
f(fra,)g(og)h(forbindelsens)h(v\346gt)g FL(w)2319 4839
y FO(i)2341 4823 y FY(,)f(s\345)f(ber)o(egningen)h(af)g(en)g(neur)o
(ons)463 4937 y(v\346r)o(di)j(inde)g(i)h(netv\346rket)g(for)o(eg\345r)e
(ved)1447 5197 y FL(y)25 b FG(=)i FL(g)1663 5055 y FF( )1763
5108 y FO(n)1736 5214 y FE(\345)1731 5288 y FO(i)r FC(=)q
FP(1)1846 5197 y FL(w)1908 5214 y FO(i)1932 5197 y FL(x)1976
5214 y FO(i)1999 5055 y FF(!)2090 5197 y FG(=)f FL(g)r
FG(\()r FT(v)2308 5163 y FO(T)2354 5197 y FT(x)r FG(\))r
FY(,)1929 5599 y(8)p eop
%%Page: 9 11
9 10 bop 1167 -129 a FY(Kapitel)21 b(2:)k(Intr)o(oduktion)c(til)h
(neurale)e(netv\346rk)p 463 -96 2975 4 v 1359 1302 a
@beginspecial 35 @llx 35 @lly 233 @urx 235 @ury 1417
@rwi @setspecial
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 @endspecial 919 1490 a FH(Figur)e(2.2:)28 b(Et)18 b(eksempel)i(p\345)f
(et)g(kunstigt)e(fr)o(emadf\370r)o(ende)i(neuralt)f(netv\346rk)463
1759 y FY(hvor)31 b FL(g)q FY(,)c FL(aktiveringsfunktionen)p
FY(,)g(har)h(til)g(form\345l)f(at)h(efterligne)f(de)g(menneskelige)i
(neur)o(o-)463 1874 y(ners)g(t\346rskelef)o(fekt)g(og)g(f.eks.)g(kunne)
h(de\002ner)o(es)f(ved)f(hj\346lp)j(af)d(en)i(diskr)o(et)e(model)i(som)
466 1989 y FL(g)r FG(\()s FL(x)s FG(\))i(=)f FY(0)25
b(for)h FL(x)33 b FS(\024)g FL(t)1152 2005 y FO(i)1198
1989 y FY(og)c FL(g)r FG(\()s FL(x)s FG(\))j(=)f FY(1)24
b(for)j FL(x)33 b FM(>)f FL(t)2003 2005 y FO(i)2050 1989
y FY(hvor)26 b FL(t)2279 2005 y FO(i)2326 1989 y FY(er)e(t\346rsklen)h
(for)g(neur)o(onen.)f(V)-8 b(al-)463 2103 y(get)22 b(af)f
(aktiveringsfunktion)i(har)f(stor)g(ind\003ydelse)h(p\345)e
(netv\346rkets)h(evner)g(og)g(behandles)463 2218 y(mer)o(e)e
(indg\345ende)g(i)h(afsnit)g(4.3.3.)588 2332 y(T)-5 b(il)33
b(slut)h(f\345s)e(ved)h(det)f(sidste)i(neur)o(onlag)f(en)g(vektor)g
(\(i)g(\002gur)g(2.2)f(ved)g(o'erne\))h(som)463 2447
y(r)o(epr\346senter)o(er)18 b(r)o(esultatet)i(af)g(netv\346rkets)h
(samlede)f(ber)o(egning.)463 2755 y FI(2.3)119 b(T)-13
b(r\346ning)29 b(og)i(anvendelse)463 2959 y FY(Man)19
b(kan)i(betragte)e(det)g(fr)o(emadf\370r)o(ende)e(netv\346rk)j(som)h
(en)g(funktion)g(der)e(afbilder)g(fra)g(en)463 3074 y(vektor)h
FT(x)f FY(i)g FB(R)893 3044 y FO(n)952 3074 y FY(til)g(en)g(vektor)h
FT(y)g FY(i)f FB(R)1588 3044 y FO(m)1661 3074 y FY(hvor)h
FL(n)g FY(er)f(antallet)f(af)g(neur)o(oner)h(i)g(f\370rste)f(lag,)h(og)
h FL(m)f FY(er)463 3188 y(antallet)g(i)h(sidste)f(lag.)g(Det)h(der)f
(er)g(til)h(r\345dighed)e(for)i(et)f(givet)h(pr)o(oblem,)e(er)i(aller)o
(ede)d(kendte)463 3303 y(datas\346t)28 b(med)g(samh\370r)o(ende)h
(v\346r)o(dier)e(af)i FT(x)g FY(og)h(\370nskede)f(v\346r)o(dier)f
FT(t)q FY(,)g FL(tr\346ningss\346ttet)p FY(,)h(og)463
3417 y(funktionen)24 b(skal)f(s\345)g(tilpasses)h(efter)e(disse)h
(s\345)g(den)g(r)o(eturner)o(er)e(en)i(rigtig)h(vektor)g
FT(y)g FY(for)f(en)463 3532 y(given)k(fr)o(emtidig)f(v\346r)o(di)h
FT(x)q FY(.)f(P\345)h(denne)g(m\345de)f(appr)o(oksimer)o(er)f
(netv\346rket)i(den)f(ukendte)463 3647 y(klassi\002kationsfunktion)19
b(som)e(det)f(var)g(for)g(komplicer)o(et)g(at)g(opstille)h(en)f
(eksplicit)h(forskrift)463 3761 y(for)-6 b(.)588 3876
y(T)h(ilpasningen)25 b(for)o(eg\345r)d(ved)i(at)f(juster)o(e)h
(v\346gtene)g(i)g(netv\346rket)g(s\345)f(forskellen)i(mellem)463
3990 y(netv\346rkets)j(faktiske)g(r)o(esultat)g FT(y)i
FY(og)e(det)g(\370nskede)g(r)o(esultat)g FT(t)h FY(minimer)o(es)g(for)e
(hele)i(tr\346-)463 4105 y(ningss\346ttet)d(\226)f(denne)g(pr)o(oces)f
(ben\346vnes)h FL(tr\346ning)g FY(eller)g FL(indl\346ring)p
FY(,)f(og)i(algoritmer)f(til)g(at)463 4219 y(for)o(etage)19
b(den)i(er)f(beskr)o(evet)g(i)h(afsnit)f(4.4.)588 4334
y(Ideen)25 b(med)i(netv\346rket)f(er)f(at)h(det)g(efter)g
(tr\346ningsfasen)g(skal)h(kunne)g FL(generaliser)o(e)g
FY(ud)463 4449 y(fra)h(tr\346ningseksemplerne)j(og)f(give)f(korr)o
(ekte)g(svar)g(p\345)g(v\346r)o(dier)f(der)h(ligger)h(\204i)f(n\346r)o
(he-)463 4563 y(den\223)f(af)g(dem.)g(Dette)h(kan)g(dels)g(lade)e(sig)j
(g\370r)o(e)e(for)o(di)g(et)g(neuralt)h(netv\346rk)g(kan)f(vises)h(at)
463 4678 y(kunne)23 b(appr)o(oksimer)o(e)e(enhver)i(funktion)g
(vilk\345rligt)g(pr\346cist,)f(og)h(dermed)f(ogs\345)h(klassi\002-)463
4792 y(kationsfunktionen;)k(dels)f(kan)g(det)f(lade)g(sig)i(g\370r)o(e)
e(for)o(di)g(netv\346rkets)g(str)o(uktur)i(bevirker)463
4907 y(at)21 b(det)f(kan)h(v\346r)o(e)f(modstandsdygtigt)i(over)f(for)g
(mindr)o(e)g(afvigelser)f(eller)g(fejl)i(i)f(dataene)e(\226)463
5021 y(begge)24 b(egenskaber)g(har)g(visse)h(for)o(uds\346tninger)g
(som)g(behandles)f(i)h(afsnit)g(4.3.)e(Generali-)463
5136 y(seringsevnen)e(kommer)f(dog)g(ikke)h(af)e(sig)h(selv)-9
b(,)20 b(og)g(et)g(f\346nomen)h(som)f(ben\346vnes)g FL(overtr\346-)463
5251 y(ning)p FY(,)f(kan)g(\370del\346gge)g(den.)f(Overtr\346ning)i(er)
e(et)h(pr)o(oblem)g(der)f(opst\345r)h(n\345r)g(netv\346rket)f(kun)1929
5599 y(9)p eop
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10 11 bop 1167 -129 a FY(Kapitel)21 b(2:)k(Intr)o(oduktion)c(til)h
(neurale)e(netv\346rk)p 463 -96 2975 4 v 463 192 a(kan)25
b(genkende)h(de)f(bogstaver)g(det)g(er)g(blevet)g(tr\346net)g(p\345.)g
(V)-5 b(i)26 b(kommer)g(n\346rmer)o(e)f(ind)h(p\345)463
306 y(overtr\346ning)21 b(og)h(hvor)o(dan)e(man)h(kan)f(formindske)h
(dette)g(f\346nomen)g(i)g(4.4.5.)588 421 y(For)27 b(at)h(anvende)f(et)g
(neuralt)g(netv\346rk)h(p\345)f(et)g(praktisk)h(pr)o(oblem)f(skal)h
(dataene)e(f\370rst)463 535 y(oms\346ttes)i(til)g(noget)g(der)f(kan)g
(sendes)g(ind)h(som)g(en)g(r)o(eel)e(vektor)h(i)h(netv\346rket.)f
(Desuden)463 650 y(skal)e(en)g(r\346kke)g(tr\346ningseksempler)g
(v\346r)o(e)f(klassi\002cer)o(et)h(p\345)f(for)o(h\345nd)g(\226)g(hver)
h(klasse)g(til-)463 765 y(deles)d(en)g(bestemt)g(kodning)h(i)f(den)g(r)
o(esulter)o(ende)e(vektor)-6 b(,)21 b(f.eks.)h(kan)g(\351n)g(af)f
(indgangene)h(i)463 879 y(vektor)o(en)e(have)h(h\370j)g(v\346r)o(di,)f
(mens)h(r)o(esten)f(er)h(lave.)588 994 y(P\345)26 b(denne)g(m\345de)f
(er)h(det)g(komplicer)o(ede)g(genkendelsespr)o(oblem)g(transformer)o
(et)f(om)463 1108 y(til)c(en)g(r\346kke)g(pr)o(oblemer)f(i)g
(forbindelse)h(med)g(design)g(af)f(et)g(neuralt)h(netv\346rk.)463
1416 y FI(2.4)119 b(Historisk)29 b(baggrund)463 1620
y FY(T)-7 b(raditionelt)22 b(set)h(har)f(der)g(v\346r)o(et)f(to)i
(hovedgr)o(upper)g(inden)f(for)h(forskning)h(i)e(neurale)g(net-)463
1735 y(v\346rk:)h(en)g(gr)o(uppe)g(der)f(var)h(inter)o(esser)o(et)f(i)h
(netv\346rkene)g(som)h(et)f(middel)g(til)g(at)g(l\370se)h(kom-)463
1850 y(plicer)o(ede)f(pr)o(oblemer)h(som)i(billed-)e(og)h
(talegenkendelse,)f(og)h(en)g(gr)o(uppe)g(der)f(var)g(inter)o(-)463
1964 y(esser)o(et)g(i)i(netv\346rkene)f(som)h(en)f(m\345de)g(at)g(skaf)
o(fe)e(en)j(st\370rr)o(e)e(indsigt)i(i)g(hjernens)g(funktion)463
2079 y(p\345)20 b([Madsen,)f(1993].)g(Udgangspunktet)h(for)h(dette)f
(pr)o(ojekt)h(er)f(den)h(f\370rstn\346vnte)f(gr)o(uppe.)588
2193 y(Forskning)f(i)f(kunstige)h(neurale)f(netv\346rk)g(startede)f(i)h
(begyndelsen)h(af)e(1940'erne)f(hvor)463 2308 y(McCulloch)28
b(og)g(Pitts)g(beskr)o(ev)f(den)h(s\345kaldte)e(neur)o(onmodel)i([Abe,)
f(1995])e(inspir)o(er)o(et)i(af)463 2422 y(hjernens)h(virkem\345de.)e
(Neur)o(onmodellen)i(benytter)f(en)g(diskontinuert)h(model)g(for)e
(akti-)463 2537 y(veringsfunktionerne)19 b(\(som)g(den)f(diskr)o(ete)f
(aktiveringsfunktion)i(n\346vnt)g(i)f(afsnit)g(2.2\))f(der)463
2652 y(efterligner)k(biologiske)h(neur)o(oner)-6 b(,)21
b(heraf)f(navnet.)h(I)h(slutningen)g(af)f(50'erne)f(og)i(starten)f(af)
463 2766 y(60'erne)e(udvikledes)g(en)h(model)h(med)e(et)h(enkelt)g(lag)
g(kaldet)g FL(per)o(ceptr)o(onen)f FY(som)i(kunne)f(op-)463
2881 y(tr\346nes)29 b(til)g(at)g(l\346r)o(e)f(simple)h(funktioner)-6
b(,)30 b(f.eks.)e(AND)h(og)g(OR.)g(Men)g(Minsky)g(og)h(Papert)463
2995 y(viste)e(i)g(1969)e(at)i(enkeltlagsnetv\346rk)g(kun)g(kunne)h
(adskille)f(line\346rt)g(opdelelige)g(data)f(og)463 3110
y(f.eks.)18 b(ikke)i(engang)f(kunne)g(br)o(uges)h(til)f(at)g(simuler)o
(e)f(en)h(simpel)h(funktion)g(som)g(XOR)g([Mit-)463 3224
y(chell,)30 b(1997,)e(s.)i(89],)f(se)h(\002gur)h(2.3.)d(Dette)i
(kombiner)o(et)h(med)f(det)f(faktum)h(at)g(man)g(ingen)463
3339 y(algoritmer)19 b(havde)f(til)i(at)e(tr\346ne)h(\003erlagsper)o
(ceptr)o(oner)e(op,)i(bet\370d)f(at)h(inter)o(essen)g(for)g(feltet)463
3454 y(og)f(r)o(esultaterne)e(for)h(en)h(tid)f(stod)h(stille)g(inden)f
(for)g(den)h(datalogisk)f(inter)o(esser)o(ede)e(gr)o(uppe.)1478
4347 y @beginspecial 0 @llx 0 @lly 119 @urx 97 @ury 1133
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 @endspecial 534 4535 a FH(Figur)26 b(2.3:)h(XOR-pr)o(oblemet)f(\226)h
(\002gur)o(en)f(viser)h(r)o(esultatet)f(fra)g(en)g(XOR-funktion)f
(\(punkterne\))g(hvor)534 4640 y(akserne)h(angiver)g(de)g(to)g
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%%BeginDefaults
%%EndDefaults

%%BeginProlog
%
% Display a color image.  The image is displayed in color on
% Postscript viewers or printers that support color, otherwise
% it is displayed as grayscale.
%
/DirectClassPacket
{
  %
  % Get a DirectClass packet.
  %
  % Parameters:
  %   red.
  %   green.
  %   blue.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  compression 0 eq
  {
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  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/DirectClassImage
{
  %
  % Display a DirectClass image.
  %
  systemdict /colorimage known
  {
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
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    { DirectClassPacket } false 3 colorimage
  }
  {
    %
    % No colorimage operator;  convert to grayscale.
    %
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
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    { GrayDirectClassPacket } image
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} bind def

/GrayDirectClassPacket
{
  %
  % Get a DirectClass packet;  convert to grayscale.
  %
  % Parameters:
  %   red
  %   green
  %   blue
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  color_packet 0 get 0.299 mul
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  color_packet 2 get 0.114 mul add
  cvi
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  compression 0 eq
  {
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  {
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  0 1 number_pixels 1 sub
  {
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  pixels 0 number_pixels getinterval
} bind def

/GrayPseudoClassPacket
{
  %
  % Get a PseudoClass packet;  convert to grayscale.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
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  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
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  /gray_packet exch def
  compression 0 eq
  {
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  {
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  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
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  pixels 0 number_pixels getinterval
} bind def

/PseudoClassPacket
{
  %
  % Get a PseudoClass packet.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  compression 0 eq
  {
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  {
    currentfile byte readhexstring pop 0 get
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  0 3 number_pixels 1 sub
  {
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/PseudoClassImage
{
  %
  % Display a PseudoClass image.
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  % Parameters:
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  %
  currentfile buffer readline pop
  token pop /class exch def pop
  class 0 gt
  {
    currentfile buffer readline pop
    token pop /depth exch def pop
    /grays columns 8 add depth sub depth mul 8 idiv string def
    columns rows depth
    [
      columns 0 0
      rows neg 0 rows
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    { currentfile grays readhexstring pop } image
  }
  {
    %
    % Parameters:
    %   colors: number of colors in the colormap.
    %   colormap: red, green, blue color packets.
    %
    currentfile buffer readline pop
    token pop /colors exch def pop
    /colors colors 3 mul def
    /colormap colors string def
    currentfile colormap readhexstring pop pop
    systemdict /colorimage known
    {
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
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      { PseudoClassPacket } false 3 colorimage
    }
    {
      %
      % No colorimage operator;  convert to grayscale.
      %
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
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      { GrayPseudoClassPacket } image
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/DisplayImage
{
  %
  % Display a DirectClass or PseudoClass image.
  %
  % Parameters:
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  %   x & y scale.
  %   label pointsize.
  %   image label.
  %   image columns & rows.
  %   class: 0-DirectClass or 1-PseudoClass.
  %   compression: 0-none or 1-RunlengthEncoded.
  %   hex color packets.
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  /pixels 768 string def

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  currentfile buffer readline pop
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  token pop /rows exch def pop
  currentfile buffer readline pop
  token pop /class exch def pop
  currentfile buffer readline pop
  token pop /compression exch def pop
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} bind def
%%EndProlog
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 @endspecial 534 2910 a FH(Figur)18 b(3.3:)28 b(Bogstavet)20
b(s)g(kan)f(genkendes)g(som)h(den)f(bane)g(der)h(g\345r)g(igennem)e
(felterne)h(3,)g(2,)h(1,)f(4,)g(5,)h(8,)f(9,)534 3015
y(12,)g(11,)g(10)g(i)g(den)g(r\346kkef\370lge)588 3238
y FY(Det)f(bevirker)g(bl.a.)g(at)g(der)g(er)g(en)h(begr\346nsning)h
(p\345)e(hvor)h(mange)g(forskellige)g(tegn)g(der)463
3352 y(kan)k(genkendes)g(for)o(di)g(banerne)f(kommer)i(til)f(at)g
(ligne)h(hinanden.)f(Desuden)g(kr\346ver)f(det)463 3467
y(at)i(br)o(ugerne)g(l\346r)o(er)g(at)f(tegne)i(den)f(rigtige)h(slags)f
(baner)g(uden)g(f.eks.)g(at)g(l\370fte)g(pennen,)h(og)463
3582 y(det)j(kan)h(v\346r)o(e)e(n\370dvendigt)i(at)f(\346ndr)o(e)f
(banerne)h(i)h(for)o(hold)f(til)h(almindelig)g(praksis)g(ved)463
3696 y(h\345ndskrift)21 b(\(i'er)f(skrives)h(f.eks.)f(nedefra)f(og)i
(op)g(for)g(at)f(undg\345)h(forveksling)g(med)g(l'er\).)463
4000 y FI(3.2)119 b(Forslag)30 b(til)g(forbedret)e(gr\346nse\003ade)463
4204 y FY(Udgangspunktet)23 b(for)g(at)g(br)o(uge)g(et)g(neuralt)f
(netv\346rk)h(til)h(tegnkendelsen)f(i)h(dette)e(pr)o(ojekt)463
4318 y(er)k(at)g(der)f(arbejdes)g(med)h(det)g(f\346r)o(dige)f(billede,)
h(selvom)h(det)e(ogs\345)i(ville)f(v\346r)o(e)f(muligt)i(at)463
4433 y(arbejde)i(med)h(andr)o(e)f(oplysninger)-6 b(,)31
b(f.eks.)f(om)g(pennens)h(bev\346gelse)f(og)h(hastighed.)f(En)463
4547 y(for)o(del)17 b(ved)h(dette)g(er)g(at)g(genkendelsen)h(ikke)g
(bliver)f(afh\346ngig)h(af)e(hvilken)i(r\346kkef\370lge)g(de)463
4662 y(forskellige)f(dele)f(af)g(bogstaverne)h(tegnes)g(i.)f(T)-5
b(il)19 b(geng\346ld)f(forkastes)f(nogle)i(informationer)463
4777 y(som)j(ellers)e(kunne)i(medvirke)e(til)h(en)g(h\370jer)o(e)f
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(den)g(h\345nd-)463 5006 y(holdtes)25 b(gr\346nse\003ade.)e(Dels)i(er)f
(der)f(mulighed)i(for)f(at)g(netv\346rket)g(kan)g(tr\346nes)g(op)h
(p\345)f(de)463 5120 y(data)e(det)h(faktisk)h(klassi\002cer)o(er)-6
b(,)22 b(og)i(dermed)e(bringes)i(til)g(at)f(tilpasse)g(sig)h(de)f
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(i)g(stand)f(til)h(at)463 5349 y(give)i(mer)o(e)f(end)g(\351t)h
(forslag)f(hvis)i(netv\346rket)e(giver)h(et)f(tvetydigt)h(r)o(esultat.)
1908 5599 y(13)p eop
%%Page: 14 16
14 15 bop 1333 -129 a FY(Kapitel)20 b(3:)25 b(H\345ndholdte)c(computer)
o(e)p 463 -96 2975 4 v 588 192 a(T)-5 b(ilpasning)21
b(til)g(den)f(individuelle)g(br)o(uger)g(for)o(dr)o(er)f(at)g(der)h
(p\345)g(den)g(h\345ndholdte)g(indf\370-)463 306 y(r)o(es)h
(underst\370ttelse)i(af)e(\003er)o(e)g(br)o(uger)o(e)h(for)g(at)f
(undg\345)h(at)g(genkendelsesevnen)h(mistes)f(hvis)463
421 y(forskellige)i(personer)f(benytter)g(computer)o(en.)g(Der)g(er)g
(alts\345)g(behov)g(for)g(en)h(facilitet)f(til)g(at)463
535 y(skifte)e(br)o(uger)-6 b(.)588 650 y(Desuden)28
b(kr\346ver)g(tilpasningen)i(at)e(det)g(korr)o(ekte)g(r)o(esultat)g
(for)g(de)g(skr)o(evne)h(tegn)f(er)463 765 y(kendt.)g(Hvis)g(br)o(uger)
o(en)f(tegner)h(et)f(a,)g(skal)h(netv\346rket)g(vide)f(at)g(tegnet)h
(faktisk)g(er)f(t\346nkt)463 879 y(som)i(et)f(a)g(for)g(at)g(kunne)h
(tr\346ne)f(sig)h(p\345)f(det.)g(Dette)g(kan)g(imidlertid)h(l\370ses)g
(ved)e(som)j(ud-)463 994 y(gangspunkt)24 b(at)g(antage)e(at)i
(netv\346rket)f(altid)g(g\346tter)g(rigtigt)i(og)f(s\345)f(bem\346rke)g
(om)h(br)o(uge-)463 1108 y(r)o(en)16 b(sletter)h(tegnet)g(efter)f(det)g
(er)h(skr)o(evet.)f(I)g(s\345)h(fald)f(kan)h(man)g(undlade)f(at)g
(tr\346ne)h(netv\346rket)463 1223 y(p\345)25 b(netop)g(det)g
(speci\002kke)h(tegn)f(\(hvis)h(man)f(tillod)h(sig)g(at)f(tr\346ne)g
(netv\346rket)g(p\345)g(det)g(bog-)463 1337 y(stav)19
b(der)g(blev)g(skr)o(evet)g(i)h(stedet,)f(var)g(der)f(risiko)j(for)e
(at)g(tr\346ne)h(netv\346rket)f(forkert)g(idet)g(det)463
1452 y(ikke)i(er)f(muligt)i(at)e(vide)h(om)g(bogstavet)g(blev)f
(slettet)h(for)o(di)f(det)g(var)g(forkert)h(genkendt\).)588
1567 y(Endelig)16 b(kan)h(det)f(neurale)g(netv\346rk)g(indr)o(ettes)g
(s\345dan)g(at)g(det)g(dels)h(kan)f(genkende)h(\003er)o(e)463
1681 y(forskellige)22 b(tegn)h(end)e(hvad)h(der)f(er)h(muligt)g(med)g
(en)g(eksister)o(ende)f(l\370sning)j(som)f(kr\346ver)463
1796 y(tr)o(e)28 b(forskellige)h(felter)f(til)i(blot)f(A\226Z,)f
(a\226z)g(og)h(0\2269,)e(dels)i(kan)g(give)g(et)g(valg)f(mellem)i
(\003er)o(e)463 1910 y(mulige)h(tegn)g(hvis)g(r)o(esultatet)e(er)i
(tvetydigt.)f(P\345)g(denne)g(m\345de)g(bliver)g(det)h(hurtiger)o(e)e
(at)463 2025 y(r)o(ette)21 b(fejl,)g(og)i(der)e(kan)g(spar)o(es)g
(plads)g(i)h(gr\346nse\003aden.)g(Det)g(stiller)g(dog)g(f\370lgende)f
(krav)h(til)463 2139 y(netv\346rkets)f(arkitektur:)588
2309 y FS(\017)41 b FY(Det)21 b(skal)h(v\346r)o(e)e(muligt)i(at)f
(afg\370r)o(e)f(om)i(der)f(er)g(tvivl)h(om)g(hvilket)g(tegn)g(der)e(er)
h(blevet)671 2423 y(skr)o(evet,)e(og)i(hvilke)h(tvivlen)f(i)g(givet)f
(fald)g(st\345r)h(mellem.)588 2593 y FS(\017)41 b FY
(Netv\346rksarkitektur)o(en)22 b(skal)h(v\346r)o(e)g(tilstr\346kkeligt)
h(skal\351rbar)f(til)g(at)h(kunne)g(skelne)671 2707 y(alle)c(de)g
(forskellige)h(tegn.)463 3010 y FI(3.3)119 b(Ydelsesm\346ssige)30
b(krav)f(til)h(netv\346rket)463 3214 y FY(En)23 b(h\345ndholdt)g
(computer)f(k\370bes)h(som)h(en)e(samlet)h(enhed,)f(og)h
(speci\002kationerne)g(er)g(der)o(-)463 3329 y(for)34
b(for)o(udbestemt)h(af)e(pr)o(oducenten.)h(De)g(kan)h(med)f(f\345)g
(undtagelser)g(ikke)h(sener)o(e)e(\346n-)463 3443 y(dr)o(es)16
b(ved)g(en)h(opgradering)g(som)h(ved)e(station\346r)o(e)g(computer)o
(e.)h(Har)o(dwar)o(en)d(s\346tter)j(derfor)463 3558 y(nogle)k
(begr\346nsninger)h(for)e(hvad)h(der)f(er)g(muligt.)588
3672 y(De)e(to)i(mest)f(begr\346nsende)f(for)o(hold)h(er)f(pr)o
(ocessor)o(ens)h(hastighed)g(og)g(st\370rr)o(elsen)g(af)f(ar)o(-)463
3787 y(bejdslager)o(et.)d(En)h(typisk)h(pr)o(ocessor)f(er)g(en)g(206)f
(MHz)h(Intel)g(Str)o(ongARM,)g(og)g(den)h(samlede)463
3902 y(lagerplads)26 b(til)h(b\345de)f(permanente)h(data)f(\(inkl.)h
(pr)o(ogrammer\))f(og)h(arbejdshukommelse)463 4016 y(svinger)22
b(mellem)g(16)f(og)i(64)e(Mb,)g(med)g(en)h(standar)o(dkon\002guration)g
(p\345)f(32)g(Mb.)g(Dette)h(stil-)463 4131 y(ler)e(alts\345)h
(f\370lgende)f(krav)g(til)h(netv\346rket:)588 4300 y
FS(\017)41 b FY(Genkendelsespr)o(ocessen)26 b(skal)g(v\346r)o(e)g(for)o
(holdsvis)g(hurtig,)h(dvs.)f(hurtig)h(nok)g(til)f(at)671
4414 y(kunne)21 b(afvikles)f(i)h(r)o(ealtid)f(p\345)g(den)g
(h\345ndholdte.)588 4584 y FS(\017)41 b FY(Netv\346rket)20
b(m\345)g(ikke)h(br)o(uge)g(for)g(meget)f(hukommelse,)i(heller)f(ikke)g
(midlertidigt.)588 4753 y(Hvor)j(gr\346nserne)g(for)h(hvad)e(der)h(er)g
(acceptabelt)f(ydelsesm\346ssigt,)i(helt)g(pr\346cist)f(lig-)463
4868 y(ger)-6 b(,)23 b(kan)g(ikke)h(pr\346ciser)o(es)e(uden)i
(n\346rmer)o(e)e(kendskab)h(til)h(implementeringsfor)o(holdene)463
4982 y(p\345)e(de)g(forskellige)h(h\345ndholdte,)f(men)h(med)f(hensyn)i
(til)e(hastighed)h(er)f(en)h(tommel\002nger)o(-)463 5097
y(r)o(egel)d([Bickfor)o(d,)f(1999])g(at)i(pr)o(ocesseringstiden)g
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5211 y(met)e(skal)g(virke)g(\003ydende,)f(og)i(med)e(hensyn)i(til)f
(hukommelse)i(skal)e(der)f(naturligvis)h(v\346r)o(e)463
5326 y(plads)h(til)h(andet)g(end)f(det)g(neurale)h(netv\346rk)f(p\345)g
(computer)o(en.)1908 5599 y(14)p eop
%%Page: 15 17
15 16 bop 463 845 a FJ(Kapitel)44 b(4)463 1298 y FZ(Konstruktion)51
b(af)463 1584 y(tegngenkendelsessystem)463 2088 y FI(4.1)119
b(Oversigt)29 b(over)h(proces)463 2292 y FY(Den)19 b(f\370lgende)f
(oversigt)h(giver)g(et)f(hurtigt)h(overblik)g(over)f(pr)o(ocessen)g(i)h
(tegnkendelsen,)g(se)463 2407 y(\002gur)i(4.1,)e(fra)h(tegnet)h
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FL(n)q FY(-dimensionale)g(vek-)463 1371 y(tor)32 b(sendes)f(vider)o(e)f
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6824 2043 m
gs 1 -1 sc (CM) col0 sh gr
7.500 slw
% Ellipse
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% Polyline
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% Polyline
45.000 slw
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% Polyline
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% Polyline
0 slj
7.500 slw
 [15 45] 45 sd
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 cp gs col0 s gr  [] 0 sd
% Polyline
2 slj
45.000 slw
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% Polyline
15.000 slw
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% Polyline
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%%EndDocument
 @endspecial 534 1559 a FH(Figur)i(4.4:)27 b(I)c(de)g(to)f
(ovenst\345ende)h(eksempler)g(fors\370ges)g(centr)o(ering)f(efter)h
(massemidtpunkt)f(og)h(midten)534 1664 y(af)i(bogstavet.)g(Det)g
(bem\346rkes)g(at)g(n\345r)g(den)g(lille)f(prik)h(\003yttes,)g
(ensartes)g(billederne)f(bedst)h(ved)h(brug)e(af)534
1770 y(massemidtpunktet.)463 2030 y FY(ger)-6 b(.)16
b(Man)h(kan)g(i)g(centr)o(eringen)f(\003ytte)i(bogstavet)f(s\345ledes)f
(at)h(massemidtpunktet)g(kommer)463 2144 y(til)i(at)f(ligge)h(i)g
(midten)g(af)f(billedet)g(og)h(dermed)f(opn\345)h(den)f(\370nskede)h
(ensartning.)g(De)f(to)h(me-)463 2259 y(toder)h(er)h(illustr)o(er)o(et)
e(p\345)i(\002gur)g(4.4.)588 2373 y(Som)31 b(det)g(ses)g(p\345)g
(\002gur)o(en)g(giver)g(centr)o(eringen)g(efter)f(massemidtpunktet)i
(det)f(bed-)463 2488 y(ste)g(r)o(esultat)g(da)f(det)h(netop)h(er)f
(\370nsket)h(at)e(ensarte)h(de)g(\003este)g(punkter)h(i)f(to)h
(n\346sten)g(ens)463 2602 y(bogstaver)-6 b(.)29 b(Man)g(kan)g(dog)h(i)g
(s\346rlige)g(situationer)g(risiker)o(e)e(at)i(miste)g(data)e(ved)h(br)
o(ug)h(af)463 2717 y(centr)o(eringen)22 b(efter)g(massemidtpunktet,)h
(f.eks.)g(hvis)g(der)f(\002ndes)h(sorte)g(punkter)g(helt)g(ud)463
2832 y(til)30 b(billedets)f(kant.)h(Dette)f(mener)h(vi)g(dog)g(er)f(et)
g(mindr)o(e)g(pr)o(oblem)h(da)e(det)i(n\346sten)g(ikke)463
2946 y(opst\345r)21 b(i)f(praksis,)h(hvorfor)g(vi)g(v\346lger)f(at)g
(centr)o(er)o(e)f(efter)h(massemidtpunket.)588 3061 y(En)30
b(algoritme)g(til)h(at)f(for)o(etage)f(centr)o(eringen)h(kan)g
(implementer)o(es)g(ved)g(at)g(betragte)463 3175 y(de)i(enkelte)g
(v\346r)o(dier)f(i)h(punkterne)g(som)h(masser)f(og)h(her)o(efter)e
(benytte)h(samme)g(formel)463 3290 y(som)21 b([Serway)e(og)h(Beichner,)
g(2000,)e(s.)i(269-273])c(til)21 b(at)f(\002nde)g(massemidtpunktet)g
(\(cm\))g(i)g(et)463 3404 y(fysisk)e(objekt.)g(Hvis)24
b FL(p)r FG(\()s FL(x)s FY(,)16 b FL(y)r FG(\))i FY(angiver)f(v\346r)o
(dien)g(i)g(punktet)i FG(\()s FL(x)s FY(,)d FL(y)r FG(\))j
FY(p\345)e(billedet)g(kan)h(formlen)463 3519 y(skrives)j(som)1273
3643 y FL(x)1316 3655 y FP(cm)1428 3643 y FG(=)1745 3587
y FY(1)p 1527 3624 479 4 v 1528 3706 a FK(\345)1594 3720
y FO(y)1645 3706 y FK(\345)1712 3720 y FO(x)1768 3700
y FL(p)r FG(\()s FL(x)s FY(,)16 b FL(y)r FG(\))2032 3643
y FS(\001)2073 3661 y FE(\345)2103 3725 y FO(x)2179 3661
y FE(\345)2208 3725 y FO(y)2285 3643 y FL(x)j FS(\001)k
FL(p)r FG(\()s FL(x)s FY(,)16 b FL(y)r FG(\))1274 3915
y FL(y)1317 3927 y FP(cm)1429 3915 y FG(=)1746 3859 y
FY(1)p 1527 3896 V 1528 3978 a FK(\345)1594 3992 y FO(y)1645
3978 y FK(\345)1712 3992 y FO(x)1769 3972 y FL(p)r FG(\()s
FL(x)s FY(,)g FL(y)r FG(\))2033 3915 y FS(\001)2074 3933
y FE(\345)2103 3997 y FO(y)2179 3933 y FE(\345)2208 3997
y FO(x)2284 3915 y FL(y)i FS(\001)23 b FL(p)r FG(\()s
FL(x)s FY(,)16 b FL(y)r FG(\))463 4134 y FY(hvor)21 b(man)g(summer)o
(er)f(over)h(alle)f(punkter)-6 b(.)588 4248 y(Det)17
b(stiller)g(dog)g(som)h(krav)f(at)f(v\346r)o(dierne)g(opf\370r)o(er)g
(sig)h(som)h(en)f(masse)g(i)h(fysikkens)f(ver)o(-)463
4363 y(den)i(ved)g(at)h(massen)f(aldrig)h(kan)f(blive)h(mindr)o(e)f
(end)g(0.)g(Det)h(opfylder)f(vor)o(es)g(tr\346ningss\346t)463
4477 y(ikke)h(da)g(gr\345tonerne)f(beskrives)h(i)h(intervallet)f
FG([)r FS(\000)q FY(1)q(,)15 b(1)q FG(])q FY(,)j(men)j(dette)e(kan)h
(l\370ses)h(ved)e(at)h(s\346tte)469 4592 y FL(p)r FG(\()s
FL(x)s FY(,)c FL(y)r FG(\))24 b(=)30 b FL(p)r FG(\()s
FL(x)s FY(,)16 b FL(y)r FG(\))g(+)h FY(1)j(ved)h(algoritmens)g(start.)
588 4706 y(N\345r)h(billedets)i(massemidtpunkt)g(er)f(fundet,)g(kan)h
(dette)e(sammenholdes)j(med)e(bille-)463 4821 y(dets)e(midtpunkt)h
FG(\()1104 4820 y FY(\257)1090 4821 y FL(x)s FY(,)q(\257)-13
b FL(y)r FG(\))21 b FY(som)h(er)e(de\002ner)o(et)g(ved)1714
5006 y(\257)1701 5007 y FL(x)26 b FG(=)f(\()r FL(b)19
b FG(+)d FY(1)q FG(\))r FY(/)q(2)1713 5191 y(\257)1699
5192 y FL(y)25 b FG(=)g(\()r FL(h)18 b FG(+)f FY(1)q
FG(\))r FY(/)q(2)463 5349 y(hvor)22 b FL(b)g FY(er)f(billedets)f(br)o
(edde,)f(og)j FL(h)g FY(er)e(billedets)h(h\370jde.)1908
5599 y(19)p eop
%%Page: 20 22
20 21 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 588 192 a(V)-5
b(i)21 b(kan)g(s\345ledes)f(ber)o(egne)f(hvor)i(mange)g(punkter)g
(billedet)g(skal)f(\003yttes)i(ved)1704 364 y FK(D)s
FL(x)k FG(=)1935 363 y FY(\257)1922 364 y FL(x)19 b FS(\000)f
FL(x)2109 376 y FP(cm)1696 536 y FK(D)r FL(y)25 b FG(=)1924
535 y FY(\257)1911 536 y FL(y)18 b FS(\000)f FL(y)2096
548 y FP(cm)2184 536 y FY(.)463 685 y(Disse)22 b(v\346r)o(dier)e(skal)i
(naturligvis)g(afr)o(undes)f(for)h(at)f(kunne)h(br)o(uges)g(p\345)f(de)
g(diskr)o(ete)g(punk-)463 800 y(ter)f(i)h(billedet.)588
914 y(For)j(at)g(illustr)o(er)o(e)g(ef)o(fekten)f(af)h(en)g(centering)h
(efter)e(massemidtpunktet)j(er)e(der)f(p\345)h(bi-)463
1029 y(lag)c(B.1)g(p\345)g(side)h(94)e(vist)i(nogle)h(bogstaver)e(der)g
(er)g(centr)o(er)o(et)f(vha.)h(vor)o(es)g(forbehandlings-)463
1143 y(pr)o(ogram.)463 1399 y Fz(4.2.3)99 b(Praktiske)24
b(overvejelser)463 1572 y FY(Da)g(der)g(i)g(begge)g(ovenst\345ende)g
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(hastighedsm\346ssige)h(konsekvenser)h(at)e(benytte)h(forbehandlingen.)
588 1916 y(Det)26 b(umiddelbar)o(e)f(input)i(best\345r)f(af)f(2880)g
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(e)f(end)h(h\370jst)i(2880)c(byte)i(ved)f(en)h(skalering)h(i)f(for)o
(holdet)g(1:2)f(p\345)g(hver)463 2374 y(led.)c(Det)h(vil)g(sige)g(at)f
(lagerforbr)o(uget)g(ikke)h(er)g(noget)g(pr)o(oblem.)588
2488 y(Ber)o(egningsm\346ssigt)28 b(er)g(centr)o(ering)f(det)g(mest)i
(kr\346vende.)e(Ber)o(egningen)g(af)g(masse-)463 2603
y(midtpunktet)d(kr\346ver)g(\351t)f(genneml\370b)i(pr)-6
b(.)23 b(billede)h(med)g(to)g(gangeoperationer)f(for)h(hvert)463
2717 y(punkt)h(\(\351n)g(for)h FL(x)21 b FS(\001)j FL(p)r
FG(\()s FL(x)s FY(,)16 b FL(y)r FG(\))25 b FY(og)g(\351n)g(for)g
FL(y)20 b FS(\001)k FL(p)r FG(\()s FL(x)s FY(,)16 b FL(y)r
FG(\))q FY(\).)23 b(Dette)i(giver)f(dog)h(ikke)g(mer)o(e)f(end)h(om-)
463 2832 y(tr)o(ent)d(2)c FS(\001)g FY(2880)26 b FG(=)i
FY(5760)21 b(operationer)-6 b(,)22 b(hvorfor)h(forbehandlingen)h
(n\346ppe)e(vil)i(udg\370r)o(e)e(no-)463 2947 y(get)f(praktisk)f
(hastighedspr)o(oblem.)463 3250 y FI(4.3)119 b(Netv\346rksarkitektur)
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b(navnet)f(antyder)-6 b(,)27 b(sendes)g(data)g(i)g(et)h(fr)o
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3761 y(v\346rket)d(i)h(den)g(forstand)g(at)f(der)g(ikke)h(g\345r)g
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3875 y(Den)c(simpleste)g(type)g(fr)o(emadf\370r)o(ende)d(netv\346rk)i
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3990 y(kelt)g(neur)o(on)f(i)h(det)f(sidste)h(lag,)f(se)g(\002gur)h(4.5)
e(p\345)h(den)h(f\370lgende)f(side.)g(Ud)g(fra)f(denne)i(model)463
4104 y(kan)g(sammens\346ttes)g(st\370rr)o(e)f(netv\346rk)h(til)g(at)f
(l\370se)h(mer)o(e)f(komplekse)i(opgaver)-6 b(.)588 4219
y(Bem\346rk)18 b(at)f(den)i(klassiske)g(per)o(ceptr)o(onmodel)e(br)o
(uger)h(fortegnsfunktionen)i(\(beskr)o(e-)463 4334 y(vet)h(i)g(afsnit)g
(2.2\))e(som)j(aktiveringsfunktion,)g(mens)g(\002gur)f(4.5)f(viser)h
(den)g(logistiske)h(funk-)463 4448 y(tion)i(\226)f(denne)g(br)o(uges)h
(i)f(dag)g(i)h(stedet)f(for)g(fortegnsfunktionen)i(da)d(den)h(har)g
(nogle)i(bedr)o(e)463 4563 y(egenskaber)20 b(\(se)g(afsnit)h(4.3.3)e
(p\345)h(side)h(24\).)588 4677 y(Per)o(ceptr)o(onen)h(best\345r)g(af)h
(en)g(r\346kke)g(input)j FL(x)2067 4693 y FP(1)2103 4677
y FY(,)17 b FL(x)2184 4692 y FP(2)2221 4677 y FY(,)f(.)g(.)g(.)e(,)j
FL(x)2449 4689 y FO(n)2512 4677 y FY(der)23 b(multiplicer)o(es)g(med)g
(en)463 4792 y(r\346kke)e(v\346gte)g FL(w)1017 4807 y
FP(1)1053 4792 y FY(,)16 b FL(w)1151 4807 y FP(2)1187
4792 y FY(,)g(.)g(.)g(.)f(,)h FL(w)1433 4804 y FO(n)1493
4792 y FY(og)21 b(der)o(efter)e(summer)o(es)1741 5010
y FL(a)26 b FG(=)1926 4920 y FO(n)1898 5027 y FE(\345)1893
5100 y FO(i)r FC(=)q FP(1)2008 5010 y FL(w)2070 5026
y FO(i)2094 5010 y FL(x)2138 5026 y FO(i)463 5235 y FY(og)20
b(en)f(aktiveringsfunktion)k FL(g)c FY(virkende)g(p\345)g(den)f
(v\346gtede)h(sum)i FL(a)q FY(,)e(hvilket)h(giver)f(aktive-)463
5349 y(ringsv\346r)o(dien)i FL(y)k FG(=)i FL(g)r FG(\()s
FL(a)r FG(\))22 b FY(af)e(neur)o(onen.)1908 5599 y(20)p
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%%Page: 21 23
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214 y FP(0)1371 199 y FG(=)k FY(1)1455 424 y FL(w)1516
439 y FP(0)1042 986 y FL(w)1104 1003 y FO(i)1042 499
y FL(w)1103 514 y FP(2)1042 199 y FL(w)1103 214 y FP(1)706
1511 y FL(x)750 1523 y FO(n)706 1099 y FL(x)750 1115
y FO(i)706 536 y FL(x)749 551 y FP(2)706 236 y FL(x)749
252 y FP(1)1042 1286 y FL(w)1104 1298 y FO(n)1180 1789
y FH(Figur)17 b(4.5:)28 b(Illustration)17 b(af)i(en)f(per)o(ceptr)o
(ons)h(opbygning)588 2059 y FY(Den)d(v\346gtede)f(sum)h(r)o
(epr\346senter)o(er)e(en)h(hyperplan)h(som)h(opdeler)e(det)h
FL(n)q FY(-dimensionale)463 2173 y(r)o(um)29 b(i)g(to,)g(men)h(da)e
(denne)h(opdeling)g(ikke)g(n\370dvendigvis)g(er)g(optimal)g(n\345r)g
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(ofte)f(den)g(v\346gtede)g(sum)h(med)f(en)h(konstant)g(i)g(form)f(af)g
(en)463 2403 y(ekstra)g(v\346gt)h(lagt)g(til)1633 2563
y FL(a)k FG(=)1817 2473 y FO(n)1790 2580 y FE(\345)1785
2653 y FO(i)r FC(=)q FP(1)1900 2563 y FL(w)1962 2579
y FO(i)1986 2563 y FL(x)2030 2579 y FO(i)2069 2563 y
FG(+)18 b FL(w)2213 2578 y FP(0)2248 2563 y FY(.)463
2780 y(V)-9 b(ed)21 b(at)h(indskyde)h(en)f(ekstra)g(inputneur)o(on)i
FL(x)1965 2795 y FP(0)2023 2780 y FY(\(en)e(\204bias\223\))f(med)h(den)
g(faste)f(v\346r)o(di)g(1)h(som)463 2895 y(illustr)o(er)o(et)e(p\345)g
(\002gur)h(4.5,)e(kan)i(dette)f(omskrives)i(til)1413
3138 y FL(a)j FG(=)1597 3048 y FO(n)1569 3155 y FE(\345)1564
3228 y FO(i)r FC(=)q FP(1)1680 3138 y FL(w)1742 3154
y FO(i)1765 3138 y FL(x)1809 3154 y FO(i)1849 3138 y
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Fz(4.3.2)99 b(Flerlagsmodeller)463 3822 y FY(Ud)19 b(fra)f(den)h
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(lag)g(udvikles,)g(se)h(f.eks.)463 3937 y(\002gur)h(4.6.)e(Den)i
(v\346gtede)f(sum)i(for)e(den)25 b FL(j)q FY('te)c(neur)o(on)f(bliver)h
(da)1708 4180 y FL(a)1749 4196 y FO(j)1795 4180 y FG(=)1917
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FP(0)2000 4180 y FL(w)2064 4196 y FO(j)r(i)2107 4180
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4546 y(enhver)k(funktion)h([Bishop,)g(1995])d(og)j(dermed)e(ogs\345)h
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(tilf\346ldet)f(med)h(booleske)h(funktioner)f(er)g(det)f(me-)463
5348 y(get)e(simpelt.)1908 5599 y(21)p eop
%%Page: 22 24
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(aktiveringsfunktion,)g(kan)463 421 y(netv\346rket)k(opfattes)f(som)i
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FL(d)e FY(input)h(er)e(antallet)h(af)463 535 y(mulige)30
b(bin\346r)o(e)e(kombinationer)j(der)d(kan)h(beskrives)g(2)2338
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650 y(m\345lv\346r)o(dien)d(1)h(tilf\370jes)h(en)f(neur)o(on)h(i)f(det)
g(skjulte)h(lag)f(som)h(\204fyr)o(er\223)d(p\345)i(lige)h(netop)f(den)
463 765 y(udvalgte)20 b(kombination)i(ved)f(at)f(v\346gten)h(i)g(en)g
(forbindelse)f(s\346ttes)h(til)h(1)e(hvis)h(v\346r)o(dien)f(fra)463
879 y(neur)o(onen)30 b(i)f(det)g(for)o(eg\345ende)f(lag)h(er)g(1,)g(og)
h(-1)e(hvis)j(v\346r)o(dien)d(er)h(et)g(0.)g(Biasv\346r)o(dien)f(for)
463 994 y(neur)o(onen)d(s\346ttes)f(s\345)g(til)h(1)18
b FS(\000)h FL(b)26 b FY(hvor)g FL(b)g FY(er)e(antallet)g(af)g(v\346r)o
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	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
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%
% Fig objects follow
%
% Polyline
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n 990 1080 m
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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n 990 2970 m
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n 990 3060 m
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n 990 3600 m
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n 990 3690 m
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% Polyline
n 990 4230 m
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% Ellipse
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% Ellipse
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% Ellipse
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% Ellipse
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% Ellipse
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% Ellipse
n 3802 2677 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3802 3802 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3802 4882 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 2632 4882 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 1552 4882 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 517 4882 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 517 3802 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 517 2722 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 517 1642 112 112 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Polyline
30.000 slw
n 0 0 m 4320 0 l 4320 5400 l 0 5400 l
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% Polyline
7.500 slw
n 0 990 m
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% Polyline
n 0 4410 m
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% Polyline
n 990 0 m
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n 3240 0 m
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n 1080 990 m
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1440 990 m
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% Polyline
n 1530 990 m
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% Polyline
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% Polyline
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15.000 slw
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-15 1095 m -15 1155 l 152 1155 l 32 1125 l 152 1095 l cp
eoclip
n 0 1125 m
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% arrowhead
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% arrowhead
n 838 1155 m 958 1125 l 838 1095 l 838 1155 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
gs  clippath
1095 1005 m 1155 1005 l 1155 838 l 1125 958 l 1095 838 l cp
1155 -15 m 1095 -15 l 1095 152 l 1125 32 l 1155 152 l cp
eoclip
n 1125 0 m
 1125 990 l gs col0 s gr gr

% arrowhead
n 1155 152 m 1125 32 l 1095 152 l 1155 152 l  cp gs 0.00 setgray ef gr  col0 s
% arrowhead
n 1095 838 m 1125 958 l 1155 838 l 1095 838 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
n 2066 0 m 2074 0 l gs col0 s gr
% Polyline
n 2070 0 m
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% Polyline
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% Polyline
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% Polyline
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eoclip
n 990 1125 m
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% arrowhead
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% arrowhead
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% Polyline
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3135 975 m 3075 975 l 3075 1142 l 3105 1022 l 3135 1142 l cp
eoclip
n 3105 990 m
 3105 4410 l gs col0 s gr gr

% arrowhead
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% arrowhead
n 3075 4258 m 3105 4378 l 3135 4258 l 3075 4258 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
n 1800 990 m
 1800 4410 l gs col0 s gr 
% Polyline
n 1890 990 m
 1890 4410 l gs col0 s gr 
% Polyline
n 1710 990 m
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% Polyline
n 1845 1215 m 2205 1215 l 2205 1440 l 1845 1440 l
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% Polyline
n 2880 990 m
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% Polyline
n 2790 990 m
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% Polyline
n 2700 2205 m 3015 2205 l 3015 2430 l 2700 2430 l
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% Polyline
15.000 slw
gs  clippath
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4470 5415 m 4530 5415 l 4530 5248 l 4500 5368 l 4470 5248 l cp
eoclip
n 4500 5400 m
 4500 0 l gs col7 1.00 shd ef gr gs col0 s gr gr

% arrowhead
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% arrowhead
n 4530 152 m 4500 32 l 4470 152 l 4530 152 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
gs  clippath
4335 -105 m 4335 -165 l 4168 -164 l 4288 -134 l 4168 -104 l cp
-15 -165 m -15 -105 l 152 -104 l 32 -134 l 152 -164 l cp
eoclip
n 0 -135 m
 4320 -135 l gs col7 1.00 shd ef gr gs col0 s gr gr

% arrowhead
n 152 -164 m 32 -134 l 152 -104 l 152 -164 l  cp gs 0.00 setgray ef gr  col0 s
% arrowhead
n 4168 -104 m 4288 -134 l 4168 -164 l 4168 -104 l  cp gs 0.00 setgray ef gr  col0 s
/Times-Roman ff 180.00 scf sf
1170 585 m
gs 1 -1 sc (12) col0 sh gr
/Times-Roman ff 180.00 scf sf
405 1305 m
gs 1 -1 sc (12) col0 sh gr
/Times-Roman ff 180.00 scf sf
1935 1395 m
gs 1 -1 sc (24) col0 sh gr
/Times-Roman ff 180.00 scf sf
2790 2385 m
gs 1 -1 sc (36) col0 sh gr
/Times-Roman ff 180.00 scf sf
495 540 m
gs 1 -1 sc (1) col0 sh gr
/Times-Roman ff 180.00 scf sf
1575 540 m
gs 1 -1 sc (2) col0 sh gr
/Times-Roman ff 180.00 scf sf
2610 540 m
gs 1 -1 sc (3) col0 sh gr
/Times-Roman ff 180.00 scf sf
3735 540 m
gs 1 -1 sc (4) col0 sh gr
/Times-Roman ff 180.00 scf sf
2610 4950 m
gs 1 -1 sc (9) col0 sh gr
/Times-Roman ff 180.00 scf sf
1485 4950 m
gs 1 -1 sc (10) col0 sh gr
/Times-Roman ff 180.00 scf sf
450 4950 m
gs 1 -1 sc (11) col0 sh gr
/Times-Roman ff 180.00 scf sf
3780 4950 m
gs 1 -1 sc (8) col0 sh gr
/Times-Roman ff 180.00 scf sf
450 2790 m
gs 1 -1 sc (13) col0 sh gr
/Times-Roman ff 180.00 scf sf
450 1710 m
gs 1 -1 sc (14) col0 sh gr
/Times-Roman ff 180.00 scf sf
450 3870 m
gs 1 -1 sc (12) col0 sh gr
/Times-Roman ff 180.00 scf sf
3780 3870 m
gs 1 -1 sc (7) col0 sh gr
/Times-Roman ff 180.00 scf sf
3780 2745 m
gs 1 -1 sc (6) col0 sh gr
/Times-Roman ff 180.00 scf sf
3780 1665 m
gs 1 -1 sc (5) col0 sh gr
/Times-Roman ff 180.00 scf sf
1980 -180 m
gs 1 -1 sc (48) col0 sh gr
/Times-Roman ff 180.00 scf sf
4545 2835 m
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$F2psEnd
rs

%%EndDocument
 @endspecial 1252 w @beginspecial 0 @llx 0 @lly 96 @urx
120 @ury 1587 @rhi @setspecial
%%BeginDocument: arkitektur/super.eps
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: (ImageMagick)
%%Title: (super.eps)
%%CreationDate: (Fri May 10 16:52:13 2002)
%%BoundingBox: 0 0 96 120
%%DocumentData: Clean7Bit
%%LanguageLevel: 1
%%Pages: 1
%%EndComments

%%BeginDefaults
%%EndDefaults

%%BeginProlog
%
% Display a color image.  The image is displayed in color on
% Postscript viewers or printers that support color, otherwise
% it is displayed as grayscale.
%
/DirectClassPacket
{
  %
  % Get a DirectClass packet.
  %
  % Parameters:
  %   red.
  %   green.
  %   blue.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  compression 0 eq
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/DirectClassImage
{
  %
  % Display a DirectClass image.
  %
  systemdict /colorimage known
  {
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { DirectClassPacket } false 3 colorimage
  }
  {
    %
    % No colorimage operator;  convert to grayscale.
    %
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { GrayDirectClassPacket } image
  } ifelse
} bind def

/GrayDirectClassPacket
{
  %
  % Get a DirectClass packet;  convert to grayscale.
  %
  % Parameters:
  %   red
  %   green
  %   blue
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/GrayPseudoClassPacket
{
  %
  % Get a PseudoClass packet;  convert to grayscale.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassPacket
{
  %
  % Get a PseudoClass packet.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  compression 0 eq
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassImage
{
  %
  % Display a PseudoClass image.
  %
  % Parameters:
  %   class: 0-PseudoClass or 1-Grayscale.
  %
  currentfile buffer readline pop
  token pop /class exch def pop
  class 0 gt
  {
    currentfile buffer readline pop
    token pop /depth exch def pop
    /grays columns 8 add depth sub depth mul 8 idiv string def
    columns rows depth
    [
      columns 0 0
      rows neg 0 rows
    ]
    { currentfile grays readhexstring pop } image
  }
  {
    %
    % Parameters:
    %   colors: number of colors in the colormap.
    %   colormap: red, green, blue color packets.
    %
    currentfile buffer readline pop
    token pop /colors exch def pop
    /colors colors 3 mul def
    /colormap colors string def
    currentfile colormap readhexstring pop pop
    systemdict /colorimage known
    {
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { PseudoClassPacket } false 3 colorimage
    }
    {
      %
      % No colorimage operator;  convert to grayscale.
      %
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { GrayPseudoClassPacket } image
    } ifelse
  } ifelse
} bind def

/DisplayImage
{
  %
  % Display a DirectClass or PseudoClass image.
  %
  % Parameters:
  %   x & y translation.
  %   x & y scale.
  %   label pointsize.
  %   image label.
  %   image columns & rows.
  %   class: 0-DirectClass or 1-PseudoClass.
  %   compression: 0-none or 1-RunlengthEncoded.
  %   hex color packets.
  %
  gsave
  /buffer 512 string def
  /byte 1 string def
  /color_packet 3 string def
  /pixels 768 string def

  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  x y translate
  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  currentfile buffer readline pop
  token pop /pointsize exch def pop
  /Times-Roman findfont pointsize scalefont setfont
  x y scale
  currentfile buffer readline pop
  token pop /columns exch def
  token pop /rows exch def pop
  currentfile buffer readline pop
  token pop /class exch def pop
  currentfile buffer readline pop
  token pop /compression exch def pop
  class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
  grestore
} bind def
%%EndProlog
%%Page:  1 1
%%PageBoundingBox: 0 0 96 120
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end
%%PageTrailer
%%Trailer
%%EOF

%%EndDocument
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b(Pr)o(oblemet)17 b(er)g(snar)o(er)o(e)f(at)h(de)g(a\003edede)g(af)f
(funktionerne)j(ved)463 3118 y(disse)25 b FL(x)r FY(-v\346r)o(dier)c
(er)i(meget)g(t\346t)g(p\345)g(nul)h(hvilket)f(f\345r)f(den)h(ene)g(af)
f(de)h(tr\346ningsalgoritmer)463 3233 y(der)h(gennemg\345s)h(i)g
(afsnit)g(4.4,)e(gradientnedstigning,)j(til)f(at)f(tage)h(sm\345)f
(skridt)h(s\345)g(tr\346nin-)463 3347 y(gen)c(er)f(lang)h(tid)g(om)g
(at)g(konver)o(ger)o(e.)588 3462 y(For)e(tanh)h(kunne)f(man)h(f.eks.)e
(v\346lge)i(m\345lv\346r)o(dierne)e(til)h(at)g(v\346r)o(e)f(enten)j
FS(\000)q FY(0)q(,)q(9)c(eller)i(0)q(,)q(9.)463 3576
y(P\345)k(denne)g(m\345de)f(er)h(m\345lv\346r)o(dierne)f(stadig)h
(t\346t)g(p\345)g(asymptoterne)g(og)h(over)f(det)g(line\346r)o(e)463
3691 y(stykke)d(s\345)f(netv\346rket)g(tvinges)g(til)h(enten)f(at)g
(v\346lge)g(enten)h(den)f(ene)g(yderlighed)g(eller)g(ogs\345)463
3805 y(den)i(anden)f(\226)g(derved)g(opn\345s)h(en)g(ef)o(fekt)e(som)i
(ved)g(fortegnsfunktionen.)463 4066 y Fz(4.3.4)99 b(Startv\346rdier)24
b(for)h(v\346gte)463 4239 y FY(T)-5 b(ildeling)21 b(af)e(startv\346r)o
(dier)f(for)h(v\346gtene)h(kan)g(principielt)f(ske)h(helt)g
(vilk\345rligt,)g(bortset)g(fra)463 4353 y(at)26 b(de)f(ikke)h(m\345)g
(v\346r)o(e)f(ens)h(for)g(at)g(undg\345)f(symmetri)i(i)f(netv\346rket)g
(og)h(ikke)f(m\345)g(v\346r)o(e)e(nul)463 4468 y(\(da)f(de)g(partielle)
g(a\003edede)g(s\345)g(bliver)h(nul,)g(se)g(afsnit)g(4.4.2\).)d(For)j
(at)g(opn\345)f(hurtigst)i(mulig)463 4582 y(konver)o(gens)17
b(under)g(optr\346ningen)h(kan)f(det)g(dog)g(v\346r)o(e)f(en)h(for)o
(del)f(af)g(begr\346nse)h(den)g(m\345de)463 4697 y(v\346gtene)k
(v\346lges)g(p\345)f([Sarle,)f(2001].)588 4812 y(Hvis)i(v\346gtene)g
(v\346lges)g(for)f(sm\345,)h(tager)f(gradientnedstigningen)i(sm\345)f
(skridt)3163 4781 y FP(2)3220 4812 y FY(\226)f(hvis)463
4926 y(v\346gtene)28 b(er)g(for)g(stor)o(e,)g(driver)g(de)g(v\346gtede)
f(summer)i(aktiveringsfunktionerne)g(ud)g(til)463 5041
y(asymptoterne)23 b(hvor)g(de)g(a\003edede)f(er)g(meget)h(sm\345)g
(s\345)g(gradientnedstigning)h(som)f(n\346vnt)463 5155
y(igen)31 b(tager)e(sm\345)h(skridt.)g(Et)g(kompr)o(omis)i(mellem)e
(disse)g(to)h(yderligheder)e(er)h(derfor)f(at)p 463 5232
1190 4 v 553 5288 a Fy(2)583 5312 y FW(Dette)15 b(kan)i(ses)g(af)g
(udtryk)f(\(4.9\))g(i)h(afsnit)g(4.4.2.)1908 5599 y FY(26)p
eop
%%Page: 27 29
27 28 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 463 192 a(v\346lge)32
b(v\346gtene)h(s\345)f(de)g(v\346gtede)f(summer)i(kommer)g(til)g(at)f
(ligge)h(i)g(intervallet)g FG([)r FS(\000)q FY(1;)14
b(1)q FG(])463 306 y FY([Bishop,)27 b(1995])e(\226)h(omkring)j(det)d
(sted)h(hvor)g(aktiveringsfunktionerne)h(kr)o(ummer)g(mest.)463
421 y(For)33 b(at)f(undg\345)h(symmetripr)o(oblemet)g(kan)g(det)f
(g\370r)o(es)h(efter)e(en)i(metode)g(der)f(gener)o(er)o(er)463
535 y(vilk\345rlige)21 b(tal)f(med)h(en)g(passende)f(for)o(deling)g
(med)h(variansen)g FN(s)2574 505 y FP(2)2630 535 y FY(og)g
(middelv\346r)o(dien)e(0.)588 650 y(Det)26 b(viser)g(sig)g(s\345)g(at)g
FN(s)i FY(skal)e(skaler)o(e)f(efter)h FN(s)35 b FK(\265)g
FL(n)2225 619 y Ft(\000)2285 594 y Fy(1)p 2285 604 25
3 v 2285 641 a(2)2350 650 y FY([Bishop,)26 b(1995],)d(se)j(evt.)g
(appen-)463 765 y(diks)21 b(C)g(p\345)f(side)g(96.)463
1073 y FI(4.4)119 b(T)-13 b(r\346ningsmetoder)463 1277
y FY(T)-7 b(r\346ning)25 b(af)f(et)h(neuralt)g(netv\346rk)g(for)o
(eg\345r)e(ved)h(at)h(\346ndr)o(e)f(p\345)g(v\346gtene)h(i)g
(forbindelserne)463 1391 y(og)g(kan)f(formaliser)o(es)f(ved)h(at)g
(de\002ner)o(e)f(en)h FL(fejlfunktion)j(E)e FY(over)f(netv\346rket)g
(som)h(udtryk-)463 1506 y(ker)30 b(hvor)g(stor)g(fejlen)g(er)f(for)h
(en)g(datavektor)g FT(x)g FY(givet)g(den)g(\370nskede)g(v\346r)o(di)g
FT(t)q FY(.)f(Fejlen)h(er)463 1620 y(naturligvis)i(en)f(funktion)i(af)d
(v\346gtene,)i FT(w)q FY(,)f(hvorfor)h(tr\346ningen)g(kan)f(anskues)h
(som)g(en)463 1735 y(tilpasning)24 b(af)d(en)i(funktion)h(af)d(\003er)o
(e)h(variable.)g(Det)g(der)g(s\370ges)h(efter)-6 b(,)21
b(er)i(den)f(v\346r)o(di)h FT(w)3309 1705 y Ft(\003)3368
1735 y FY(af)463 1850 y(v\346gtene)18 b(som)g(giver)g(den)g(mindste)g
(fejl)g(over)f(alle)h(tr\346ningseksemplerne,)g(hvilket)h(svar)o(er)463
1964 y(til)i(at)f(minimer)o(e)j FL(E)r FG(\()r FT(w)r
FG(\))q FY(.)588 2079 y(Minimeringen)e(af)f(funktionen)k
FL(E)e FY(kan)e(ikke)h(l\370ses)g(analytisk)g(for)g(et)f
(ikke-line\346rt)h(net-)463 2193 y(v\346rk.)28 b(I)g(stedet)g(kan)g
(iterative,)g(numeriske)g(algoritmer)h(br)o(uges,)f(men)h(disse)f
(garanter)o(er)463 2308 y(dog)k(kun)h(at)e(\002nde)h(et)g(lokalt)g
(minimum.)i(F\346lles)e(for)g(algoritmerne)g(er)f(at)h(de)f(benytter)
463 2422 y(sig)23 b(af)f(de)h(partielle)f(a\003edede)f(af)j
FL(E)g FY(med)f(hensyn)h(til)f(de)f(forskellige)h(v\346gte)g(hvilket)g
(giver)464 2537 y FS(r)s FL(E)i FG(=)g(\()764 2504 y
FQ(\266)s FO(E)p 745 2518 109 4 v 746 2568 a FQ(\266)r
FO(w)826 2581 y Fy(1)864 2537 y FY(,)928 2504 y FQ(\266)s
FO(E)p 910 2518 V 911 2568 a FQ(\266)r FO(w)991 2580
y Fy(2)1029 2537 y FY(,)16 b(.)g(.)g(.)f(,)1242 2504
y FQ(\266)s FO(E)p 1221 2518 113 4 v 1222 2568 a FQ(\266)r
FO(w)1303 2577 y Fr(n)1345 2537 y FG(\))q FY(.)k(Denne)g(kan)h(ber)o
(egnes)e(ef)o(fektivt)g(vha.)h(tilbagef\370rselsalgorit-)463
2652 y(men)26 b(der)-6 b(,)25 b(som)i(n\346vnt)g(i)f(afsnit)g(2.2,)f
(blev)g(populariser)o(et)g(af)h(en)g(artikel)f(af)h(Rummelhart,)463
2766 y(Hinton)c(og)f(W)-5 b(illiams)22 b(i)f(1986.)588
2881 y(I)34 b(Rummelhart)h(et)g(al.'s)f(artikel)h(benyttes)f
(gradientvektor)o(en)g(dir)o(ekte)g(til)h(at)f(opda-)463
2995 y(ter)o(e)20 b(v\346gtene)i(ved)f FL(gradientnedstigning)f
FY([Madsen,)g(1993])g(som)i(blot)g(g\345r)f(ud)h(p\345)f(for)g(hvert)
463 3110 y(skridt)33 b(at)g(g\345)g(i)g(r)o(etningen)h
FS(\000)r(r)s FL(E)q FY(,)f(men)h(der)e(\002ndes)i(andr)o(e)d
(algoritmer)j(som)g(r)o(educer)o(er)463 3224 y(fejlen)h(hurtiger)o(e)e
([Bishop,)i(1995],)d(bl.a.)i FL(konjuger)o(et)g(gradient)g
FY(som)h(behandles)f(sener)o(e)g(i)463 3339 y(dette)20
b(kapitel.)463 3599 y Fz(4.4.1)99 b(Fejlfunktioner)463
3772 y FY(For)31 b(alle)f(de)g(betragtede)f(fejlfunktioner)i(g\346lder)
f(at)g(den)g(samlede)g(fejl)g(som)h(netv\346rket)463
3887 y(for\345rsager)19 b(over)h(et)g(tr\346ningss\346t,)h(er)f(summen)
i(af)d(fejlene)h(ved)g(for)g(de)g(enkelte)g(tr\346nings-)463
4001 y(eksempler)i FG(\()r FT(x)944 4019 y FO(d)981 4001
y FY(,)r FT(t)1034 4019 y FO(d)1071 4001 y FG(\))q FY(:)1761
4116 y FL(E)j FG(=)1927 4133 y FE(\345)1956 4208 y FO(d)2033
4116 y FL(E)2084 4134 y FO(d)2120 4116 y FY(.)463 4335
y(Fejlfunktionen)32 b(for)e(et)g(tr\346ningseksempel)h(kan)f(de\002ner)
o(es)f(p\345)h(mange)g(m\345der)-6 b(,)29 b(en)h(ofte)463
4449 y(benyttet)21 b(er)f FL(sum)h(af)g(kvadrater)1352
4678 y(E)1403 4696 y FO(d)1441 4678 y FG(\()r FT(w)r
FG(\))k(=)1702 4622 y FY(1)p 1702 4659 42 4 v 1702 4735
a(2)1768 4695 y FE(\345)1799 4770 y FO(k)1860 4678 y
FG(\()r FL(t)1924 4696 y FO(k)r(d)2007 4678 y FS(\000)18
b FL(y)2134 4696 y FO(k)r(d)2200 4678 y FG(\()r FT(x)2278
4696 y FO(d)2316 4678 y FY(,)e FT(w)r FG(\))r(\))2491
4644 y FP(2)2528 4678 y FY(,)729 b(\(4.5\))463 4932 y(hvor)22
b(man)f(summer)o(er)g(over)h(kvadratet)d(p\345)i(hver)h(komposant)g(i)g
(vektor)o(en)g FT(t)2959 4950 y FO(d)3012 4932 y FS(\000)c
FT(y)3143 4950 y FO(d)3181 4932 y FG(\()r FT(x)3259 4950
y FO(d)3296 4932 y FY(,)e FT(w)r FG(\))463 5047 y FY(for)21
b(et)g(givet)h(tr\346ningseksempel)h FL(d)q FY(;)f FT(y)1713
5065 y FO(d)1751 5047 y FG(\()r FT(x)1829 5065 y FO(d)1866
5047 y FY(,)16 b FT(w)r FG(\))22 b FY(angiver)f(outputtet)h(fra)f
(netv\346rket)g(for)g(in-)463 5162 y(puttet)h FT(x)750
5179 y FO(d)806 5162 y FY(og)g(v\346gtene)f FT(w)q FY(.)1908
5599 y(27)p eop
%%Page: 28 30
28 29 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 463 192 a
Fz(4.4.2)99 b(Beregning)25 b(af)h Fq(r)s Fo(E)h Fz(med)d
(tilbagef\370rselsalgoritmen)463 365 y FY(I)f(et)g
(\003erlagsnetv\346rk)h(kan)f(det)g(sidste)g(lag)h(tr\346nes)f(som)h
(ved)f(en)g(model)h(uden)f(skjulte)h(lag)463 479 y(for)o(di)e(det)h
(for)o(eg\345ende)e(skjulte)i(lag)g(kan)g(betragtes)f(som)i(et)f
(inputlag.)g(Dette)g(g\370r)g(det)f(mu-)463 594 y(ligt)32
b(at)e(ber)o(egne)g(fejlen)h(for)g(hver)g(af)f(neur)o(onerne)h(i)g(det)
f(sidste)i(lag)e(dir)o(ekte)g(idet)h(der)f(er)463 708
y(adgang)e(til)h(b\345de)e(outputv\346r)o(di)h(og)h(\370nsket)g(v\346r)
o(di.)e(Det)i(er)f(straks)g(sv\346r)o(er)o(e)f(at)h(ber)o(egne)463
823 y(fejlene)22 b(i)h(de)e(skjulte)i(lag)g(for)o(di)e(der)h(i)g(dem)g
(ikke)h(er)e(adgang)h(til)h(de)f(\370nskede)g(v\346r)o(dier)-6
b(.)20 b(Her)463 938 y(kommer)i(den)e(algoritme)i(der)e(beskrives)h(i)g
(det)f(f\370lgende)h([Bishop,)g(1995;)e(Mitchell,)h(1997])463
1052 y(ind,)d(idet)g(man)g(ved)g(hj\346lp)h(af)e(den)h(kan)g(pr)o
(opager)o(e)e(et)i(udtryk)g(for)g(fejlen)g(bagl\346ns)g(gennem)463
1167 y(netv\346rket)k(p\345)f(en)h(ef)o(fektiv)e(m\345de.)588
1281 y(Det)28 b(der)f(\370nskes)i(ber)o(egnet)f(til)g(de)g(forskellige)
g(tr\346ningsalgoritmer)-6 b(,)29 b(er)f(de)g(partielle)463
1396 y(a\003edede)19 b(af)h(fejlfunktionen)i(med)f(hensyn)h(til)f(hver)
f(v\346gt)h(i)g(netv\346rket:)1892 1571 y FN(\266)s FL(E)p
1871 1608 139 4 v 1872 1685 a FN(\266)r FL(w)1976 1702
y FO(k)2019 1628 y FY(.)463 1854 y(Bem\346rk)30 b(at)f(vi)h(i)h(det)e
(f\370lgende)h(n\370jes)h(med)f(at)f(betragte)g(den)h(a\003edede)f(af)g
(fejlen)h(for)g(et)463 1969 y(enkelt)e(tr\346ningseksempel)h
FL(d)g FY(\226)e(da)g(fejlen)g(over)h(hele)f(tr\346ningss\346ttet)i
(blot)f(er)f(summen)463 2083 y(af)j(de)g(fejlene)h(for)g(hver)f
(tr\346ningseksempel,)i(kan)f(den)f(partielle)g(a\003edede)g(af)i
FL(E)g FY(\002ndes)463 2198 y(ved)20 b(at)h(summer)o(e)f(de)h
(partielle)f(a\003edede)f(for)i(hvert)g(tr\346ningseksempel)g(som)h(en)
f(f\370lge)g(af)463 2313 y(r)o(eglerne)f(for)g(dif)o(fer)o(entiation:)
1703 2416 y FN(\266)s FL(E)p 1681 2454 V 1682 2530 a
FN(\266)r FL(w)1786 2547 y FO(k)1853 2473 y FG(=)1975
2383 y FO(m)1955 2490 y FE(\345)1943 2564 y FO(d)r FC(=)q
FP(1)2085 2416 y FN(\266)s FL(E)2179 2434 y FO(d)p 2081
2454 V 2082 2530 a FN(\266)r FL(w)2186 2547 y FO(k)463
2691 y FY(hvor)34 b FL(m)g FY(er)e(antallet)h(af)f
(tr\346ningseksempler)-6 b(.)33 b(Algoritmen)h(har)f(som)g(for)o
(uds\346tning)h(at)463 2806 y(aktiveringsfunktionerne)27
b(er)e(dif)o(fer)o(entiable)f(samt)i(at)f(netv\346rket)h(er)f(fr)o
(emadf\370r)o(ende)e(s\345)463 2921 y(de)d(partielle)g(a\003edede)g
(kan)h(udledes)f(ved)g(hj\346lp)h(af)f(k\346der)o(eglen.)588
3035 y(Idet)k(fejlfunktionen)i(ved)e(en)h(neur)o(on)g(afh\346nger)g(af)
f(b\345de)g(aktiveringsv\346r)o(dierne)g(og)463 3150
y(de)18 b(tilknyttede)h(v\346gte)g(i)g(neur)o(onens)g(tilg\345ende)f
(forbindelser)-6 b(,)18 b(jvf.)h(\(4.1\))n(,)f(kan)h(k\346der)o(eglen)
463 3264 y(benyttes)28 b(p\345)g(den)g(partielle)f(a\003edede)g(af)i
FL(E)1909 3282 y FO(d)1973 3264 y FY(med)f(hensyn)h(til)f(v\346gten)h
FL(w)2926 3281 y FO(j)r(i)2996 3264 y FY(mellem)f(den)463
3379 y FL(i)r FY('te)21 b(neur)o(on)g(i)g(det)f(for)o(eg\345ende)f(lag)
i(og)g(neur)o(onen)k FL(j)d FY(i)e(det)h(betragtede)e(lag)1112
3574 y FN(\266)s FL(E)1206 3592 y FO(d)p 1103 3611 149
4 v 1104 3687 a FN(\266)r FL(w)1210 3704 y FO(j)r(i)1285
3630 y FG(=)1385 3574 y FN(\266)s FL(E)1479 3592 y FO(d)p
1384 3611 132 4 v 1397 3687 a FN(\266)s FL(a)1481 3704
y FO(j)1557 3564 y FN(\266)s FL(a)1641 3581 y FO(j)p
1536 3611 149 4 v 1537 3687 a FN(\266)r FL(w)1643 3704
y FO(j)r(i)1718 3630 y FG(=)1818 3574 y FN(\266)s FL(E)1912
3592 y FO(d)p 1817 3611 132 4 v 1830 3687 a FN(\266)s
FL(a)1914 3704 y FO(j)1970 3564 y FN(\266)2026 3497 y
FF(\000)2066 3570 y FK(\345)2132 3530 y FO(n)2132 3589
y(i)r FC(=)q FP(1)2251 3564 y FL(w)2315 3581 y FO(j)r(i)2358
3564 y FL(x)2402 3581 y FO(i)2426 3497 y FF(\001)p 1969
3611 497 4 v 2143 3687 a FN(\266)r FL(w)2249 3704 y FO(j)r(i)2499
3630 y FG(=)2598 3574 y FN(\266)s FL(E)2692 3592 y FO(d)p
2597 3611 132 4 v 2610 3687 a FN(\266)s FL(a)2694 3704
y FO(j)2741 3630 y FL(x)2785 3647 y FO(i)463 3873 y FY(hvor)g
FL(n)f FY(er)f(antallet)g(af)g(neur)o(oner)g(i)h(det)f(for)o
(eg\345ende)e(lag,)20 b FL(x)2341 3889 y FO(i)2380 3873
y FY(betegner)d(den)g FL(i)r FY('te)h(neur)o(ons)g(ak-)463
3987 y(tiveringsv\346r)o(di,)24 b FL(a)1090 4004 y FO(j)1135
3987 y FY(betegner)e(den)27 b FL(j)q FY('te)22 b(neur)o(ons)h
(v\346gtede)f(sum,)h(og)g(andet)f(skridt)g(f\370lger)463
4102 y(af)e(\(4.1\))n(.)588 4216 y(De\002ner)o(es)g(nu)1776
4343 y FN(d)1815 4360 y FO(j)1862 4343 y FG(=)1961 4287
y FN(\266)s FL(E)2055 4305 y FO(d)p 1960 4324 V 1973
4400 a FN(\266)s FL(a)2057 4417 y FO(j)2103 4343 y FY(,)463
4552 y(f\345r)g(man)1763 4622 y FN(\266)s FL(E)1857 4639
y FO(d)p 1754 4659 149 4 v 1755 4735 a FN(\266)r FL(w)1861
4751 y FO(j)r(i)1937 4678 y FG(=)k FN(d)2065 4694 y FO(j)2090
4678 y FL(x)2134 4694 y FO(i)3278 4678 y FY(\(4.6\))463
4887 y(hvoraf)29 b(et)g(gener)o(elt)g(udtryk)g(for)g(den)g(partielle)g
(a\003edede)f(er)g(fundet.)h(Bestemmelsen)h(af)464 5002
y FN(d)503 5018 y FO(j)558 5002 y FY(afh\346nger)i(af)g(hvor)o(dan)f
(netv\346rket)i(er)f(opbygget)h(\226)f(for)g(et)g(simpelt)h(fr)o
(emadf\370r)o(ende)463 5116 y(netv\346rk)e(er)g(der)f(to)h
(tilf\346lde:)g(enten)g(sidder)g(neur)o(on)k FL(j)d FY(i)f(et)g(skjult)
h(lag)f(eller)g(ogs\345)g(i)g(det)463 5231 y(sidste)21
b(lag.)1908 5599 y(28)p eop
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29 30 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 463 192 a
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FY(2)q FG(\()r FL(t)1912 1707 y FO(j)1951 1691 y FS(\000)f
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1747 a FN(\266)r FL(o)2336 1764 y FO(j)1595 1895 y FG(=)25
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FS(\000)q FY(1)q FG(\))1595 2034 y(=)25 b FL(o)1726 2051
y FO(j)1765 2034 y FS(\000)18 b FL(t)1880 2051 y FO(j)588
2229 y FY(Inds\346ttelse)i(i)h(\(4.7\))e(giver)1579 2424
y FN(d)1618 2440 y FO(j)1665 2424 y FG(=)27 b FL(g)1801
2390 y Ft(0)1823 2424 y FG(\()s FL(a)1899 2440 y FO(j)1923
2424 y FG(\))19 b FS(\001)f FG(\()r FL(o)2090 2440 y
FO(j)2130 2424 y FS(\000)f FL(t)2244 2440 y FO(j)2268
2424 y FG(\))q FY(.)956 b(\(4.8\))463 2619 y(Dvs.)21
b(at)g FN(d)q FY(-v\346r)o(dien)f(kan)h(vha.)f(af)g(dif)o(fer)o(ensen)f
(mellem)i(outputtet)h FL(o)2691 2635 y FO(j)2734 2619
y FY(og)f(m\345lv\346r)o(dien)g FL(t)3343 2635 y FO(j)3365
2619 y FY(.)463 2859 y FT(Skjult)e(neuron)463 3032 y
FY(M\345den)24 b(hvorp\345)h(den)g(v\346gtede)f(sum)i(i)f(en)g(skjult)h
(neur)o(on,)h FL(a)2464 3049 y FO(j)2487 3032 y FY(,)e(kan)g
(p\345virke)f(r)o(esultatet)g(af)463 3147 y(netv\346rket)17
b(og)i(dermed)d(\346ndr)o(e)h(fejlfunktionen,)h(er)g(igennem)g(de)f
(neur)o(oner)g(hvis)i(indgang)463 3262 y(afh\346nger)f(af)i
FL(a)960 3278 y FO(j)982 3262 y FY(,)f(dvs.)f(de)g(neur)o(oner)h
FL(k)h FY(som)f(er)g(dir)o(ekte)e(forbundet)h(med)h(neur)o(onen)j
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(summer)i(betegnes)f(med)i FL(a)2273 3394 y FO(k)2306
3376 y FY(,)f(kan)f(k\346der)o(eglen)f(benyttes)i(til)f(at)463
3491 y(bestemme)j(fejlen:)1483 3720 y FN(d)1522 3736
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4195 y FG(\))463 4446 y FY(hvor)e FL(o)696 4463 y FO(j)743
4446 y FG(=)26 b FL(g)r FG(\()s FL(a)954 4463 y FO(j)978
4446 y FG(\))19 b FY(angiver)e(aktiveringen)h(af)f(neur)o(on)22
b FL(j)q FY(.)c(For)g(en)g(skjult)h(neur)o(on)f(g\346lder)g(alts\345)
1587 4641 y FN(d)1626 4657 y FO(j)1672 4641 y FG(=)27
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y FO(k)2198 4641 y FL(w)2260 4659 y FO(k)t(j)3278 4641
y FY(\(4.9\))463 4891 y(Som)25 b(det)f(ses,)h(kan)h FN(d)1149
4908 y FO(j)1196 4891 y FY(udtrykkes)f(p\345)f(en)h(simpel)h(m\345de)e
(ved)h FN(d)2526 4909 y FO(k)2559 4891 y FY(,)g(og)g(s\345fr)o(emt)f
(neur)o(onen)i FL(k)463 5006 y FY(ikke)d(er)f(en)g(neur)o(on)h(i)f(det)
g(sidste)h(lag,)f(kan)h(pr)o(ocedur)o(en)d(gentages)j(for)f(det)g
(n\346ste)h(lag)f(osv)-9 b(.)463 5120 y(til)21 b(man)g(n\345r)f(en)h
(neur)o(on)g(i)g(det)f(sidste)h(lag)g(hvor)h FN(d)g FY(kan)e(ber)o
(egnes)g(dir)o(ekte.)588 5235 y(I)15 b(praksis)h(starter)f(man)h
(naturligvis)g(bagerst)f(i)h(netv\346rket)g(med)f(at)h(pr)o(opager)o(e)
e FN(d)q FY(-v\346r)o(dierne)463 5349 y(tilbage)20 b(gennem)i(neur)o
(onerne,)e(hvoraf)g(navnet)h(p\345)f(algoritmen:)1908
5599 y(29)p eop
%%Page: 30 32
30 31 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 567 192 a(1.)41
b(For)21 b(hver)f(neur)o(on)h(i)g(det)f(sidste)h(lag)g(bestem)f(fejlen)
i FN(d)2376 209 y FO(k)2430 192 y FY(ved)e(hj\346lp)i(af)e(\(4.7\))n(.)
567 373 y(2.)41 b(For)23 b(hver)f(neur)o(on)h(i)h(det)e(for)o
(eg\345ende)f(skjulte)j(lag)f(\(lag)g FL(n)c FS(\000)f
FY(1\))k(ber)o(egn)g(fejlen)i FN(d)3254 389 y FO(j)3300
373 y FY(ved)671 487 y(hj\346lp)d(af)f(\(4.9\))n(.)h(Bem\346rk)f(at)g
(formlen)h(benytter)h FN(d)2280 505 y FO(k)2313 487 y
FY(-v\346r)o(dierne)d(fra)h(lag)i FL(n)q FY(.)567 668
y(3.)41 b(Gentag)26 b(det)g(for)o(eg\345ende)f(skridt)h(med)i
FL(n)f FY(\351n)g(mindr)o(e)f(indtil)h(alle)f(de)g(skjulte)h(lag)f(er)
671 783 y(gennemg\345et.)463 964 y(Den)21 b(partielle)f(a\003edede)g(i)
h(hver)f(v\346gt)h(kan)g(der)o(efter)d(\002ndes)j(ved)f(\(4.6\))n(.)463
1205 y FT(T)-5 b(idskompleksitet)463 1378 y FY(T)g
(ilbagef\370rselsalgoritmen)19 b(har)f(vist)h(sig)g(i)g(praksis)f(at)g
(v\346r)o(e)g(meget)g(anvendelig)g(pga.)g(tids-)463 1492
y(kompleksiten)26 b(af)f(algoritmen)g([Bishop,)g(1995].)e(T)-5
b(idskompleksiteten)26 b(for)f(en)g(ber)o(egning)463
1607 y(af)c FL(E)605 1625 y FO(d)661 1607 y FY(som)g(funktion)g(af)f
FT(w)h FY(er)d FL(O)r FG(\()r FL(n)r FG(\))j FY(hvor)g
FL(n)g FY(er)f(antallet)f(af)g(v\346gte,)h(for)o(di)f(den)g(tunge)h
(del)g(af)463 1722 y(ber)o(egningen)k(ligger)h(i)f(fr)o(emadpr)o
(opageringen)e(af)i(v\346r)o(dier)f(hvor)i(der)e(er)h(\351n)h
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b(.)20 b(v\346gt.)588 1951 y(Uden)d(tilbagef\370rselsalgoritmen)h(kan)g
(bestemmelsen)h(af)e(de)h(partielle)f(a\003edede)g(af)i
FL(E)3401 1968 y FO(d)463 2065 y FY(g\370r)o(es)h(numerisk)i(for)e
(hver)h(enkelt)g(v\346gt)g(ved)f(hj\346lp)h(af)f(dif)o(fer)o
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2309 y FS(\031)1688 2242 y FL(E)1739 2260 y FO(d)1776
2242 y FG(\()r FL(w)1874 2259 y FO(j)r(i)1933 2242 y
FG(+)e FN(e)r FG(\))g FS(\000)h FL(E)2242 2260 y FO(d)2279
2242 y FG(\()r FL(w)2377 2259 y FO(j)r(i)2419 2242 y
FG(\))p 1686 2289 767 4 v 2050 2365 a FN(e)463 2553 y
FY(hvor)31 b FN(e)g FY(er)f(en)g(lille)h(v\346r)o(di.)e(For)h(at)g(ber)
o(egne)h FL(E)2019 2571 y FO(d)2056 2553 y FG(\()r FL(w)2154
2569 y FO(j)r(i)2216 2553 y FG(+)21 b FN(e)r FG(\))32
b FY(kr\346ves)d(en)i(ekstra)e(fr)o(emadf\370-)463 2667
y(ring)22 b(for)f(hver)g(v\346gt,)g(dvs.)g(tidskompleksiteten)i(pr)-6
b(.)20 b(v\346gt)i(er)f(af)f(st\370rr)o(elsesor)o(denen)g
FL(O)r FG(\()r FL(n)r FG(\))463 2782 y FY(hvilket)h(giver)f
FL(O)r FG(\()r FL(n)1098 2752 y FP(2)1135 2782 y FG(\))i
FY(i)f(alt.)588 2897 y(T)-5 b(ilbagef\370rselsalgoritmen)22
b(derimod)g(kan)g(br)o(uge)g(delr)o(esultater)e(i)j(form)f(af)g
FN(d)q FY('erne)g(be-)463 3011 y(r)o(egnet)h(undervejs)i(til)f(at)g
(bestemme)g(de)g(partielle)f(a\003edede)g(ved)g(\(4.6\))n(.)h(Dette)g
(kr\346ver)g(en)463 3126 y(enkelt)29 b(gangeoperation)f(og)h(en)f
(addition)g(pr)-6 b(.)28 b(v\346gt)g(s\345)g(tidskompleksiteten)i(af)d
(algorit-)463 3240 y(men)22 b(samlet)g(bliver)e FL(O)r
FG(\()q FY(2)q FL(n)r FG(\))j FY(\(en)f(fr)o(emadf\370ring)e(og)i(en)g
(tilbagef\370ring\))f(i)h(stedet)g(for)e FL(O)r FG(\()r
FL(n)3346 3210 y FP(2)3383 3240 y FG(\))q FY(.)463 3355
y(Da)25 b(antallet)f(af)g(v\346gte)h(som)g(oftest)g(er)g(meget)g
(stort,)g(f.eks.)f(1.000-500.000,)19 b(giver)25 b(tilbage-)463
3469 y(f\370rselsalgoritmen)20 b(derfor)e(en)h(meget)h(stor)f
(hastighedsforbedring.)g(Desuden)h(er)f(der)f(den)463
3584 y(for)o(del)i(at)g(ber)o(egningen)h(ikke)g(bygger)f(p\345)h(en)f
(appr)o(oksimation.)463 3844 y Fz(4.4.3)99 b(Gradientnedstigning)463
4017 y FY(Dif)o(fer)o(entier)o(es)20 b(fejlfunktionen)k(med)e(hensyn)i
(til)f(samtlige)g(v\346gte)f(f\345s)g(gradientvektor)o(en)463
4132 y(som)f(peger)e(i)h(den)g(r)o(etning)g(hvor)g(fejlfunktionen)i
(stiger)e(mest,)g(s\345)f(ved)h(at)f(g\345)h(i)g(modsat)g(r)o(et-)463
4246 y(ning)g(af)e(gradienten)h(minimer)o(es)j FL(E)q
FY(.)d(Hver)f(v\346gt)h(kan)h(opdater)o(es)d(ved)i(hj\346lp)h(af)e
(udtrykket)1671 4478 y FK(D)r FL(w)1793 4495 y FO(j)r(i)1858
4478 y FG(=)25 b FS(\000)r FN(h)2081 4422 y(\266)s FL(E)2175
4440 y FO(d)p 2071 4459 149 4 v 2072 4535 a FN(\266)r
FL(w)2178 4552 y FO(j)r(i)3236 4478 y FY(\(4.10\))463
4720 y(hvor)i FK(D)r FL(w)785 4737 y FO(j)r(i)853 4720
y FY(er)f(den)g(v\346r)o(di)f(der)g(skal)h(l\346gges)h(til)f(v\346gten)
h FL(w)2425 4737 y FO(j)r(i)2467 4720 y FY(,)e(og)i
(tr\346ningskonstanten)i FN(h)463 4835 y FY(fasts\346tter)20
b(skridtst\370rr)o(elsen.)588 4949 y(V\346r)o(dien)26
b(af)i FN(h)k FY(kan)27 b(dog)h(v\346r)o(e)f(lidt)h(sv\346r)f(at)g
(fastl\346gge.)g(Holdes)i FN(h)j FY(konstant)c(under)463
5064 y(hele)20 b(optr\346ningen,)g(kan)f(tr\346ning)i(med)e(en)g(lille)
h(v\346r)o(di)f(af)h FN(h)j FY(v\346r)o(e)18 b(meget)i(langsom)g(om)g
(at)463 5179 y(konver)o(ger)o(e)g(mod)h(et)f(minimum)j(som)e(det)g(ses)
f(p\345)h(\002gur)g(4.10)e(p\345)h(n\346ste)h(side.)1908
5599 y(30)p eop
%%Page: 31 33
31 32 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 612 1982 a
@beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3212
@rwi @setspecial
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5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
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4 0 V
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4 0 V
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4 0 V
5 0 V
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4 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
currentpoint stroke M
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
1.000 UL
LT1
6311 4599 M
(0.01) Rshow
6395 4599 M
399 0 V
687 4872 M
4 -350 V
4 -341 V
5 -269 V
5 -210 V
4 -164 V
5 -130 V
5 -106 V
4 -89 V
5 -77 V
5 -70 V
4 -64 V
5 -60 V
5 -57 V
4 -55 V
5 -52 V
5 -49 V
4 -48 V
5 -44 V
5 -41 V
4 -39 V
5 -36 V
5 -32 V
4 -30 V
5 -28 V
5 -25 V
4 -23 V
5 -21 V
5 -19 V
4 -17 V
5 -16 V
5 -15 V
4 -13 V
5 -13 V
5 -11 V
4 -11 V
5 -10 V
5 -9 V
4 -8 V
5 -8 V
5 -8 V
4 -7 V
5 -6 V
5 -6 V
4 -6 V
5 -5 V
5 -5 V
4 -5 V
5 -4 V
5 -5 V
4 -4 V
5 -3 V
5 -4 V
5 -4 V
4 -3 V
5 -3 V
5 -3 V
4 -3 V
5 -3 V
5 -2 V
4 -3 V
5 -2 V
5 -3 V
4 -2 V
5 -2 V
5 -2 V
4 -2 V
5 -2 V
5 -2 V
4 -2 V
5 -2 V
5 -2 V
4 -1 V
5 -2 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
currentpoint stroke M
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
currentpoint stroke M
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
currentpoint stroke M
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
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4 0 V
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5 0 V
5 0 V
4 0 V
5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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5 -1 V
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4 0 V
5 -1 V
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4 0 V
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4 0 V
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5 -1 V
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4 0 V
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4 -1 V
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4 -1 V
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5 -1 V
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5 0 V
4 0 V
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5 0 V
4 0 V
5 0 V
1.000 UL
LT2
6311 4459 M
(0.001) Rshow
6395 4459 M
399 0 V
945 4872 M
3 -27 V
4 -44 V
5 -43 V
5 -43 V
4 -41 V
5 -41 V
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5 -37 V
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4 -3 V
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4 -2 V
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5 -2 V
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5 -2 V
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5 -3 V
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4 -1 V
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4 -2 V
5 -1 V
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4 -2 V
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5 -2 V
4 -1 V
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4 -1 V
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5 -2 V
4 -1 V
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5 -2 V
4 -1 V
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4 -1 V
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5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
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4 -2 V
5 -1 V
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4 -2 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
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4 -1 V
5 -1 V
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4 -1 V
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4 -1 V
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4 -1 V
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5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
currentpoint stroke M
4 0 V
5 -1 V
5 -1 V
4 -1 V
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5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
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5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
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4 -1 V
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5 -1 V
4 0 V
5 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
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4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
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5 -1 V
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4 0 V
5 -1 V
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4 -1 V
5 0 V
5 0 V
4 -1 V
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5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
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5 -1 V
4 0 V
5 -1 V
5 0 V
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5 -1 V
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5 -1 V
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4 -1 V
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4 0 V
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4 -1 V
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5 -1 V
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4 0 V
5 -1 V
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4 0 V
5 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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4 -1 V
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4 -1 V
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4 -1 V
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5 -1 V
4 0 V
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4 0 V
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5 -1 V
4 0 V
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5 -1 V
4 0 V
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4 -1 V
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4 -1 V
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4 -1 V
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5 -1 V
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4 0 V
5 -1 V
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5 -1 V
4 0 V
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4 -1 V
5 0 V
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4 0 V
5 -1 V
5 0 V
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5 0 V
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4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
currentpoint stroke M
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
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4 0 V
5 0 V
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4 0 V
5 -1 V
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4 0 V
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5 -1 V
4 0 V
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5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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4 0 V
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5 -1 V
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4 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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4 0 V
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4 0 V
5 0 V
5 -1 V
4 0 V
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5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
currentpoint stroke M
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
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4 0 V
5 -1 V
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5 0 V
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4 -1 V
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5 0 V
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5 0 V
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5 0 V
5 0 V
4 -1 V
5 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
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4 0 V
5 0 V
5 0 V
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5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
stroke
grestore
end
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%%EndDocument
 @endspecial 534 2170 a FH(Figur)26 b(4.10:)i(Fejlen)e(som)i(funktion)e
(af)h(antallet)f(af)h(iterationer)g(med)h(forskellige)f(v\346rdier)h
(af)g FA(h)i FH(for)d(et)534 2275 y(netv\346rk)f(med)h(\351t)g(skjult)e
(lag)i(med)g(to)g(neur)o(oner)e(og)i(\351n)f(input-)g(og)h(\351n)f
(outputneur)o(on;)e(bem\346rk)i(at)h(en)534 2380 y(iteration)16
b(svar)o(er)h(til)g(\351n)f(komplet)i(genneml\370bning)d(af)i
(tr\346ningss\346ttet.)e(Netv\346rket)i(fors\370ger)g(at)g(appr)o(ok-)
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%%Page: 32 34
32 33 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
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5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
currentpoint stroke M
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
currentpoint stroke M
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
currentpoint stroke M
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
1.000 UL
LT2
6311 4599 M
(0.5) Rshow
6395 4599 M
399 0 V
663 4629 M
4 -489 V
5 -526 V
5 -734 V
4 -238 V
5 -154 V
5 -165 V
4 -139 V
5 -89 V
5 -52 V
4 -26 V
5 -10 V
5 -1 V
4 -1 V
5 -5 V
5 -11 V
4 -16 V
5 -22 V
5 -25 V
4 -26 V
5 -25 V
5 -23 V
4 -19 V
5 -14 V
5 -10 V
4 -4 V
5 0 V
5 4 V
4 7 V
5 10 V
5 11 V
4 12 V
5 12 V
5 12 V
4 10 V
5 9 V
5 7 V
4 6 V
5 4 V
5 2 V
4 1 V
5 -1 V
5 -1 V
4 -4 V
5 -4 V
5 -5 V
4 -6 V
5 -6 V
5 -7 V
4 -8 V
5 -7 V
5 -9 V
4 -8 V
5 -8 V
5 -9 V
4 -9 V
5 -9 V
5 -8 V
5 -9 V
4 -9 V
5 -8 V
5 -9 V
4 -8 V
5 -8 V
5 -8 V
4 -8 V
5 -8 V
5 -8 V
4 -7 V
5 -8 V
5 -7 V
4 -7 V
5 -6 V
5 -7 V
4 -7 V
5 -6 V
5 -6 V
4 -6 V
5 -6 V
5 -6 V
4 -5 V
5 -6 V
5 -5 V
4 -5 V
5 -5 V
5 -5 V
4 -5 V
5 -5 V
5 -4 V
4 -5 V
5 -4 V
5 -5 V
4 -4 V
5 -4 V
5 -4 V
4 -4 V
5 -4 V
5 -4 V
4 -4 V
5 -4 V
5 -3 V
4 -4 V
5 -4 V
5 -3 V
4 -4 V
5 -3 V
5 -3 V
4 -4 V
5 -3 V
5 -3 V
4 -3 V
5 -3 V
5 -3 V
4 -2 V
5 -3 V
5 -3 V
4 -2 V
5 -2 V
5 -2 V
4 -2 V
5 -2 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 -1 V
4 1 V
5 -3 V
5 6 V
4 -15 V
5 39 V
5 -76 V
4 187 V
5 -44 V
5 -93 V
4 -45 V
5 142 V
5 -104 V
4 -8 V
5 25 V
5 -33 V
5 99 V
4 -106 V
5 86 V
5 -104 V
4 176 V
5 -126 V
5 -39 V
4 123 V
5 -126 V
5 138 V
4 -135 V
5 119 V
5 -132 V
4 185 V
5 -165 V
5 86 V
4 -114 V
5 242 V
5 -205 V
4 45 V
5 -63 V
5 183 V
4 -182 V
5 123 V
5 -144 V
4 237 V
5 -218 V
5 134 V
4 -155 V
5 245 V
5 -232 V
4 170 V
5 -182 V
5 238 V
4 -233 V
5 207 V
5 -212 V
4 236 V
5 -236 V
5 232 V
4 -234 V
5 242 V
5 -242 V
4 245 V
5 -246 V
5 250 V
4 -251 V
5 254 V
5 -254 V
4 258 V
5 -259 V
5 262 V
4 -262 V
5 264 V
5 -265 V
4 268 V
5 -268 V
5 270 V
4 -270 V
5 272 V
5 -272 V
4 274 V
5 -275 V
5 277 V
4 -277 V
5 278 V
5 -278 V
4 279 V
5 -279 V
5 280 V
4 -279 V
5 280 V
5 -280 V
4 281 V
5 -281 V
5 281 V
4 -281 V
5 281 V
5 -280 V
4 280 V
5 -280 V
5 280 V
4 -280 V
5 280 V
5 -279 V
4 279 V
5 -279 V
5 278 V
4 -277 V
5 276 V
5 -276 V
4 275 V
5 -274 V
5 273 V
4 -272 V
5 271 V
5 -270 V
4 268 V
5 -268 V
5 266 V
4 -265 V
5 264 V
5 -263 V
4 261 V
5 -260 V
5 257 V
4 -256 V
5 254 V
5 -253 V
4 251 V
5 -250 V
5 247 V
5 -245 V
4 242 V
5 -241 V
5 238 V
4 -237 V
5 234 V
5 -232 V
4 229 V
5 -228 V
5 224 V
4 -222 V
5 219 V
5 -218 V
4 214 V
5 -212 V
5 208 V
4 -206 V
5 202 V
5 -200 V
4 196 V
5 -194 V
5 190 V
4 -188 V
5 183 V
5 -181 V
4 176 V
5 -173 V
5 169 V
4 -166 V
5 161 V
5 -159 V
4 153 V
5 -150 V
5 145 V
4 -141 V
5 136 V
5 -133 V
4 127 V
5 -123 V
5 117 V
4 -113 V
5 108 V
5 -104 V
4 98 V
5 -94 V
5 88 V
4 -83 V
5 77 V
5 -72 V
4 67 V
5 -62 V
5 56 V
4 -52 V
5 47 V
5 -42 V
4 38 V
5 -33 V
5 29 V
4 -25 V
5 22 V
5 -18 V
4 16 V
5 -13 V
5 11 V
4 -8 V
5 7 V
5 -5 V
4 5 V
5 -3 V
5 3 V
4 -2 V
5 2 V
5 -1 V
4 2 V
5 -1 V
5 1 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
currentpoint stroke M
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -2 V
5 -2 V
5 -3 V
4 -4 V
5 -5 V
5 -6 V
4 -10 V
5 -13 V
5 -18 V
4 -24 V
5 -31 V
5 -36 V
4 -37 V
5 -39 V
5 -38 V
4 -39 V
5 -34 V
5 -44 V
4 -12 V
5 -91 V
5 106 V
4 -303 V
5 101 V
5 88 V
4 -186 V
5 144 V
5 -175 V
5 118 V
4 -119 V
5 71 V
5 -76 V
4 38 V
5 -46 V
5 17 V
4 -27 V
5 6 V
5 -17 V
4 1 V
5 -11 V
5 -1 V
4 -10 V
5 -1 V
5 -8 V
4 -2 V
5 -7 V
5 -1 V
4 -7 V
5 -1 V
5 -7 V
4 1 V
5 -10 V
5 5 V
4 -14 V
5 13 V
5 -27 V
4 34 V
5 -60 V
5 88 V
4 -148 V
5 204 V
5 -280 V
4 193 V
5 -92 V
5 152 V
4 -246 V
5 239 V
5 -240 V
4 245 V
5 -254 V
5 232 V
4 -213 V
5 260 V
5 -301 V
4 132 V
5 110 V
5 -190 V
4 260 V
5 -320 V
5 69 V
4 246 V
5 -315 V
5 118 V
4 155 V
5 -245 V
5 255 V
4 -267 V
5 235 V
5 -199 V
4 259 V
5 -313 V
5 150 V
4 76 V
currentpoint stroke M
5 -126 V
5 197 V
4 -288 V
5 215 V
5 -125 V
4 197 V
5 -285 V
5 227 V
4 -158 V
5 227 V
5 -302 V
4 211 V
5 -98 V
5 157 V
4 -237 V
5 259 V
5 -281 V
4 244 V
5 -200 V
5 246 V
4 -292 V
5 239 V
5 -179 V
4 228 V
5 -281 V
5 248 V
4 -211 V
5 243 V
5 -275 V
4 251 V
5 -223 V
5 245 V
4 -265 V
5 251 V
5 -235 V
4 246 V
5 -255 V
5 250 V
4 -242 V
5 245 V
5 -246 V
4 245 V
5 -242 V
5 243 V
4 -241 V
5 241 V
5 -239 V
5 239 V
4 -236 V
5 236 V
5 -234 V
4 234 V
5 -232 V
5 231 V
4 -228 V
5 228 V
5 -226 V
4 226 V
5 -224 V
5 224 V
4 -221 V
5 220 V
5 -217 V
4 216 V
5 -214 V
5 214 V
4 -211 V
5 210 V
5 -207 V
4 206 V
5 -203 V
5 202 V
4 -199 V
5 198 V
5 -195 V
4 194 V
5 -191 V
5 189 V
4 -185 V
5 184 V
5 -181 V
4 179 V
5 -175 V
5 173 V
4 -169 V
5 167 V
5 -163 V
4 161 V
5 -157 V
5 155 V
4 -151 V
5 149 V
5 -145 V
4 142 V
5 -137 V
5 134 V
4 -130 V
5 127 V
5 -122 V
4 119 V
5 -114 V
5 110 V
4 -105 V
5 102 V
5 -97 V
4 93 V
5 -87 V
5 83 V
4 -78 V
5 75 V
5 -70 V
4 66 V
5 -61 V
5 57 V
4 -52 V
5 49 V
5 -44 V
4 40 V
5 -36 V
5 33 V
4 -29 V
5 27 V
5 -23 V
4 21 V
5 -18 V
5 16 V
4 -14 V
5 12 V
5 -9 V
4 8 V
5 -7 V
5 7 V
4 -5 V
5 4 V
5 -3 V
4 3 V
5 -2 V
5 3 V
4 -2 V
5 2 V
5 -1 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
5 -2 V
5 -1 V
4 -2 V
currentpoint stroke M
5 -1 V
5 -2 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -2 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
1.000 UL
LT0
6311 4459 M
(0.9) Rshow
6395 4459 M
399 0 V
673 4872 M
4 -2206 V
4 156 V
5 582 V
5 -697 V
4 -471 V
5 -280 V
5 -93 V
4 107 V
5 164 V
5 113 V
4 30 V
5 -49 V
5 -97 V
4 -106 V
5 -72 V
5 -22 V
4 17 V
5 63 V
5 107 V
4 135 V
5 143 V
5 139 V
4 129 V
5 114 V
5 94 V
4 74 V
5 54 V
5 34 V
4 17 V
5 2 V
5 -9 V
4 -18 V
5 -25 V
5 -29 V
4 -32 V
5 -33 V
5 -35 V
4 -35 V
5 -35 V
5 -35 V
4 -33 V
5 -33 V
5 -32 V
4 -31 V
5 -29 V
5 -29 V
4 -27 V
5 -27 V
5 -25 V
4 -24 V
5 -23 V
5 -22 V
4 -21 V
5 -20 V
5 -20 V
5 -18 V
4 -17 V
5 -16 V
5 -16 V
4 -14 V
5 -14 V
5 -13 V
4 -12 V
5 -11 V
5 -10 V
4 -10 V
5 -9 V
5 -7 V
4 -7 V
5 -7 V
5 -5 V
4 -5 V
5 -4 V
5 -3 V
4 -3 V
5 -2 V
5 -1 V
4 -1 V
5 -1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -2 V
4 -3 V
5 -2 V
5 -4 V
4 -3 V
5 -4 V
5 -4 V
4 -5 V
5 -5 V
5 -5 V
4 -5 V
5 -6 V
5 -6 V
4 -6 V
5 -6 V
5 -7 V
4 -7 V
5 -7 V
5 -7 V
4 -7 V
5 -8 V
5 -8 V
4 -7 V
5 -8 V
5 -9 V
4 -8 V
5 -8 V
5 -9 V
4 -9 V
5 -9 V
5 -9 V
4 -9 V
5 -9 V
5 -10 V
4 -9 V
5 -10 V
5 -10 V
4 -11 V
5 -10 V
5 -11 V
4 -12 V
5 -12 V
5 -12 V
4 -14 V
5 -14 V
5 -16 V
4 -17 V
5 -20 V
5 -25 V
4 -31 V
5 -43 V
5 -67 V
4 -84 V
5 269 V
5 -239 V
4 63 V
5 117 V
5 -203 V
4 223 V
5 -186 V
5 -29 V
4 372 V
5 -22 V
5 11 V
4 -16 V
5 -20 V
5 -24 V
4 -29 V
5 -34 V
5 -41 V
4 -56 V
5 -83 V
5 -89 V
5 383 V
4 -277 V
5 -106 V
5 320 V
4 -252 V
5 -52 V
5 414 V
4 -231 V
5 -82 V
5 -109 V
4 210 V
5 -215 V
5 386 V
4 -236 V
5 -113 V
5 16 V
4 289 V
5 -224 V
5 -122 V
4 258 V
5 -241 V
5 199 V
4 -217 V
5 379 V
5 -244 V
4 -131 V
5 220 V
5 -225 V
4 371 V
5 -265 V
5 -120 V
4 399 V
5 -289 V
5 -108 V
4 428 V
5 -316 V
5 -114 V
4 426 V
5 -331 V
5 -80 V
4 425 V
5 -345 V
5 -78 V
4 419 V
5 -354 V
5 -43 V
4 363 V
5 -340 V
5 26 V
4 195 V
5 -275 V
5 331 V
4 -319 V
5 241 V
5 -257 V
4 398 V
5 -372 V
5 179 V
4 -178 V
5 379 V
5 -382 V
4 185 V
5 -186 V
5 381 V
4 -388 V
5 230 V
5 -243 V
4 406 V
5 -403 V
5 285 V
4 -296 V
5 395 V
5 -393 V
4 343 V
5 -347 V
5 382 V
4 -382 V
5 376 V
5 -377 V
4 386 V
5 -386 V
5 389 V
4 -389 V
5 392 V
5 -391 V
4 394 V
5 -393 V
5 394 V
4 -393 V
5 394 V
5 -393 V
4 393 V
5 -392 V
5 391 V
4 -389 V
5 387 V
5 -385 V
4 383 V
5 -381 V
5 377 V
4 -374 V
5 370 V
5 -367 V
4 361 V
5 -358 V
5 352 V
4 -348 V
5 341 V
5 -337 V
4 329 V
5 -325 V
5 316 V
5 -311 V
4 301 V
5 -296 V
5 285 V
4 -280 V
5 269 V
5 -264 V
4 252 V
5 -246 V
5 235 V
4 -229 V
5 217 V
5 -210 V
4 197 V
5 -191 V
5 179 V
4 -172 V
5 160 V
5 -152 V
4 140 V
5 -133 V
5 122 V
4 -114 V
5 103 V
5 -95 V
4 85 V
5 -77 V
5 67 V
4 -59 V
5 51 V
5 -44 V
4 37 V
5 -31 V
5 26 V
4 -20 V
5 16 V
5 -12 V
4 10 V
5 -7 V
5 5 V
4 -3 V
5 3 V
5 -2 V
4 2 V
5 -1 V
5 1 V
4 0 V
5 1 V
5 0 V
4 1 V
5 0 V
5 0 V
4 1 V
5 0 V
5 1 V
4 0 V
5 0 V
5 1 V
4 0 V
5 1 V
5 0 V
4 1 V
5 0 V
5 1 V
4 0 V
5 1 V
5 0 V
4 0 V
5 1 V
5 0 V
4 1 V
5 0 V
5 1 V
4 0 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 0 V
5 1 V
4 0 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 0 V
5 1 V
4 1 V
5 0 V
5 1 V
4 0 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 0 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
currentpoint stroke M
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 1 V
4 0 V
5 1 V
5 1 V
4 1 V
5 0 V
5 1 V
4 0 V
5 1 V
5 0 V
4 -1 V
5 0 V
5 -2 V
4 -3 V
5 -5 V
5 -7 V
4 -13 V
5 -20 V
5 -30 V
4 -38 V
5 -29 V
5 -23 V
5 -26 V
4 -15 V
5 -64 V
5 142 V
4 -265 V
5 1009 V
5 -210 V
4 -56 V
5 -22 V
5 -19 V
4 -17 V
5 -14 V
5 -12 V
4 -7 V
5 -5 V
5 -1 V
4 1 V
5 2 V
5 1 V
4 -4 V
5 -11 V
5 -22 V
4 -37 V
5 -44 V
5 -41 V
4 -31 V
5 -25 V
5 -21 V
4 -19 V
5 -15 V
5 -13 V
4 -11 V
5 -10 V
5 -7 V
4 -8 V
5 -3 V
5 -7 V
4 3 V
5 -13 V
5 19 V
4 -38 V
5 69 V
5 -118 V
4 200 V
5 -303 V
5 416 V
4 -392 V
5 368 V
5 -312 V
4 258 V
5 -228 V
5 193 V
4 -153 V
5 116 V
5 -81 V
4 54 V
5 -31 V
5 19 V
4 -7 V
5 4 V
5 -1 V
4 1 V
5 0 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
4 -2 V
5 -2 V
5 -2 V
4 -3 V
5 -2 V
5 -3 V
4 -3 V
5 -3 V
5 -3 V
4 -3 V
5 -3 V
5 -4 V
4 -3 V
5 -4 V
5 -3 V
4 -4 V
5 -3 V
5 -4 V
4 -4 V
5 -3 V
5 -4 V
4 -4 V
5 -4 V
5 -3 V
4 -4 V
5 -4 V
5 -4 V
4 -3 V
5 -4 V
5 -4 V
4 -4 V
5 -3 V
5 -4 V
4 -4 V
5 -4 V
5 -4 V
4 -3 V
5 -4 V
5 -4 V
4 -4 V
5 -4 V
5 -4 V
4 -4 V
5 -4 V
5 -5 V
5 -4 V
4 -5 V
5 -5 V
5 -5 V
4 -5 V
5 -6 V
5 -6 V
4 -7 V
5 -8 V
5 -9 V
4 -11 V
5 -12 V
5 -16 V
4 -19 V
5 -21 V
5 -23 V
4 -23 V
5 -20 V
5 -15 V
4 -10 V
5 -6 V
5 -4 V
4 -3 V
5 -2 V
5 -1 V
4 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 1 V
5 0 V
5 1 V
4 1 V
5 1 V
5 1 V
4 2 V
5 1 V
5 2 V
4 1 V
5 2 V
5 2 V
4 2 V
5 2 V
5 2 V
4 3 V
5 2 V
5 2 V
4 2 V
5 2 V
5 2 V
4 3 V
5 2 V
5 2 V
4 1 V
5 2 V
5 2 V
4 1 V
5 2 V
5 1 V
4 1 V
5 1 V
5 0 V
4 1 V
5 0 V
5 0 V
4 -1 V
5 -1 V
5 -2 V
4 -2 V
5 -3 V
5 -3 V
4 -5 V
5 -7 V
5 -8 V
4 -11 V
5 -13 V
5 -18 V
4 -22 V
5 -32 V
5 -54 V
4 -160 V
5 -262 V
5 1869 V
4 -325 V
5 -138 V
5 -765 V
4 -517 V
5 -58 V
5 8 V
4 1 V
5 1 V
5 2 V
4 4 V
5 9 V
5 19 V
4 42 V
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y(N\345r)d FA(h)i FH(s\346ttes)d(h\370jt,)g(opf\370r)o(er)g(algoritmen)
g(sig)g(mer)o(e)g(ustabilt,)e(og)j(netv\346rket)e(ender)h(i)g
(forskellige)g(minima.)567 2543 y FY(2.)41 b(Ber)o(egn)20
b FN(d)q FY(-v\346r)o(dierne)g(ved)g(tilbagef\370rsel.)567
2723 y(3.)41 b(Opdater)19 b(v\346gtene)i(efter)f(\(4.10\))e(og)j
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b(for)g(hele)h(tr\346ningss\346ttet)h(f\370r)e(v\346gtene)h(opdater)o
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19 b(et)h(tr\346ningseksempel.)567 4013 y(2.)41 b(Ber)o(egn)20
b FN(d)q FY(-v\346r)o(dierne)g(ved)g(tilbagef\370rsel.)567
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(l\346g)g(den)g(til)g(den)f(summer)o(ede)g(gradient.)567
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%%Page: 33 35
33 34 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
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(Desuden)g(kan)h(stokastisk)g(nedstigning)h(konver)o(ger)o(e)d
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(tr\346ningss\346t)h(med)e(megen)h(r)o(edundans)f([LeCun)g(et)h(al.,)f
(1998].)f(Dette)h(kan)h(forst\345s)463 421 y(intuitivt)21
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(gradienten)f(kun)463 650 y(tage)22 b(h\370jde)h(for)f(den)g(r)o
(edundante)f(data)g(\351n)h(gang)h(idet)f(r)o(esultatet)f(blot)i
(bliver)f(en)h(l\346nger)o(e)463 765 y(gradient)k(som)h(man)f(s\345)g
(m\345)g(kompenser)o(e)g(for)g(ved)f(en)i(laver)o(e)d(v\346r)o(di)h(af)
i FN(h)j FY(for)26 b(at)h(undg\345)463 879 y(ustabilitet.)19
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(\003er)o(e)g(lokale)g(minima)h(og)g(simpel)g(gra-)463
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(sidde)h(fast)f(i)g(et)h(af)f(disse)g([Mitchell,)h(1997].)463
1981 y(Desuden)f(er)g(skridtst\370rr)o(elsen)g(for)g
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2095 y(tor)o(en)i(for)g(fejlfunktionen)i(s\345)e(p\345)g(stejle)h
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2210 y(dientnedstigningen)f(for)o(eg\345)d(hurtigt,)i(mens)g(den)f
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b FY(som)c(funktion)463 2553 y(af)d(gradientvektor)o(en)h(til)g
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(begge)463 2668 y(ovenst\345ende)16 b(pr)o(oblemer)-6
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(for)o(eg\345ende)f(opdatering.)463 2783 y(V)-9 b(ed)25
b(at)h(lade)g(en)g(v\346gtopdatering)h FK(D)r FL(w)1760
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FN(m)e FY(som)g(bestemmer)f(graden)g(af)f(inerti,)h(skal)g(opfylde)g(0)
k FS(\024)h FN(m)h FS(\024)e FY(1.)463 3496 y(Inertief)o(fekten)18
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(\003ade)h(str\346kninger)463 3725 y(acceler)o(er)o(er)18
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3840 y(Ef)o(fekten)e(af)h(den)g(tilf\370jede)h(inerti)g(skulle)g(med)f
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3954 y(ver)o(gens)25 b(mod)i(et)f(laver)o(e)e(minimum.)j(Hvis)g
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b(Konjugeret)25 b(gradient)463 4846 y FY(Nedstigning)k(efter)e(den)g(r)
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4 0 V
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5 -1 V
4 0 V
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4 0 V
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4 0 V
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4 0 V
5 0 V
currentpoint stroke M
5 0 V
4 0 V
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currentpoint stroke M
4 0 V
5 0 V
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5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
1.000 UL
LT1
6311 4599 M
(0.7) Rshow
6395 4599 M
399 0 V
579 1157 M
4 -27 V
5 -105 V
5 -37 V
5 -28 V
4 -17 V
5 -13 V
5 -11 V
5 -10 V
4 -8 V
5 -7 V
5 -7 V
5 -6 V
4 -5 V
5 -6 V
5 -4 V
4 -4 V
5 -4 V
5 -4 V
5 -3 V
4 -3 V
5 -3 V
5 -3 V
5 -2 V
4 -2 V
5 -2 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
5 -2 V
4 -1 V
5 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 -1 V
5 0 V
4 -1 V
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5 -1 V
4 0 V
5 -1 V
5 0 V
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4 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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5 -1 V
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4 0 V
5 -1 V
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5 0 V
5 -1 V
5 0 V
4 0 V
5 0 V
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5 0 V
4 0 V
currentpoint stroke M
5 0 V
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4 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
5 -1 V
5 -1 V
5 -1 V
4 -1 V
5 -2 V
5 -1 V
5 -2 V
4 -2 V
5 -2 V
5 -3 V
5 -3 V
4 -4 V
5 -4 V
5 -12 V
4 -30 V
5 -41 V
5 7 V
5 113 V
4 96 V
5 35 V
5 52 V
5 -250 V
4 136 V
5 66 V
5 39 V
5 -220 V
4 181 V
5 41 V
5 -114 V
4 -23 V
5 17 V
5 0 V
5 -2 V
4 -1 V
5 -2 V
5 -1 V
5 -1 V
4 -1 V
5 -1 V
5 -1 V
5 -1 V
4 0 V
5 -1 V
5 -1 V
4 0 V
5 0 V
5 -1 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
currentpoint stroke M
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
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5 -1 V
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4 -1 V
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4 0 V
5 0 V
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4 -1 V
5 0 V
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4 0 V
5 0 V
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5 0 V
4 0 V
5 0 V
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4 0 V
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4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
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5 0 V
5 -1 V
5 0 V
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4 0 V
5 -1 V
5 0 V
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4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
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%%Trailer
%%DocumentFonts: Helvetica

%%EndDocument
 @endspecial 534 2539 a FH(Figur)26 b(4.12:)i(Fejlen)d(som)i(funktion)e
(af)h(antallet)g(af)g(iterationer)g(med)h(de)g(angivne)f
(inertikonstanter)-6 b(.)25 b(I)534 2644 y(nogle)18 b(tilf\346lde)g
(\204overskyder\223)j(tr\346ningen)c(minimum)h(og)h(netv\346rket)g
(havner)g(i)f(et)h(andet)g(minimum.)909 4667 y @beginspecial
0 @llx 0 @lly 337 @urx 194 @ury 2498 @rwi @setspecial
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%%Page: 35 37
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%%Page: 39 41
39 40 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 588 192 a(Gener)o(elt)31
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(de)f(br)o(uger)i FT(H)463 306 y FY(eller)e(skaler)o(et)g(konjuger)o
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765 y(og)23 b(af)f(denne)h(gr)o(und)g(kan)g(man)f(ikke)h(v\346r)o(e)f
(sikker)h(p\345)f(at)g(den)h(indbyr)o(des)f(konjuger)o(ethed)463
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994 y FL(n)d FY(iterationer)-6 b(.)20 b(Man)f(bliver)h(alts\345)g
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FL(n)h FY(iterationer)-6 b(,)463 1108 y(og)18 b(dette)e(g\370r)o(es)h
(blot)g(ved)g(at)f(genstarte)h(algoritmen)h(efter)e(hver)i
FL(n)g FY(iterationer)-6 b(.)16 b(Algoritmen)463 1223
y(har)k(dog)h(vist)g(sig)h(at)e(opr)o(etholde)g(sin)i(ef)o(fektivitet)d
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/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
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/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
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/PentE { stroke [] 0 setdash gsave
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  closepath stroke grestore } def
/CircE { stroke [] 0 setdash 
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/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
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  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
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  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
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/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
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  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
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/CircW { stroke [] 0 setdash 
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/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
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1.000 UL
LTb
574 280 M
63 0 V
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490 280 M
(0) Rshow
574 1428 M
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(200) Cshow
2467 280 M
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(400) Cshow
3413 280 M
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(800) Cshow
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(1000) Cshow
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1.000 UL
LTb
574 280 M
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LT0
6311 4739 M
(Skaleret konjugeret gradient) Rshow
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5 -31 V
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currentpoint stroke M
5 0 V
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4 0 V
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4 0 V
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4 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
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5 -1 V
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5 0 V
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4 0 V
5 0 V
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4 0 V
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4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
1.000 UL
LT1
6311 4599 M
(Gradientnedstigning) Rshow
6395 4599 M
399 0 V
579 2473 M
4 -68 V
5 -262 V
5 -94 V
5 -69 V
4 -42 V
5 -33 V
5 -28 V
5 -23 V
4 -21 V
5 -19 V
5 -16 V
5 -15 V
4 -14 V
5 -13 V
5 -11 V
4 -11 V
5 -10 V
5 -9 V
5 -8 V
4 -8 V
5 -7 V
5 -6 V
5 -6 V
4 -5 V
5 -5 V
5 -4 V
4 -4 V
5 -4 V
5 -4 V
5 -3 V
4 -3 V
5 -3 V
5 -3 V
5 -2 V
4 -3 V
5 -2 V
5 -2 V
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5 -2 V
5 -1 V
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5 -1 V
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5 -1 V
5 -1 V
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5 -1 V
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5 -1 V
4 -1 V
5 0 V
5 -1 V
4 0 V
5 -1 V
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5 -1 V
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5 -1 V
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5 -1 V
4 -1 V
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5 -1 V
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4 -1 V
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4 -1 V
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5 -1 V
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4 -1 V
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5 -1 V
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5 -1 V
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4 -1 V
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4 -1 V
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4 -1 V
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5 -1 V
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5 0 V
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5 -1 V
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5 0 V
5 -1 V
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4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 -1 V
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4 0 V
5 -1 V
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4 0 V
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5 -1 V
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4 -1 V
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5 0 V
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4 0 V
5 0 V
5 -1 V
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5 0 V
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4 0 V
5 -1 V
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5 0 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 -1 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 -1 V
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4 0 V
5 0 V
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5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
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5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
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5 0 V
4 -1 V
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5 0 V
4 0 V
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4 0 V
5 0 V
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5 0 V
4 -1 V
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4 0 V
5 0 V
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4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 -1 V
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5 0 V
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5 -1 V
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5 0 V
5 0 V
5 0 V
4 0 V
currentpoint stroke M
5 -1 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
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4 0 V
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4 0 V
5 0 V
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5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 -1 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
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4 0 V
5 0 V
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5 0 V
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4 0 V
5 0 V
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5 0 V
4 0 V
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5 0 V
4 0 V
5 0 V
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4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
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4 0 V
5 0 V
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4 0 V
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5 0 V
4 0 V
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4 0 V
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4 0 V
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4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
4 0 V
5 0 V
5 0 V
5 0 V
4 0 V
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 @endspecial 534 3408 a FH(Figur)23 b(4.15:)28 b(T)-8
b(r\346ningsfejlen)21 b(som)k(funktion)d(af)i(antallet)f(af)h
(iterationer)f(ved)i(brug)e(af)h(skaler)o(et)g(konju-)534
3513 y(ger)o(et)29 b(gradient)g(sammenlignet)g(med)g
(gradientnedstigning)f(med)i(inerti)e(med)i(de)f(bedste)g(v\346rdier)g
(af)534 3618 y(konstanterne)16 b(fra)h(\002gur)g(4.12.)h(Funktionen)c
(som)k(apr)o(oksimer)o(es)f(beskrives)g(i)g(4.4.5)g(p\345)h(den)f
(f\370lgende)g(side)588 3829 y FY(P\345)27 b(\002gur)h(4.15)e(ses)i(en)
g(iterationsm\346ssig)h(sammenligning)g(af)e(gradientnedstigning)463
3944 y(og)19 b(skaler)o(et)e(konjuger)o(et)i(gradient.)f(Skaler)o(et)f
(konjuger)o(et)i(gradient)f(konver)o(ger)o(er)f(her)h(me-)463
4058 y(get)k(hurtiger)o(e)f(end)g(gradientnedstigning,)i(men)f(er)f
(samtidig)h(mer)o(e)f(ber)o(egningsintensiv)463 4173
y(pr)-6 b(.)24 b(iteration.)h(Dette)g(skyldes)h(at)e(der)h(skal)g(ber)o
(egnes)f(to)i(fejlv\346r)o(dier)d(samt)i(to)h(gradienter)463
4287 y(pr)-6 b(.)20 b(iteration.)588 4402 y(Gradienterne)30
b(ber)o(egnes)h(i)g(\370vrigt)h(over)f(hele)g(tr\346ningss\346ttet)h
(som)g(ved)f(den)g(ikke-)463 4516 y(stokastiske)22 b
(gradientnedstigning)g(da)e(det)h(hverken)g(giver)g(mening)h(at)f
(tr\346ne)g(netv\346rket)463 4631 y(helt)26 b(op)h(p\345)e(de)h
(enkelte)g(tr\346ningseksempler)h(hver)e(for)h(sig,)h(eller)e(tr\346ne)
h(med)g(skiftende)463 4746 y(eksempler)20 b(for)g(hvert)g(skridt)g
(idet)f(gr)o(undlaget)h(for)g(algoritmen)h(s\345)f(bryder)f(sammen)h
(med)463 4860 y(de)g(skiftende)h(gradienter)-6 b(.)1908
5599 y(39)p eop
%%Page: 40 42
40 41 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
(tegngenkendelsessystem)p 463 -96 2975 4 v 463 192 a
Fz(4.4.5)99 b(Generalisering,)24 b(overtr\346ning)h(og)h
(termineringskriteriet)463 365 y FY(En)d(af)f(de)g(st\346rke)g(sider)h
(ved)f(at)g(br)o(uge)h(neurale)f(netv\346rk)h(som)g(et)g
(v\346rkt\370j,)g(er)f(at)g(netv\346r)o(-)463 479 y(ket)28
b(kan)g(tr\346nes)g(op)g(til)g(at)g(generaliser)o(e)e(ud)i(over)g(de)f
(pr\346senter)o(ede)f(tr\346ningseksemp-)463 594 y(ler)-6
b(.)25 b(Der)g(er)g(imidlertid)h(to)g(f\346nomener)g(der)e(kan)i
(spoler)o(e)f(generaliseringsevnen)h(\226)f(man)463 708
y(kan)20 b(opn\345)g(en)h(intuitiv)g(fornemmelse)f(for)g(hvad)g(disse)g
(g\370r)h(ved)e(netv\346rk)i(ved)e(at)h(betragte)463
823 y(r)o(egr)o(essionspr)o(oblemer)k(hvor)h(netv\346rket)f(skal)h
(appr)o(oksimer)o(e)e(kontinuerte)j(funktioner)-6 b(.)463
938 y(Som)30 b(eksempel)g(benyttes)f(funktionen)j FL(h)r
FG(\()s FL(x)s FG(\))42 b(=)e FY(0)q(,)q(9)19 b FS(\001)i
FY(sin)r FG(\()s FL(x)s FG(\))g(+)g FN(e)31 b FY(fra)d(\002gur)i(4.10)e
(hvor)j FN(e)463 1052 y FY(r)o(epr\346senter)o(er)18
b(normalfor)o(delt)h(st\370j)i(med)g FN(m)k FG(=)f FY(0)c(og)h
FN(s)k FG(=)f FY(0)q(,)q(07)19 b(og)h(et)g(netv\346rk)g(med)g(\351n)g
(in-)463 1167 y(putneur)o(on,)g(\351n)h(outputneur)o(on)g(og)g(et)f
(varier)o(ende)e(antal)i(neur)o(oner)h(i)f(det)g(skjulte)h(lag,)f(alle)
463 1281 y(med)g(tanh)g(som)h(aktiveringsfunktion,)g(optr\346net)g(med)
f(gradientnedstigning.)g(Konstan-)463 1396 y(ten)d(0,9)f(er)h(valgt)g
(for)o(di)f(den)i(v\346gtede)e(sum)i(til)f(aktiveringsfunktionen)i(i)e
(outputneur)o(onen)463 1510 y(skal)22 b(g\345)g(mod)g(uendelig)g(for)g
(at)g(r)o(esultatet)f(bliver)h(1)f(s\345)h(netv\346rket)g(f\345r)f
(sv\346rt)h(ved)f(at)h(klar)o(e)463 1625 y(appr)o(oksimationen.)588
1739 y(N\345r)c(et)h(neuralt)g(netv\346rk)g(skal)g(optr\346nes)g(til)h
(at)f(kunne)g(generaliser)o(e)f(ud)h(fra)f(de)h(30)f(da-)463
1854 y(tapunkter)g(der)g(gener)o(er)o(es)f(med)j FL(h)q
FY(,)f(er)f(det)g(meningen)i(at)e(netv\346rket)h(skal)g(kunne)g(ignor)o
(er)o(e)463 1969 y(den)h(tillagte)g(st\370j)g(og)h(i)f(stedet)f(appr)o
(oksimer)o(e)1973 1947 y(\210)1966 1969 y FL(h)r FG(\()s
FL(x)s FG(\))26 b(=)e FY(0)q(,)q(9)15 b FS(\001)h FY(sin)r
FG(\()s FL(x)s FG(\))q FY(.)k(Der)f(kan)h(nu)h(g\345)e(to)h(ting)463
2083 y(galt:)i(enten)g(er)g(der)g(for)g(f\345)f(neur)o(onerne)h(og)h
(v\346gte)f(s\345)g(netv\346rket)g(ikke)g(har)g(tilstr\346kkeligt)463
2198 y(med)d FL(frihedsgrader)g FY(til)g(over)o(hovedet)f(at)h(v\346r)o
(e)f(i)h(stand)g(til)g(at)g(r)o(epr\346senter)o(e)2922
2176 y(\210)2915 2198 y FL(h)q FY(,)g(eller)f(ogs\345)h(er)463
2312 y(der)h(for)h(mange)h(frihedsgrader)d(s\345)i(netv\346rket)g(appr)
o(oksimer)o(er)f(st\370jen)i(i)f(dataene)f(i)h(stedet)463
2427 y(for)598 2405 y(\210)591 2427 y FL(h)q FY(.)588
2541 y(Det)16 b(sidste)g(yderpunkt,)g(ben\346vnt)g(overtr\346ning,)h
(er)f(karakteriser)o(et)e(ved)h(at)h(netv\346rket)463
2656 y(kan)27 b(give)g(funktionsv\346r)o(dier)f(der)g(er)h(helt)g(i)g
(over)o(enstemmelse)g(med)f(tr\346ningseksemp-)463 2771
y(lerne)k(hvor)g(de)f(er)g(placer)o(et,)f(men)i(til)h(geng\346ld)f
(ikke)g(er)f(i)h(stand)g(til)g(at)f(\204udj\346vne\223)g(den)463
2885 y(appr)o(oksimer)o(ede)d(funktion)j(mellem)g(datapunkterne)e
(s\345)h(den)g(svar)o(er)e(til)2947 2863 y(\210)2940
2885 y FL(h)q FY(.)i(Modsat)f(r)o(e-)463 3000 y(pr\346senter)o(er)19
b(det)h(f\370rste)g(yderpunkt)g(den)g(situation)i(at)d(netv\346rket)i
(udj\346vner)f(punkterne)463 3114 y(for)27 b(meget)g(s\345)g(den)g
(appr)o(oksimer)o(ede)e(funktion)j(er)f(for)g(j\346vn)g(til)h(at)e
(v\346r)o(e)g(i)i(over)o(enstem-)463 3229 y(melse)21
b(med)889 3207 y(\210)882 3229 y FL(h)g FY([Bishop,)g(1995].)588
3343 y(Et)e(eksempel)g(p\345)f(for)h(f\345)g(frihedsgrader)e(kan)j(ses)
f(p\345)g(\002gur)g(4.16)f(p\345)g(den)h(f\370lgende)g(side.)463
3458 y(Da)d(netv\346rket)g(kun)h(best\345r)e(af)g(to)i(v\346gte)f(i)g
(hvert)g(lag,)g(en)g(fra)f(det)h(forrige)g(lag)g(og)g(en)g(fra)g(bias-)
463 3573 y(neur)o(onen,)k(kan)g(netv\346rket)f(kun)i(skaler)o(e)d
(inputtet)j(og)f(l\346gge)g(en)g(konstant)h(til.)f(Resultatet)463
3687 y(er)g(ikke)g(overraskende)g(at)g(netv\346rkets)g(r)o(esultat)f
(ligner)i(aktiveringsfunktionen)h(tanh)q FG(\()s FL(x)s
FG(\))q FY(,)463 3802 y(blot)f(skaler)o(et)f(med)g(et)h(negativt)g(tal)
f(s\345ledes)g(at)h(den)f(er)h(blevet)f(spejlet)h(om)i
FL(x)r FY(-aksen.)588 3916 y(For)17 b(at)f(forbedr)o(e)f(appr)o
(oksimationen)i(er)f(det)g(n\370dvendigt)h(at)g(give)f(netv\346rket)h
(\003er)o(e)f(fri-)463 4031 y(hedsgrader)f(i)g(form)h(af)f(\003er)o(e)g
(neur)o(oner)h(og)g(dermed)f(v\346gte)g(i)h(det)g(skjulte)g(lag.)g
(Figur)g(4.17)e(p\345)463 4145 y(n\346ste)28 b(side)g(viser)g(hvad)g
(der)f(sker)h(hvis)h(der)e(tilf\370jes)i(for)f(mange)g(frihedsgrader)-6
b(,)26 b(i)j(dette)463 4260 y(tilf\346lde)20 b(10)g(skjulte)i(neur)o
(oner)-6 b(.)20 b(Det)h(ses)f(at)h(netv\346rkets)g(r)o(esultat)e
(g\345r)i(meget)f(pr\346cist)h(gen-)463 4375 y(nem)31
b(n\346sten)g(alle)f(datapunkter)-6 b(,)29 b(men)i(at)f(dette)g(er)g
(en)h(d\345rlig)f(appr)o(oksimation)h(af)e(den)463 4489
y(underliggende)e(funktion)1382 4467 y(\210)1375 4489
y FL(h)q FY(.)g(Netv\346rket)f(har)g(tilstr\346kkeligt)i(med)e
(frihedsgrader)f(til)i(at)463 4604 y(\204huske\223)k(hvert)g(datapunkt)
g(i)h(stedet)f(for)g(at)g(udtr\346kke)g(den)h(\370nskede)f(information)
i(og)463 4718 y(bliver)20 b(dermed)g(meget)h(f\370lsomt)h(over)e(for)h
(st\370j.)588 4833 y(For)30 b(at)f(opn\345)i(den)e(bedste)h(appr)o
(oksimation)g(skal)g(netv\346rket)g(alts\345)f(have)h(for)g(f\345)f
(fri-)463 4947 y(hedsgrader)22 b(til)h(at)g(det)f(kan)i(g\345)e(gennem)
i(alle)f(punkterne,)g(men)g(samtidig)g(nok)h(til)g(at)e(dets)463
5062 y(str)o(uktur)d(er)g(i)g(stand)g(til)g(at)g(r)o(epr\346senter)o(e)
d(den)j(underliggende)g(funktion.)h(At)e(\002nde)h(denne)463
5177 y(balance)k(er)g(netop)h(det)f(vanskelig)h(ved)f(neurale)g
(netv\346rk)h(da)e(det)i(gener)o(elt)f(ikke)g(kan)h(g\370-)463
5291 y(r)o(es)18 b(analytisk,)g(men)h(m\345)g(afg\370r)o(es)e
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3 4 V 4 3 V 3 4 V 3 3 V 3 4 V 4 3 V 3 4 V 3 3 V 3 3 V 4 4 V 3 3 V 3
3 V 3 4 V 4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 4 V 3 3 V 4 3 V 3 3 V 3 3
V 3 2 V 4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 2 V 3 3 V 3 3 V 4 2 V 3 3 V
3 3 V 3 2 V 4 3 V 3 2 V 3 3 V 3 2 V 4 2 V 3 3 V 3 2 V 3 2 V 4 3 V 3
2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2
V 3 2 V 3 1 V 3 2 V 4 2 V 3 2 V 3 1 V 3 2 V 4 1 V 3 2 V 3 1 V 3 2 V
4 1 V 3 2 V 3 1 V 4 1 V 3 2 V 3 1 V 3 1 V 4 1 V 3 1 V 3 2 V 3 1 V 4
1 V 3 1 V 3 1 V 3 1 V 4 0 V 3 1 V 3 1 V 3 1 V 4 1 V 3 0 V 3 1 V 3 1
V 4 0 V 3 1 V 3 0 V 3 1 V 4 0 V 3 1 V 3 0 V 3 0 V 4 1 V 3 0 V 3 0 V
3 1 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V 3
-1 V 3 0 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 -1 V 3 0 V 3 -1 V 4 0 V 3 -1
V 3 0 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 3 -1
V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -2 V 3 -1 V 4 -1 V 3
-1 V 3 -2 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V
3 -2 V 4 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -2
V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -3 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3
-3 V 4 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 4 -3 V
3 -2 V 3 -3 V 3 -2 V 4 -3 V 3 -3 V 3 -3 V 3 -2 V 4 -3 V 3 -3 V 3 -3
V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4
-4 V 3 -3 V 3 -3 V 3 -3 V 4 -4 V 3 -3 V 3 -3 V 3 -4 V 4 -3 V 3 -4 V
3 -3 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 4 -4
V 3 -3 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -3 V 3 -4 V 3
-4 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V
4 -5 V 3 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5
V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3
-4 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V
currentpoint stroke M 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -5 V 4 -5
V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3
-5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -6 V
4 -5 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5
V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3
-5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V
3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6
V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3
-6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V
3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6
V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3
-5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V
3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5
V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4
-5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V
3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5
V 3 -4 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3
-4 V 3 -5 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V
4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4
V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3
-4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -3 V
3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -4 V 4 -3 V 3 -4 V 3 -4 V 3 -3
V 4 -4 V 3 -3 V 3 -4 V 3 -3 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 4 -4 V 3
-3 V 3 -3 V 3 -3 V 4 -4 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V
3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -3 V 4 -2
V 3 -3 V 3 -2 V 3 -3 V 4 -3 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -3 V 3
-2 V 3 -3 V 4 -2 V 3 -2 V 3 -2 V 4 -3 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V
3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -1 V 4 -2 V 3 -2 V 3 -2
V 3 -1 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 4
-2 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V
3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 0 V 3 -1
V 4 -1 V 3 0 V 3 -1 V 3 0 V 4 -1 V 3 0 V 3 -1 V 4 0 V 3 0 V 3 -1 V
3 0 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V
3 0 V 3 0 V 4 1 V 3 0 V 3 0 V 3 0 V 4 1 V 3 0 V 3 0 V 3 1 V 4 0 V 3
1 V 3 0 V 3 1 V 4 1 V 3 0 V 3 1 V 3 1 V 4 1 V 3 0 V 3 1 V 3 1 V 4 1
V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V 4 2 V 3 1 V 3 1 V 4 1 V
3 2 V 3 1 V 3 2 V 4 1 V 3 1 V 3 2 V 3 2 V 4 1 V 3 2 V 3 1 V 3 2 V 4
2 V 3 2 V 3 1 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2
V 4 2 V 3 2 V 3 3 V 3 2 V 4 2 V 3 2 V 3 3 V 3 2 V 4 3 V 3 2 V 3 2 V
3 3 V 4 2 V 3 3 V 3 3 V 3 2 V 4 3 V 3 3 V 3 2 V 3 3 V 4 3 V 3 3 V 3
3 V 3 2 V 4 3 V currentpoint stroke M 3 3 V 3 3 V 4 3 V 3 3 V 3 3 V
3 3 V 4 4 V 3 3 V 3 3 V 3 3 V 4 3 V 3 4 V 3 3 V 3 3 V 4 4 V 3 3 V 3
3 V 3 4 V 4 3 V 3 4 V 3 3 V 3 4 V 4 4 V 3 3 V 3 4 V 3 4 V 4 3 V 3 4
V 3 4 V 3 3 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V
3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 5 V 3 4 V 3 4 V 3 4 V 4
5 V 3 4 V 3 4 V 4 5 V 3 4 V 3 5 V 3 4 V 4 4 V 3 5 V 3 4 V 3 5 V 4 4
V 3 5 V 3 5 V 3 4 V 4 5 V 3 4 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V
4 4 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 4 V 3 5 V 4 5 V 3 5 V 3 5 V 3
5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5
V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 6 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5 V
3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 6 V 3 5 V 3 5 V 3 6 V 4 5 V 3 5 V 3
6 V 3 5 V 4 6 V 3 5 V 3 5 V 3 6 V 4 5 V 3 6 V 3 5 V 3 6 V 4 5 V 3 5
V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V
3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 4 5 V 3
6 V 1 2 V 1.000 UL LT0 350 1177 M 3 3 V 4 2 V 3 3 V 3 2 V 3 3 V 4 2
V 3 3 V 3 3 V 3 2 V 4 3 V 3 3 V 3 3 V 3 2 V 4 3 V 3 3 V 3 3 V 3 3 V
4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 4 V 3 3 V 3 3 V 4 3 V 3 4 V 3 3 V 3
3 V 4 4 V 3 3 V 3 4 V 3 3 V 4 4 V 3 3 V 3 4 V 3 4 V 4 3 V 3 4 V 3 4
V 3 4 V 4 4 V 3 3 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 5 V 3 4 V
3 4 V 4 4 V 3 4 V 3 5 V 3 4 V 4 4 V 3 5 V 3 4 V 3 5 V 4 4 V 3 5 V 3
4 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V 4 4 V 3 5
V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V
3 5 V 3 5 V 3 6 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 6 V 3 5 V 4
5 V 3 5 V 3 5 V 3 6 V 4 5 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 5
V 3 5 V 3 5 V 3 5 V 4 6 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V
4 5 V 3 5 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V 4 5 V 3 4 V 3 5 V 3
5 V 4 4 V 3 5 V 3 4 V 3 5 V 4 4 V 3 5 V 3 4 V 3 4 V 4 5 V 3 4 V 3 4
V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 3 V
3 4 V 4 4 V 3 3 V 3 4 V 3 3 V 4 4 V 3 3 V 3 3 V 3 4 V 4 3 V 3 3 V 3
3 V 3 3 V 4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 3 V 3 2 V 3 3 V 4 3 V 3 2
V 3 3 V 3 2 V 4 3 V 3 2 V 3 3 V 3 2 V 4 2 V 3 2 V 3 3 V 3 2 V 4 2 V
3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 1 V 4 2 V 3 2 V 3 2 V 3 1 V 4
2 V 3 1 V 3 2 V 4 1 V 3 2 V 3 1 V 3 2 V 4 1 V 3 1 V 3 2 V 3 1 V 4 1
V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V
4 1 V 3 1 V 3 0 V 3 1 V 4 1 V 3 0 V 3 1 V 3 0 V 4 1 V 3 1 V 3 0 V 3
1 V 4 0 V 3 0 V 3 1 V 3 0 V 4 0 V 3 1 V 3 0 V 3 0 V 4 0 V 3 1 V 3 0
V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V 3 0 V 4 0 V 3 0 V 3 -1
V 3 0 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 0 V 3 -1 V 3 0 V 3 -1 V 4 0 V 3
-1 V 3 0 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 3 -1 V 4 0 V 3 -1 V 3 -1 V 3
-1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V
3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -2 V 3 -1 V 3 -2
V 3 -2 V 4 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -3 V 4 -2 V 3
-2 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V
3 -2 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 4 -3
V 3 -4 V 3 -3 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V 3 -4 V 4 -4 V 3 -4 V 3
-4 V 3 -3 V 4 -4 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -5 V
4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -4
V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4
-5 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V
3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5
V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3
-5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V currentpoint
stroke M 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3
-5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V
4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6
V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3
-5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V
3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5
V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3
-6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V
3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6
V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3
-6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V
3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6
V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4
-6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V
3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5
V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3
-6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V
4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -4
V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3
-4 V 4 -5 V 3 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -5 V
3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -3 V 3 -4 V 4 -4 V 3 -4 V 3 -3
V 4 -4 V 3 -4 V 3 -3 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 4 -4 V 3
-3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V
3 -3 V 4 -3 V 3 -2 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 4 -2
V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -3 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3
-2 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V
4 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1
V 3 -2 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4
-1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 0 V 4 -1 V 3 -1 V
3 0 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 3 0 V 4 -1 V 3 0 V 3 -1 V 3 0 V 4
-1 V 3 0 V 3 0 V 3 -1 V 4 0 V 3 0 V 3 0 V 3 -1 V 4 0 V 3 0 V 3 0 V
3 0 V 4 0 V 3 -1 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 1 V 3 0 V
3 0 V 4 0 V 3 0 V 3 0 V 3 1 V 4 0 V 3 0 V 3 0 V 3 1 V 4 0 V 3 0 V 3
1 V 3 0 V 4 1 V 3 0 V 3 1 V 3 0 V 4 1 V 3 0 V 3 1 V 3 0 V 4 1 V 3 1
V 3 0 V 3 1 V 4 1 V 3 1 V 3 1 V 3 0 V 4 1 V 3 1 V 3 1 V 3 1 V 4 1 V
3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 4 1 V 3 2 V 3 1 V 3 1 V 4 1 V 3
2 V 3 1 V 3 1 V 4 2 V 3 1 V 3 2 V 3 1 V 4 2 V 3 1 V 3 2 V 3 1 V 4 2
V 3 2 V 3 1 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 1 V 3 2 V 3 2 V 3 2 V
4 3 V 3 2 V 3 2 V 3 2 V 4 2 V 3 3 V 3 2 V 3 2 V 4 3 V 3 2 V 3 2 V 3
3 V 4 3 V 3 2 V 3 3 V 3 2 V 4 3 V 3 3 V 3 3 V 3 3 V 4 2 V currentpoint
stroke M 3 3 V 3 3 V 4 3 V 3 3 V 3 4 V 3 3 V 4 3 V 3 3 V 3 3 V 3 4
V 4 3 V 3 4 V 3 3 V 3 3 V 4 4 V 3 4 V 3 3 V 3 4 V 4 4 V 3 3 V 3 4 V
3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 5 V 3 4 V 3
4 V 3 5 V 4 4 V 3 5 V 3 4 V 3 5 V 4 4 V 3 5 V 3 5 V 3 4 V 4 5 V 3 5
V 3 5 V 3 4 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 4 6 V 3 5 V
3 5 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V 3
5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 6 V 4 5 V 3 6 V 3 6 V 3 5 V 4 6
V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V
4 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 6 V 3
6 V 4 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6
V 4 6 V 3 5 V 3 6 V 3 6 V 4 5 V 3 6 V 3 5 V 3 6 V 4 5 V 3 6 V 3 5 V
3 6 V 4 5 V 3 6 V 3 5 V 3 6 V 4 5 V 3 5 V 3 6 V 3 5 V 4 5 V 3 5 V 3
6 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5
V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 4 4 V 3 5 V 1 2 V stroke grestore end
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@endspecial 3287 1853 a(2)q
FN(p)2653 1816 y FP(3)q FQ(p)p 2653 1830 78 4 v 2676
1877 a FP(2)2017 1845 y FN(p)1379 1810 y FQ(p)p 1378
1824 46 4 v 1385 1871 a FP(2)734 1853 y FY(0)671 221
y(1)609 607 y(0.5)671 995 y(0)582 1382 y(-0.5)644 1771
y(-1)534 2097 y FH(Figur)15 b(4.18:)28 b(Et)16 b(netv\346rk)f(med)i(to)
f(skjulte)f(neur)o(oner)g(er)h(optimalt)g(til)f(at)h(appr)o(oksimer)o
(e)2879 2077 y FU(\210)2873 2097 y FH(h)h(i)f(dette)g(tilf\346lde.)534
2202 y(Bem\346rk)i(hvor)i(n\370je)e(netv\346rket)h(f\370lger)f(den)h
(unnderliggende)f(funktion.)463 2471 y FY(skjulte)k(lag)e(var)g(nok)i
(til)f(appr)o(oksimer)o(e)1793 2449 y(\210)1786 2471
y FL(h)q FY(,)g(se)f(\002gur)h(4.18.)588 2586 y(Dette)e(betyder)f(dog)i
(ikke)g(at)f(to)g(skjulte)i(neur)o(oner)e(er)g(nok)h(til)g(at)f(appr)o
(oksimer)o(e)f(sinus)i(i)463 2701 y(ethvert)i(tilf\346lde.)g(Hvis)h
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(antal)g(neur)o(o-)463 2930 y(ner)j(\346ndr)o(e)g(sig.)h(Faktisk)g(kan)
g(man)f(blot)h(ved)f(at)g(for)o(\370ge)g(antallet)h(af)e
(tr\346ningseksempler)463 3044 y(undg\345)e(pr)o(oblemet)f(med)g
(overtr\346ning.)h(Dette)g(kan)g(ses)g(p\345)f(\002gur)h(4.19)e(p\345)h
(n\346ste)h(side.)588 3159 y(V)-9 b(ed)15 b(en)g(for)o(\370gelse)h(af)f
(antallet)g(af)g(tr\346ningseksempler)h(for)o(\370ger)f(man)h(antallet)
g(af)e(punk-)463 3273 y(ter)33 b(som)h(netv\346rket)f(fors\370ger)h(at)
f(ramme)g(med)g(appr)o(oksimationen.)g(N\345r)g(datas\346ttets)463
3388 y(st\370rr)o(else)18 b(er)h(tilstr\346kkeligt)h(stort,)f(har)g
(netv\346rket)g(ikke)g(l\346nger)o(e)g(nok)h(frihedsgrader)d(nok)463
3502 y(til)26 b(at)f(ramme)g(alle)g(punkter)h(som)g(p\345)f(\002gur)h
(4.17)d(p\345)i(forrige)h(side)f(og)h(tvinges)g(i)g(stedet)f(til)463
3617 y(v\346lge)c(en)g(middelvej)f(som)i(ligger)f(t\346tter)o(e)e
(p\345)2007 3595 y(\210)2000 3617 y FL(h)q FY(.)588 3732
y(Desv\346rr)o(e)27 b(er)i(st\370rr)o(elsen)f(af)g(datas\346ttet)g(som)
i(r)o(egel)e(ikke)h(let)f(at)h(for)o(\370ge)f(selvom)h(det)463
3846 y(kan)19 b(v\346r)o(e)e(muligt)j(at)e(g\370r)o(e)g(det)g(st\370rr)
o(e)g(kunstigt)i(ved)e(at)g(dupliker)o(e)g(nogle)h(af)f(eksemplerne)463
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V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3
-4 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V
currentpoint stroke M 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -5 V 4 -5
V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3
-5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -6 V
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V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3
-5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V
3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6
V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3
-6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V
3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6
V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3
-5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V
3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5
V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4
-5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V
3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5
V 3 -4 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3
-4 V 3 -5 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -4 V 3 -5 V
4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4
V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3
-4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -3 V
3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -4 V 4 -3 V 3 -4 V 3 -4 V 3 -3
V 4 -4 V 3 -3 V 3 -4 V 3 -3 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 4 -4 V 3
-3 V 3 -3 V 3 -3 V 4 -4 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V
3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -3 V 4 -2
V 3 -3 V 3 -2 V 3 -3 V 4 -3 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -3 V 3
-2 V 3 -3 V 4 -2 V 3 -2 V 3 -2 V 4 -3 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V
3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -1 V 4 -2 V 3 -2 V 3 -2
V 3 -1 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 4
-2 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V
3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 0 V 3 -1
V 4 -1 V 3 0 V 3 -1 V 3 0 V 4 -1 V 3 0 V 3 -1 V 4 0 V 3 0 V 3 -1 V
3 0 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V
3 0 V 3 0 V 4 1 V 3 0 V 3 0 V 3 0 V 4 1 V 3 0 V 3 0 V 3 1 V 4 0 V 3
1 V 3 0 V 3 1 V 4 1 V 3 0 V 3 1 V 3 1 V 4 1 V 3 0 V 3 1 V 3 1 V 4 1
V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V 4 2 V 3 1 V 3 1 V 4 1 V
3 2 V 3 1 V 3 2 V 4 1 V 3 1 V 3 2 V 3 2 V 4 1 V 3 2 V 3 1 V 3 2 V 4
2 V 3 2 V 3 1 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2
V 4 2 V 3 2 V 3 3 V 3 2 V 4 2 V 3 2 V 3 3 V 3 2 V 4 3 V 3 2 V 3 2 V
3 3 V 4 2 V 3 3 V 3 3 V 3 2 V 4 3 V 3 3 V 3 2 V 3 3 V 4 3 V 3 3 V 3
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3 V 3 4 V 4 3 V 3 4 V 3 3 V 3 4 V 4 4 V 3 3 V 3 4 V 3 4 V 4 3 V 3 4
V 3 4 V 3 3 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V
3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 5 V 3 4 V 3 4 V 3 4 V 4
5 V 3 4 V 3 4 V 4 5 V 3 4 V 3 5 V 3 4 V 4 4 V 3 5 V 3 4 V 3 5 V 4 4
V 3 5 V 3 5 V 3 4 V 4 5 V 3 4 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V
4 4 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 4 V 3 5 V 4 5 V 3 5 V 3 5 V 3
5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5
V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 6 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5 V
3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 6 V 3 5 V 3 5 V 3 6 V 4 5 V 3 5 V 3
6 V 3 5 V 4 6 V 3 5 V 3 5 V 3 6 V 4 5 V 3 6 V 3 5 V 3 6 V 4 5 V 3 5
V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V
3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 4 5 V 3
6 V 1 2 V 1.000 UL LT0 350 1184 M 3 1 V 4 2 V 3 1 V 3 2 V 3 2 V 4 1
V 3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 3 V 3 2 V 3 2 V
4 3 V 3 2 V 3 3 V 3 3 V 4 2 V 3 3 V 3 3 V 3 3 V 4 3 V 3 3 V 3 3 V 3
3 V 4 3 V 3 4 V 3 3 V 3 3 V 4 4 V 3 3 V 3 4 V 3 4 V 4 3 V 3 4 V 3 4
V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 5 V 3 4 V 3 4 V 4 5 V 3 4 V
3 5 V 4 4 V 3 5 V 3 5 V 3 4 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3
5 V 3 5 V 4 5 V 3 5 V 3 6 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 6 V 3 5
V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 6 V 4 5 V 3 6 V 3 5 V 3 6 V 4 6 V
3 6 V 3 5 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 4
5 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 5 V 4 6 V 3 5 V 3 6 V 3 5 V 4 6
V 3 5 V 3 6 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V
4 5 V 3 5 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 4 V 4 5 V 3 5 V 3 4 V 3
5 V 4 4 V 3 4 V 3 5 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4
V 3 4 V 4 3 V 3 4 V 3 3 V 3 4 V 4 3 V 3 4 V 3 3 V 4 3 V 3 4 V 3 3 V
3 3 V 4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 2 V 3 3 V 3 3 V 4 2 V 3 3 V 3
2 V 3 3 V 4 2 V 3 2 V 3 3 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 2
V 3 2 V 3 2 V 4 2 V 3 2 V 3 1 V 3 2 V 4 2 V 3 1 V 3 2 V 3 1 V 4 2 V
3 1 V 3 2 V 3 1 V 4 1 V 3 2 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 2 V 4
1 V 3 1 V 3 1 V 4 0 V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 0 V 4 1
V 3 1 V 3 0 V 3 1 V 4 1 V 3 0 V 3 1 V 3 0 V 4 1 V 3 0 V 3 1 V 3 0 V
4 0 V 3 1 V 3 0 V 3 1 V 4 0 V 3 0 V 3 0 V 3 1 V 4 0 V 3 0 V 3 0 V 3
0 V 4 0 V 3 1 V 3 0 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 -1 V 3
0 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 -1 V 3 0 V 3 0 V 4 -1 V 3 0 V 3
0 V 3 -1 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 -1 V 3 0 V 3 -1 V 3 0 V 4 -1
V 3 -1 V 3 0 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 3 -1 V 4 -1 V 3 0 V 3 -1
V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4
-1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V
3 -2 V 3 -1 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 4 -2
V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -3 V 4 -2 V 3 -2 V 3
-2 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -2 V
4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -4 V 4 -3 V 3 -3
V 3 -4 V 3 -3 V 4 -3 V 3 -4 V 3 -4 V 3 -3 V 4 -4 V 3 -4 V 3 -4 V 3
-3 V 4 -4 V 3 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -5 V
3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 3 -5
V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3
-5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V
3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6
V 3 -6 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V currentpoint
stroke M 3 -5 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3
-6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V
4 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6
V 3 -5 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -5 V 3
-6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V
3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5
V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3
-5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V
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V 3 -5 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 3
-4 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 4 -5 V
3 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5
V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 3 -4 V 4
-4 V 3 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V
3 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 3 -4
V 4 -5 V 3 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3
-4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 3 -4 V
4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4
V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 3
-4 V 4 -4 V 3 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V
3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4
V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -3 V 3 -4 V 3 -4 V 3 -4 V 4 -3 V 3
-4 V 3 -4 V 3 -3 V 4 -4 V 3 -4 V 3 -3 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V
3 -4 V 4 -3 V 3 -3 V 3 -4 V 3 -3 V 4 -3 V 3 -3 V 3 -4 V 3 -3 V 4 -3
V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3
-3 V 3 -2 V 4 -3 V 3 -3 V 3 -2 V 3 -3 V 4 -3 V 3 -2 V 3 -3 V 3 -2 V
4 -3 V 3 -2 V 3 -2 V 4 -3 V 3 -2 V 3 -2 V 3 -3 V 4 -2 V 3 -2 V 3 -2
V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 4
-1 V 3 -2 V 3 -2 V 3 -1 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V
3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -2 V 3 -1 V 4 -1 V 3 -1 V 3 -2 V 3 -1
V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 4 -1 V 3
-1 V 3 -1 V 3 -1 V 4 -1 V 3 0 V 3 -1 V 4 -1 V 3 -1 V 3 0 V 3 -1 V 4
-1 V 3 0 V 3 -1 V 3 -1 V 4 0 V 3 -1 V 3 0 V 3 -1 V 4 0 V 3 -1 V 3 0
V 3 0 V 4 -1 V 3 0 V 3 -1 V 3 0 V 4 0 V 3 -1 V 3 0 V 3 0 V 4 0 V 3
-1 V 3 0 V 3 0 V 4 0 V 3 0 V 3 -1 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4
0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 0 V 3 0 V 3 0 V 4 0 V 3 1 V 3 0 V 4 0
V 3 0 V 3 0 V 3 0 V 4 1 V 3 0 V 3 0 V 3 1 V 4 0 V 3 0 V 3 1 V 3 0 V
4 0 V 3 1 V 3 0 V 3 1 V 4 0 V 3 1 V 3 0 V 3 1 V 4 0 V 3 1 V 3 0 V 3
1 V 4 0 V 3 1 V 3 1 V 3 1 V 4 0 V 3 1 V 3 1 V 3 1 V 4 0 V 3 1 V 3 1
V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V 3 1 V 3 1 V 4 1 V 3 1 V
3 1 V 3 2 V 4 1 V currentpoint stroke M 3 1 V 3 1 V 4 2 V 3 1 V 3 2
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3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3 2 V 3 2 V 3 2 V 4 2 V 3
3 V 3 2 V 3 2 V 4 3 V 3 2 V 3 3 V 3 2 V 4 3 V 3 2 V 3 3 V 3 3 V 4 2
V 3 3 V 3 3 V 3 3 V 4 3 V 3 3 V 3 3 V 3 3 V 4 3 V 3 3 V 3 3 V 3 4 V
4 3 V 3 3 V 3 4 V 4 3 V 3 4 V 3 3 V 3 4 V 4 3 V 3 4 V 3 4 V 3 3 V 4
4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 4 V 3 4 V 4 4 V 3 4 V 3 5 V 3 4
V 4 4 V 3 4 V 3 5 V 3 4 V 4 5 V 3 4 V 3 5 V 3 4 V 4 5 V 3 4 V 3 5 V
3 5 V 4 4 V 3 5 V 3 5 V 3 5 V 4 4 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3
5 V 3 4 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5
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3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3
5 V 3 5 V 3 5 V 4 5 V 3 4 V 3 5 V 3 5 V 4 5 V 3 5 V 3 4 V 3 5 V 4 5
V 3 5 V 3 4 V 3 5 V 4 4 V 3 5 V 3 5 V 3 4 V 4 5 V 3 4 V 3 5 V 4 4 V
3 5 V 1 1 V stroke grestore end showpage


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FN(p)1379 4727 y FQ(p)p 1378 4741 46 4 v 1385 4788 a
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h FL(k)20 b FS(\000)f FY(1)26 b(dele)f(hvor)i(en)f(anden)f(del)h
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(v\346lge)h(det)f(samme)h(bogstav)-9 b(.)588 3512 y(Hvis)23
b(til)g(geng\346ld)h(`a')f(optr\346der)f(med)h(en)g(fr)o(ekvens)f
(p\345)h(0)q(,)q(75)e(og)j(`b')f(dermed)f(med)h(en)463
3627 y(fr)o(ekvens)28 b(p\345)g(1)19 b FS(\000)h FY(0)q(,)q(75)37
b FG(=)i FY(0)q(,)q(25,)26 b(vil)j(det)g(v\346r)o(e)e(mer)o(e)h(for)o
(delagtigt)g(hvis)h(man)g(v\346lger)f(at)463 3741 y(klassi\002cer)o(e)g
(et)g(eksempel)g(som)h(`a')f(da)g(man)g(derved)f(opn\345r)h(en)h
(genkendelsesfr)o(ekvens)463 3856 y(p\345)21 b(0)q(,)q(75.)f(Et)h
(klassi\002kationssystem)j(der)c(altid)i(g\346tter)f(p\345)g(det)g
(samme)h(bogstav)-9 b(,)21 b(er)g(natur)o(-)463 3970
y(ligvis)g(ikke)g(meget)f(v\346r)o(d,)f(men)i(n\345r)f(man)h(tager)f
(yderliger)o(e)f(oplysninger)j(om)f(forskellige)463 4085
y(visuelle)c(egenskaber)f(ved)g(eksemplerne)g(med)h(i)g(ber)o
(egningen,)f(kan)h(ideerne)e(fra)h(en)h(s\345dan)463
4199 y(sandsynlighedsanskuelse)22 b(genbr)o(uges.)588
4314 y(Ber)o(egningerne)f(kan)h(for)o(etages)f(ved)g(hj\346lp)i(af)f
FL(Bayes')f(s\346tning)p FY(.)h(Begynd)g(med)g(at)g(de\002-)463
4429 y(ner)o(e:)588 4610 y FS(\017)44 b FL(P)r FG(\()r
FL(C)816 4627 y FO(k)849 4610 y FG(\))26 b FY(som)f(sandsynligheden)g
(for)f(et)g(eksempel)g(tilh\370r)o(er)g(klassen)h FL(C)2977
4627 y FO(k)3034 4610 y FY(og)g(dermed)671 4724 y(sandsynligheden)c
(for)g(at)f(bogstavet)h(er)f(bogstav)i FL(k)q FY(.)588
4905 y FS(\017)44 b FL(P)r FG(\()s FL(X)823 4923 y FO(l)848
4905 y FG(\))25 b FY(som)g(sandsynligheden)h(for)e(at)f(observer)o(e)g
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(datas\346t-)671 5020 y(tet.)588 5201 y FS(\017)44 b
FL(P)r FG(\()r FL(C)816 5218 y FO(k)849 5201 y FY(,)17
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FL(X)3412 5218 y FO(l)671 5315 y FY(og)f(tilh\370r)o(er)f(klassen)i
FL(C)1424 5333 y FO(k)1457 5315 y FY(.)1908 5599 y(45)p
eop
%%Page: 46 48
46 47 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
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y FS(j)s FL(X)936 209 y FO(l)962 192 y FG(\))27 b FY(som)f
(sandsynligheden)g(for)f(at)f(tr\346ningseksemplet)j(tilh\370r)o(er)e
(klasse)h FL(C)3404 209 y FO(k)671 306 y FY(givet)20
b(at)h(vi)g(har)f(observer)o(et)f(egenskaben)k FL(X)2119
324 y FO(l)2144 306 y FY(.)463 487 y(Bayes')d(s\346tning)i(er)e(da)1210
720 y FL(P)r FG(\()r FL(C)1352 738 y FO(k)1386 720 y
FS(j)s FL(X)1473 738 y FO(l)1498 720 y FG(\))26 b(=)1657
664 y FL(P)r FG(\()r FL(C)1799 682 y FO(k)1833 664 y
FY(,)17 b FL(X)1932 682 y FO(l)1957 664 y FG(\))p 1654
701 337 4 v 1720 777 a FL(P)r FG(\()s FL(X)1869 795 y
FO(l)1895 777 y FG(\))2025 720 y(=)2126 664 y FL(P)r
FG(\()s FL(X)2275 682 y FO(l)2301 664 y FS(j)r FL(C)2381
682 y FO(k)2415 664 y FG(\))t FL(P)r FG(\()r FL(C)2593
682 y FO(k)2627 664 y FG(\))p 2123 701 537 4 v 2290 777
a FL(P)r FG(\()s FL(X)2439 795 y FO(l)2464 777 y FG(\))2671
720 y FY(,)544 b(\(4.22\))463 950 y(hvor)1459 1064 y
FL(P)r FG(\()s FL(X)1608 1082 y FO(l)1633 1064 y FG(\))26
b(=)1780 1081 y FE(\345)1811 1156 y FO(k)1887 1064 y
FL(P)r FG(\()s FL(X)2036 1082 y FO(l)2062 1064 y FS(j)r
FL(C)2142 1082 y FO(k)2176 1064 y FG(\))t FL(P)r FG(\()r
FL(C)2354 1082 y FO(k)2388 1064 y FG(\))r FY(,)463 1283
y(og)k(kan)g(br)o(uges)g(til)h(at)e(\002nde)h(sandsynligheden)h(for)e
(at)h(bogstavet)f(tilh\370r)o(er)h(en)g(bestemt)463 1397
y(klasse)d(n\345r)g(egenskaben)i FL(X)1383 1415 y FO(l)1434
1397 y FY(er)e(observer)o(et,)e(idet)i(sandsynlighederne)h(p\345)e
(h\370jr)o(e)h(side)g(af)463 1512 y(lighedstegnet)22
b(i)g(\(4.22\))d(alle)i(kan)g(bestemmes)h(ud)f(fra)g(observationer)g
(der)g(er)g(for)o(etaget)f(p\345)463 1627 y(for)o(h\345nd.)g(En)i
(egenskab)f(kunne)h(f.eks.)e(v\346r)o(e)g(om)i(et)f(bestemt)h(punkt)g
(var)e(sort.)i(For)f(at)g(vise)463 1741 y(hvor)o(dan)f(det)g(kan)h(br)o
(uges,)g(udbygger)f(vi)h(eksemplet)g(fra)f(f\370r)-6
b(.)588 1856 y(Bogstavet)25 b(kan)i(som)g(f\370r)e(enten)i(v\346r)o(e)e
(et)h(`a')g(eller)g(et)g(`b')g(hvor)g(fr)o(ekvensen)g(for)g(`a')f(er)
463 1970 y(0)q(,)q(25.)17 b(For)j(tr\346ningseksempler)g(p\345)f(`a')f
(er)h(fr)o(ekvensen)g(0)q(,)q(80)e(for)i(at)g(observer)o(e)f
(egenskaben)465 2085 y FL(X)525 2100 y FP(1)579 2085
y FY(og)g(0)q(,)q(20)f(for)g(egenskaben)j FL(X)1488 2100
y FP(2)1524 2085 y FY(.)e(T)-5 b(ilsvar)o(ende)17 b(er)g(p\345)h
(tr\346ningseksempler)g(af)g(`b')f(fr)o(ekven-)463 2199
y(sen)k(0)q(,)q(40)e(for)i(at)f(observer)o(e)g(egenskaben)i
FL(X)1882 2215 y FP(1)1939 2199 y FY(og)f(0)q(,)q(60)e(for)i(at)f
(observer)o(e)g(egenskaben)i FL(X)3326 2214 y FP(2)3363
2199 y FY(.)588 2314 y(Hvis)33 b(man)g(for)g(et)g(testeksempel)h
(observer)o(er)d(egenskaben)k FL(X)2690 2330 y FP(1)2727
2314 y FY(,)e(kan)g(man)g(ber)o(egne)463 2429 y(sandsynlighederne)21
b(for)g(at)f(eksemplet)h(tilh\370r)o(er)f(\351n)h(af)f(klasserne:)1082
2662 y FL(P)r FG(\()r FL(C)1225 2674 y FO(a)1259 2662
y FS(j)s FL(X)1345 2677 y FP(1)1382 2662 y FG(\))25 b(=)1541
2606 y FL(P)r FG(\()s FL(X)1689 2621 y FP(1)1726 2606
y FS(j)r FL(C)1807 2618 y FO(a)1841 2606 y FG(\))t FL(P)r
FG(\()r FL(C)2020 2618 y FO(a)2053 2606 y FG(\))p 1538
2643 549 4 v 1705 2719 a FL(P)r FG(\()s FL(X)1853 2734
y FP(1)1889 2719 y FG(\))2121 2662 y(=)2219 2606 y FY(0)q(,)q(80)16
b FS(\001)h FY(0)q(,)q(25)p 2219 2643 353 4 v 2322 2719
a(0)q(,)q(50)2605 2662 y FG(=)24 b FY(0)q(,)q(4)1082
2891 y FL(P)r FG(\()r FL(C)1224 2909 y FO(b)1259 2891
y FS(j)s FL(X)1345 2907 y FP(1)1382 2891 y FG(\))h(=)1541
2835 y FL(P)r FG(\()s FL(X)1689 2850 y FP(1)1726 2835
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y FP(1)1890 2948 y FG(\))2121 2891 y(=)2220 2835 y FY(0)q(,)q(40)15
b FS(\001)i FY(0)q(,)q(75)p 2220 2872 353 4 v 2322 2948
a(0)q(,)q(50)2606 2891 y FG(=)23 b FY(0)q(,)q(6.)463
3120 y(For)33 b(at)g(opn\345)g(\003est)g(korr)o(ekte)f
(klassi\002kationer)i(skal)f(man)g(alts\345)f(klassi\002cer)o(e)g
(bogstaver)463 3235 y(hvor)d(egenskaben)h FL(X)1186 3250
y FP(1)1251 3235 y FY(er)e(observer)o(et)f(som)i(`b')g(\226)f(dermed)f
(opn\345s)i(en)f(genkendelsesfr)o(e-)463 3349 y(kvens)21
b(p\345)f(0)q(,)q(60.)588 3464 y(Hvis)25 b(de)f(egenskaber)h(man)g
(ekstraher)o(er)e(fra)h(inputtet)i(ikke)f(er)f(diskr)o(ete,)g(bliver)h
(man)463 3579 y(n\370dt)20 b(til)h(at)f(benytte)g(sandsynlighedsfor)o
(delingsfunktioner)i(hvor)e(sandsynligheden)h(for)463
3693 y(at)h FL(x)h FY(ligger)e(i)g(intervallet)g FG([)s
FL(a)q FY(;)15 b FL(b)r FG(])22 b FY(er)f(givet)g(ved)1467
3943 y FL(P)r FG(\()s FL(x)k FS(2)h FG([)s FL(a)q FY(;)15
b FL(b)r FG(])r(\))25 b(=)2015 3830 y FF(Z)2100 3854
y FO(b)2064 4017 y(a)2154 3943 y FL(p)r FG(\()s FL(x)s
FG(\))16 b FL(d)s(x)s FY(,)463 4172 y(hvor)26 b FL(p)r
FG(\()s FL(x)s FG(\))21 b FY(er)f(sandsynlighedsfor)o
(delingsfunktionen.)h(Det)f(kan)g(vises)h([Bishop,)f(1995])d(at)463
4286 y(Bayes')j(s\346tning)i(med)f(sandsynlighedsfor)o
(delingsfunktioner)h(bliver)1489 4519 y FL(P)r FG(\()r
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4463 y FL(p)r FG(\()s FL(x)s FS(j)r FL(C)2102 4481 y
FO(k)2135 4463 y FG(\))t FL(P)r FG(\()r FL(C)2313 4481
y FO(k)2348 4463 y FG(\))p 1892 4500 489 4 v 2061 4576
a FL(p)r FG(\()s FL(x)s FG(\))2392 4519 y FY(,)463 4751
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FY(.)588 5084 y(Dette)c(kan)h(br)o(uges)h(til)f(at)g(vise)g(at)f(man)h
(kan)g(udlede)g(fejlfunktionen)h(summen)g(af)e(kva-)463
5199 y(drater)27 b(ved)h(at)h(fors\370ge)g(at)f(maksimer)o(e)g
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5211 y FO(n)3413 5199 y FY(i)463 5313 y(tr\346ningss\346ttet)c(r)o
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FT(t)2226 5325 y FO(n)2290 5313 y FY(for)f(samme)h(eksempel,)f(svar)o
(ende)1908 5599 y(46)p eop
%%Page: 47 49
47 48 bop 1007 -129 a FY(Kapitel)20 b(4:)25 b(Konstr)o(uktion)e(af)d
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b(at)g(maksimer)o(e)f(de)h(f\346lles)g(sandsynlighedsfor)o(delinger)33
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b FT(t)2697 204 y FO(n)2737 192 y FG(\))28 b FY(for)f(hver)g(v\346r)o
(di)f(af)464 306 y FL(n)q FY(.)19 b(Hvis)g(man)g(antager)f(at)g
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(sand-)463 421 y(synlighedsfor)o(deling)i(skrives)g(som)974
594 y FL(p)r FG(\()r FT(x)1095 609 y FP(1)1131 594 y
FY(,)16 b FT(t)1197 609 y FP(1)1233 594 y FY(,)g(.)g(.)g(.)e(,)i
FT(x)1460 606 y FO(n)1501 594 y FY(,)g FT(t)1568 606
y FO(n)1608 594 y FG(\))25 b(=)1755 611 y FE(\325)1789
675 y FO(n)1878 594 y FL(p)r FG(\()r FT(x)2000 606 y
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594 y FL(p)r FG(\()r FT(t)2525 606 y FO(n)2565 594 y
FS(j)r FT(x)2634 606 y FO(n)2675 594 y FG(\))7 b FL(p)r
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1040 y(ring)24 b(af)e(\(4.23\))f(vil)j(v\346r)o(e)e(\346kvivalent)i
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1155 y(\(4.23\))n(.)c(Dette)h(giver)f(fejlfunktionen)k
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1248 y FF(!)1404 1607 y FG(=)25 b FS(\000)1575 1624 y
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2143 y(gige,)g(kan)f(udtrykket)h(skrives)g(om)h(til)1260
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g(vil)h(der)e(v\346r)o(e)g(forbundet)h(en)h(vis)f(st\370j)i(eller)e
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FL(t)f FY(op)g(som)h(gener)o(er)o(et)c(af)i(en)h(underliggende)f
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3078 y FO(k)r(n)1944 3061 y FG(\()r FT(x)r FG(\))18 b
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463 3348 y(kes)g(ved)g(hj\346lp)h(af)e(det)h(gener)o(elle)f(udtryk)h
(for)g(sandsynlighedsfor)o(delingsfunktion)i(for)463
3463 y(normalfor)o(delingen)1519 3626 y FL(p)r FG(\()s
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a(2)q Fs(s)2280 3602 y Fy(2)2366 3626 y FY(,)849 b(\(4.26\))463
3806 y(og)21 b(bliver)1540 3983 y FL(p)r FG(\()r FN(e)1658
4000 y FO(k)r(n)1728 3983 y FG(\))k(=)1974 3926 y FY(1)p
1884 3964 223 4 v 1885 4050 a FN(s)1935 3980 y FS(p)p
2004 3980 102 4 v 70 x FY(2)q FN(p)2133 3983 y FS(\001)19
b FL(e)2209 3937 y Ft(\000)2271 3890 y Fs(e)2295 3872
y Fy(2)2296 3917 y Fr(k)s(n)p 2269 3922 84 3 v 2269 3973
a Fy(2)q Fs(s)2324 3959 y Fy(2)463 4163 y FY(idet)h(middelv\346r)o
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4278 y(Det)h(ses)g(af)f(\(4.25\))e(at)k FL(t)1303 4295
y FO(k)r(n)1393 4278 y FY(for)f(en)g(given)g(v\346r)o(di)f(af)h
FT(x)h FY(ikke)f(er)g(helt)g(fastlagt,)f(men)h(i)g(stedet)463
4392 y(har)31 b(en)g(sandsynlighedsfor)o(deling)37 b
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FS(j)r FT(x)r FG(\))32 b FY(som)f(svar)o(er)f(til)h(sandsynlighedsfor)o
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3340 y(vejs)32 b(s\345)f(det)f(f\346r)o(dige)g(tr\346ningspr)o(ogram)i
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(at)g(for)o(etage)463 3798 y(denne)d(afg\370r)o(else,)e(kan)i
(nedarvede)f(klasser)g(overskrive)h(den)g(abstrakte)f(funktion)i(og)g
(der)o(-)463 3913 y(med)h(implementer)o(e)f(forskellige)h
(termineringskriterier)g(og)g(logf\370ringer)-6 b(.)588
4027 y(I)27 b(praksis)g(viste)h(det)f(sig)h(kun)g(n\370dvendigt)g(at)f
(implementer)o(e)g(\351n)h(nedarvet)e(klasse)h(\226)463
4142 y(den)c(tager)g(bl.a.)f(mod)i(et)f(\002lnavn)h(til)g(logf\370ring)
g(og)g(det)f(maksimale)g(antal)g(iterationer)-6 b(,)23
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(n\345et.)f(V)-8 b(alideringsfejlen)20 b(kan)e(ber)o(eg-)463
4486 y(nes)29 b(ved)f(at)g(kalde)i FW(E)t(R)t(R)t(O)t(R)i
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(iteration)463 4600 y(bliver)23 b(netv\346rket)g(gemt)g(\226)f
(medmindr)o(e)h(valideringsfejlen)g(er)f(steget)i(\226)e(s\345)h(det)f
(er)h(muligt)463 4715 y(at)d(afbryde)g(tr\346ningspr)o(ogrammet)h(uden)
f(at)h(miste)g(det)f(hidtidige)h(arbejde.)463 4956 y
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(endelig)f(forbundne)g(neur)o(oner)-6 b(.)25 b(For)1908
5599 y(51)p eop
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52 53 bop 1465 -129 a FY(Kapitel)20 b(5:)25 b(Implementering)p
463 -96 2975 4 v 463 192 a(at)32 b(muligg\370r)o(e)i(forbindelser)e
(mellem)h(vilk\345rlige)g(typer)g(neur)o(oner)f(benyttes)i(et)e
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h(\370verst)f(\226)g(denne)h(underst\370tter)f(ikke)h(andet)f(end)463
421 y(opbevaring)f(af)g(v\346r)o(dien)g(for)g(neur)o(onen)h(samt)f(en)h
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535 y(Herfra)29 b(nedarves)g(en)i(inputneur)o(on)g(som)g(indeholder)g
(en)f(funktion)i(til)e(at)g(inds\346tte)h(en)463 650
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(blot)f(s\346tter)g(sin)h(v\346r)o(di)e(til)h(1)g(n\345r)g(den)g(kon-)
463 765 y(str)o(uer)o(es,)g(samt)h(en)g(neur)o(onklasse)g(med)f
(forbindelser)-6 b(.)588 879 y(Sidstn\346vnte)23 b(klasse)g(indeholder)
h(dels)f(en)h(vektor)f(af)g(forbindelser)g(\226)g(en)g(forbindelse)463
994 y(best\345r)k(blot)g(af)g(en)g(v\346gt)h(og)g(en)f(r)o(efer)o(ence)
e(til)j(en)f(anden)g(neur)o(on)g(\226)g(som)h(peger)f(bagud)g(i)463
1108 y(netv\346rket)h(\(dvs.)f(f.eks.)h(fra)f(andet)h(lag)g(til)g
(f\370rste)g(lag)g(og)h(aldrig)e(omvendt\),)h(og)h(dels)f(en)463
1223 y(funktion,)c FW(C)t(O)t(M)t(P)t(U)t(T)t(E)t FY(-)t
FW(V)m(A)t(L)t(U)t(E)r FY(,)c(til)h(at)g(ber)o(egne)g(og)h(s\346tte)f
(den)g(forbundne)g(neur)o(ons)h(v\346r)o(di)463 1337
y(ved)j(at)g(l\370be)g(alle)g(forbindelserne)g(igennem,)i(ber)o(egne)d
(den)h(v\346gtede)g(sum)h(af)f(de)g(tilkob-)463 1452
y(lede)i(neur)o(oners)g(v\346r)o(dier)e(og)j(anvende)f
(aktiveringsfunktionen)h(p\345)f(r)o(esultatet.)f(De)h(for)o(-)463
1567 y(bundne)h(neur)o(oner)f(indeholder)g(desuden)g(hver)h(en)g
FN(d)q FY(-v\346r)o(di)f(til)g(br)o(ug)h(for)f(ber)o(egning)h(af)463
1681 y(de)20 b(partielle)g(a\003edede)g(med)g
(tilbagef\370ringsalgoritmen.)588 1796 y(Neur)o(onobjekterne)f(for)f
(det)h(f\370rste)f(lag,)h(inputlaget,)g(opbevar)o(es)f(i)h(en)g
(s\346rskilt)g(vektor)-6 b(,)463 1910 y(mens)21 b(r)o(esten)e(af)h
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(en)f(vektor)g(med)463 2025 y(neur)o(oner)g(for)h(hvert)g(lag.)463
2281 y Fz(5.1.4)99 b(Beskrivelse)26 b(af)e(netv\346rksalgoritmer)463
2454 y FY(De)j(mest)g(inter)o(essante)f(af)g(de)g(algoritmer)h(som)h
(benyttes)f(i)g(netv\346rket,)f(bliver)h(gennem-)463
2569 y(g\345et)20 b(med)h(pseudokode)g(i)f(det)h(f\370lgende.)463
2806 y FT(Fremadf\370ring)f(af)h(v\346rdier)463 2979
y FY(Fr)o(emadf\370ring)f(af)h(v\346r)o(dier)f(for)o(eg\345r)g(ved)h
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3094 y(igennem)d(og)h(inds\346tte)f(en)f(v\346r)o(di)g(fra)g
(inputvektor)o(en)g(i)h(hver)g(neur)o(on,)f(hvor)o(efter)g(r)o(esten)g
(af)463 3208 y(lagene)h(kan)h(l\370bes)f(igennem)i(fra)d(den)i(ene)f
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(dierne)f(gradvist)h(bliver)g(ber)o(egnet)463 3438 y(efter)o(h\345nden)
17 b(som)i(lagene)g(gennemg\345s.)f(T)-5 b(il)20 b(sidst)e(indeholder)h
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g(af)g(netv\346rkets)h(samlede)f(ber)o(egning)g(som)i(derved)d(let)i
(kan)g(indsamles.)588 3667 y(I)f(pseudokode)h(ser)f(algoritmen)i
(s\345ledes)e(ud:)629 3959 y FX(l\370b)h(gennem)g(neur)o(onerne)g(i)g
(f\370rste)f(lag:)795 4074 y(s\346t)h(v\346r)o(dien)f(for)g(hver)h
(neur)o(on)g(til)g(v\346r)o(dien)f(i)h(inputvektor)o(en)629
4188 y(l\370b)g(gennem)g(lagene)g(fra)f(f\370rste)g(skjulte)i(lag:)795
4303 y(l\370b)f(gennem)g(neur)o(onerne)g(for)f(hvert)h(lag:)961
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4769 y FT(T)-5 b(ilbagef\370ring)20 b(af)i FN(d)q FT(-v\346rdier)463
4942 y FY(T)-5 b(ilbagef\370ring)26 b(af)f FN(d)q FY(-v\346r)o(dier)f
(forl\370ber)h(under)h(antagelse)f(af)f(at)h(der)g(f\370rst)h(er)f(for)
o(eg\345et)f(en)463 5057 y(fr)o(emadf\370ring)d(s\345)h(alle)h(neur)o
(onerne)f(indeholder)h(der)o(es)e(r)o(espektive)h(aktiveringsv\346r)o
(dier)-6 b(.)464 5172 y FN(d)q FY(-v\346r)o(dierne)19
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FG(\()r FL(o)2089 5366 y FO(j)2129 5349 y FS(\000)f FL(t)2243
5366 y FO(j)2267 5349 y FG(\))r FY(,)1908 5599 y(52)p
eop
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53 54 bop 1465 -129 a FY(Kapitel)20 b(5:)25 b(Implementering)p
463 -96 2975 4 v 463 192 a(mens)c(summen)h(i)f(udtryk)g(\(4.9\))e(for)h
(skjulte)i(neur)o(oner)1587 381 y FN(d)1626 397 y FO(j)1672
381 y FG(=)27 b FL(g)1808 347 y Ft(0)1830 381 y FG(\()s
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398 y FE(\345)2053 473 y FO(k)2127 381 y FN(d)2164 399
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FY(ikke)25 b(er)e(mulig)j(at)d(ber)o(egne)h(dir)o(ekte)f(da)h(der)f(i)i
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739 y(neur)o(oner)c(i)g(det)g FL(for)o(eg\345ende)g FY(lag)g(og)h(ikke)
f(til)h(neur)o(onerne)f(som)h(genneml\370bes)g(med)f(indek-)463
854 y(set)i FL(k)g FY(da)e(de)g(be\002nder)g(sig)i(i)f
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(alle)f FN(d)q FY('er)g(untagen)g(det)f(sidste)h(lag)g(og)h(starte)e
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(\226)g(for)g(hver)h(neur)o(on)f(i)h(det)e(sidste)i(lag,)g
FL(k)q FY(,)f(kan)h(hvert)f(lille)h(bidrag)463 1197 y(til)d(summen)h(i)
e(\(4.9\))f(for)27 b FL(j)e FY(ber)o(egnes.)d(V)-9 b(ed)23
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1312 y(ber)o(egnet)19 b(og)i FN(d)952 1328 y FO(j)995
1312 y FY(kan)f(\002ndes,)f(idet)h(aktiveringsfunktionerne)h(er)e
(kendt.)h(Pr)o(ocedur)o(en)e(gen-)463 1426 y(tages)i(for)h(alle)f(r)o
(ester)o(ende)f(lag.)588 1541 y(P\345)31 b(pseudokodeform)g(ser)g
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(og)3316 1635 y(\210)3303 1656 y FN(d)g FY(er)464 1770
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(ende)g(af)g(forbindelsen\):)629 2074 y FX(l\370b)h(gennem)g(neur)o
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2205 y FO(i)629 2303 y FX(nulstil)22 b(alle)f FN(d)q
FX('er)g(p\345n\346r)f(i)h(det)f(sidste)h(lag)629 2418
y(l\370b)g(gennem)g(lagene)g(bagfra)e(p\345n\346r)h(det)h(f\370rste)f
(mellemlag:)795 2532 y(l\370b)h(gennem)g(neur)o(onerne)g(for)f(hvert)h
(lag:)961 2647 y(s\346t)h FN(d)g FX(for)e(hver)h(neur)o(on)g(til)j
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(hver)h(neur)o(on:)1127 2876 y(s\346t)1287 2856 y FY(\210)1274
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2856 y FY(\210)2598 2876 y FN(d)e FG(+)g FN(d)g FS(\001)g
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(f\370rste)h(mellemlag:)795 3105 y(s\346t)h FN(d)g FX(for)e(hver)h
(neur)o(on)g(til)j FL(g)1744 3075 y Ft(0)1766 3105 y
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FY(Gradientnedstigning)32 b(for)o(etages)d(ved)i(at)f(igangs\346tte)h
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y(for)20 b(et)f(givet)h(tr\346ningseksempel,)h(efterfulgt)d(af)h(en)h
(tilbagef\370rsel)g(af)f FN(d)q FY('erne.)h(Der)o(efter)e(kan)463
3861 y(alle)i(de)f(forbundne)h(neur)o(oner)g(l\370bes)g(igennem,)h(og)f
(for)g(hver)g(forbindelse)g(kan)g(den)g(parti-)463 3976
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(tr\346ningsfunktionen)i(idet)e(den)g(normaliser)o(ede)f(v\346r)o(di)
3077 4776 y(\257)3063 4777 y FN(h)34 b FY(som)d(be-)463
4891 y(skr)o(evet)22 b(i)g(afsnit)h(4.4.3)d(ber)o(egnes)i(og)h
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5006 y(indf\370rt)29 b(efter)g(en)g(r\346kke)h(indledende)e(test)i
(hvor)g(nedstigningsalgoritmen)h(opf\370rte)e(sig)463
5120 y(yderst)20 b(ustabilt.)588 5235 y(Ovenst\345ende)g(pr)o(ocedur)o
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(af)g(om)g(tr\346ningen)h(skal)g(stoppes)f(eller)g(forts\346tte)1908
5599 y(53)p eop
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54 55 bop 1465 -129 a FY(Kapitel)20 b(5:)25 b(Implementering)p
463 -96 2975 4 v 463 192 a(\(og)j(evt.)g(ogs\345)g(for)g(at)g(udskrive)
g(loginformationer\).)h(Hvis)f(tr\346ningen)h(forts\346ttes,)f(gen-)463
306 y(nemf\370r)o(es)20 b(operationerne)h(blot)g(en)g(gang)g(til)g
(over)f(hele)h(tr\346ningss\346ttet.)588 421 y(I)e(pseudekode)f(ser)h
(algoritmen)h(ud)f(p\345)g(f\370lgende)g(m\345de)f(\()r
FL(x)j FY(betegner)e(den)g(tilknyttede)463 535 y(neur)o(ons)i
(aktiveringsv\346r)o(di)f(i)h(en)g(forbindelse\):)629
848 y FX(gentag)795 962 y(l\370b)g(gennem)g(tr\346ningss\346ttet:)961
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b FX(p\345)20 b(inputvektor)o(en)961 1191 y(kald)32 b
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1535 y(l\370b)h(gennem)h(forbindelserne)e(for)h(hver)f(neur)o(on:)1459
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b(r\346ning)20 b(med)g(skaleret)e(konjugeret)h(gradient)463
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(gradient)g(f\370lger)f(den)h(matematiske)g(formulering)463
2522 y(fra)21 b(appendiks)h(A.2.)f(V)-9 b(ed)22 b(en)g(oms\346ttelse)h
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2636 y(ninger)k(af)e(gradienten)h(af)f(fejlfunktionen)i(og)g(selve)f
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y(Mer)o(e)28 b(speci\002kt)h(har)h(man)f(ved)g(udr)o(egning)g(af)h
FS(\000)r(r)s FL(E)r FG(\()r FT(w)2343 2767 y FO(i)2366
2751 y FG(\))h FY(i)f(appendiks)f(A.2)f(p\345)h(side)g(92)463
2866 y(aller)o(ede)18 b(ber)o(egnet)j FL(E)r FG(\()r
FT(w)1272 2882 y FO(i)1296 2866 y FG(\))g FY(ved)f(fr)o(emadf\370ring.)
g FL(E)r FG(\()r FT(w)2197 2882 y FO(i)2236 2866 y FG(+)d
FN(a)2364 2882 y FO(i)2388 2866 y FT(d)2441 2882 y FO(i)2464
2866 y FG(\))k FY(benyttes)f(to)h(gange)e(hvilket)463
2980 y(kun)i(kr\346ver)g(en)g(ber)o(egning.)f(T)-5 b(il)22
b(sidst)f(kan)i FL(E)r FG(\()r FT(w)2057 2997 y FO(i)2097
2980 y FG(+)18 b FN(a)2226 2997 y FO(i)2249 2980 y FT(d)2302
2997 y FO(i)2325 2980 y FG(\))k FY(benyttes)f(i)g(det)g(n\346ste)g
(iteration)463 3095 y(af)c(algoritmen.)i(Dette)f(betyder)f(at)h
(tidskompleksiteten)h(havner)f(p\345)f FL(O)r FG(\()q
FY(4)q FL(n)r FG(\))i FY(i)f(stedet)g(for)g(de)462 3209
y FL(O)r FG(\()q FY(8)q FL(n)r FG(\))k FY(som)f(umiddelbart)f(ser)h(ud)
f(til)h(at)g(v\346r)o(e)e(tilf\346ldet.)588 3324 y(Ber)o(egningerne)24
b(med)h(gradienten)g(involver)o(er)f(\003er)o(e)h(vektor)o(operationer)
-6 b(,)24 b(f.eks.)h FN(a)3317 3340 y FO(i)3371 3324
y FG(=)474 3413 y Fc(d)516 3390 y Fr(T)515 3436 y(i)552
3413 y Fc(g)589 3427 y Fr(i)p 473 3443 134 4 v 517 3491
a FQ(d)546 3505 y Fr(i)617 3462 y FY(,)c(s\345)h(det)g(er)g
(n\370dvendigt)g(at)g(udtr\346kke)f(den)h(fra)f(netv\346rket.)h(Dette)g
(kan)g(g\370r)o(es)g(ved)f(at)463 3577 y(l\370be)i(neur)o(onerne)f
(igennem)i(p\345)e(samme)h(m\345de)f(som)h(ved)f
(gradientnedstigningen,)i(men)463 3691 y(blot)k(indsamle)g(de)f
(partielle)f(a\003edede)h(til)g(en)h(vektor)g(i)f(stedet)g(for)h(dir)o
(ekte)e(at)h(\346ndr)o(e)g(p\345)463 3806 y(v\346gtene.)j(Gradienterne)
g FS(r)s FL(E)1467 3824 y FO(d)1534 3806 y FY(ber)o(egnes)f(s\345)h
(for)g(hvert)g(tr\346ningseksempel)h(og)g(l\346gges)463
3920 y(sammen)g(til)g(den)f(f\346r)o(dige)f(gradient)i
FS(r)s FL(E)43 b FG(=)2009 3926 y FK(\345)2088 3920 y
FS(r)s FL(E)2211 3938 y FO(d)2278 3920 y FY(\226)30 b(gyldigheden)h(af)
f(dette)g(skyldes)463 4035 y(r)o(egner)o(eglerne)20 b(for)i(dif)o(fer)o
(entiation)e(som)j(tillader)e(dif)o(fer)o(entiation)f(af)h(de)h
(enkelte)f(led)h(i)g(en)463 4150 y(summering)g(for)e(sig.)588
4264 y(Da)d(alle)h(v\346gtene)g(ligger)g(spr)o(edt)f(r)o(undt)i
(omkring)g(i)f(netv\346rket)g(og)h(ikke)f(kan)g(udskiftes)463
4379 y(p\345)32 b(en)g(simpel)g(m\345de)g(p\345)f(\351n)i(gang,)f
(viste)g(det)g(sig)g(lettest)g(at)g(for)o(etage)f(ber)o(egningen)h(af)
463 4493 y(f.eks.)27 b FS(r)s FL(E)r FG(\()r FT(w)22
b FG(+)e FN(s)s FT(d)r FG(\))27 b FY(ved)f(at)h(inds\346tte)g
(v\346gtene)g(i)g(en)h(kopi)f(af)f(netv\346rket)h(s\345)g(v\346gtene)g
(i)463 4608 y(det)20 b(rigtige)h(netv\346rk)g(ikke)g(blev)g(forstyrr)o
(et.)588 4722 y(Implementeringen)g(f\370lger)g(punktopstillingen)i(i)d
(appendiks)h(A.2)e(p\345)i(side)f(92.)463 5031 y FI(5.2)119
b(Indsamling)30 b(af)f(datas\346t)463 5235 y FY(T)-5
b(il)20 b(testene)g(skulle)g(der)f(pr)o(oducer)o(es)f(et)i(datas\346t.)
f(V)-5 b(i)20 b(besluttede)f(os)i(p\345)e(for)o(h\345nd)g(til)h(at)f
(ind-)463 5349 y(skr\346nke)e(os)h(til)f(de)g(sm\345)g(bogstaver)f(i)h
(det)g(danske)g(alfabet,)e(alts\345)h(29)g(forskellige)i(bogstaver)-6
b(.)1908 5599 y(54)p eop
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55 56 bop 1465 -129 a FY(Kapitel)20 b(5:)25 b(Implementering)p
463 -96 2975 4 v 463 192 a(For)i(at)f(kunne)h(f\345)f(et)g(r)o
(ealistisk)g(estimat)h(af)f(netv\346rkets)g(genkendelse)h(p\345)f(en)h
(h\345ndholdt)463 306 y(computer)-6 b(,)20 b(fr)o(emstillede)g(vi)h
(selv)g(datas\346ttet)e(p\345)i(en)f(h\345ndholdt.)588
421 y(T)-5 b(il)29 b(dette)f(blev)g(et)h(tegnepr)o(ogram)f(p\345)g(en)g
(h\345ndholdt)h(benyttet)g(\226)f(ved)g(at)h(benytte)f(et)463
535 y(startbillede)d(med)h(et)f(indtegnet)h(net)g(med)g(i)f(alt)h(40)f
(felter)g(p\345)g(hver)g(60)g(x)g(48)g(punkter)h(kan)463
650 y(der)g(indtegnes)h(f.eks.)f(40)g(a'er)g(hvor)o(efter)f(billedet)h
(gemmes)i(og)f(et)f(nyt)h(billede)g(fr)o(emstil-)463
765 y(les)c(med)f(f.eks.)g(40)g(b'er)-6 b(.)22 b(Med)f(6)i(gr)o
(uppemedlemmer)f(r)o(esulter)o(ede)f(dette)h(i)g(et)h(datas\346t)f
(p\345)463 879 y(40)d FS(\001)g FY(29)g FS(\001)h FY(6)36
b FG(=)g FY(6960)25 b(tegn.)j(Desuden)f(fr)o(emstillede)g(et)g(af)g(gr)
o(uppemedlemmer)g(tr)o(e)g(ekstra)463 994 y(datas\346t)e(med)h(hver)g
(40)f(tegn)i(af)e(hvert)h(bogstav)-9 b(.)26 b(V)-8 b(alget)26
b(af)f(h\370jde)i(og)f(br)o(edde)e(blev)i(for)o(e-)463
1108 y(taget)21 b(ud)h(fra)f(hvad)g(der)g(passede)g(med)h(en)g
(behagelig)g(st\370rr)o(else)f(skrivefelt.)g(Instr)o(ukserne)463
1223 y(til)30 b(fors\370gspersonerne)g(gik)g(ud)f(p\345)g(at)g(skrive)g
(i)h(almindeligt)g(tempo)g(uden)f(over)o(dr)o(even)463
1337 y(akkuratesse)20 b(som)i(hvis)f(det)f(var)g(en)h(r)o(ealistisk)f
(situation.)588 1452 y(Billederne)31 b(blev)i(der)o(efter)d
(overf\370rt)i(til)h(en)g(pc,)f(og)h(de)f(enkelte)h(tegn)g(udtr)o
(ukket)g(og)463 1567 y(oms\346t)22 b(til)f(\002ler)f(indeholdende)h(-1)
f(og)h(1.)463 1875 y FI(5.3)119 b(Beskrivelse)29 b(af)g
(hj\346lpeprogrammer)463 2079 y FY(Det)22 b(viste)f(n\370dvendigt)h(at)
f(fr)o(emstille)g(et)h(antal)f(hj\346lpepr)o(ogrammer)-6
b(,)21 b(bl.a.)f(til)i(forbehand-)463 2193 y(lingen.)27
b(De)f(v\346sentligste)g(af)f(disse)h(omtales)h(i)f(det)f(f\370lgende)h
(og)g(kan)g(\002ndes)g(p\345)g(vedlagt)463 2308 y(cd.)463
2568 y Fz(5.3.1)99 b(Opdeling)25 b(i)g(validerings-)h(og)f
(tr\346ningss\346t)463 2741 y FY(Pr)o(ogrammet)19 b Fb(splitsamples)f
FY(blev)i(fr)o(emstillet)f(til)h(at)f(dele)g(datas\346ttet)g(op)h(i)g
(tr\346ningss\346t)463 2856 y(og)h(valideringss\346t.)f(Det)h(tager)e
(mod)i(en)g(liste)f(af)g(\002lnavne)g(og)h(tilf\370jer)g(en)f(vis)h
(andel,)f(f.eks.)463 2970 y(10\045,)f(af)g(hvert)h(bogstav)h(fra)e
(hver)h(person)g(til)h(en)f(liste)h(over)f(valideringss\346ttet)g(\226)
g(r)o(esten)g(af)463 3085 y(navnene)f(blandes)f(og)i(udg\370r)f
(tr\346ningss\346ttet.)g(Hvilke)g(bogstaver)g(inden)g(for)g(en)g
(bestemt)463 3199 y(klasse)f(der)g(udv\346lges,)f(sker)h
(tilf\346ldigt,)h(men)f(alts\345)g(p\345)f(en)i(s\345dan)e(m\345de)g
(s\345)h(det)g(f.eks.)f(altid)463 3314 y(er)j(10\045)g(af)g(a'erne.)463
3574 y Fz(5.3.2)99 b(Centrering)24 b(af)h(bogstaverne)463
3747 y FY(Pr)o(ogrammet)18 b Fb(center)f FY(har)h(til)g(form\345l)g(at)
g(centr)o(er)o(e)e(bogstaverne.)i(Det)g(benytter)g(ligningen)463
3862 y(beskr)o(evet)f(i)i(afsnit)f(4.2.2)f(til)h(at)g(bestemme)h
(massemidtpunktet)g(af)f(bogstavet)g(for)g(der)o(efter)463
3976 y(at)e(centr)o(er)o(e)f(efter)h(dette.)g(Det)g(benytter)h(sig)g
(af)f(et)g(underpr)o(ogram,)g Fb(moveimage)p FY(,)f(der)h(br)o(uges)463
4091 y(til)29 b(at)f(\003ytte)h(billedet)g(efter)e(centr)o(um)i(er)g
(fundet.)e(Pr)o(ogrammet)i(\003ytter)g(billedet)f(et)g(antal)463
4205 y(punkter)19 b(ved)f(at)h(bortkaste)f(punkter)h(der)g(sk\346r)o
(es)f(fra,)g(og)h(tilf\370je)g(blanke)g(punkter)g(til)g(de)g(to)463
4320 y(kanter)h(hvor)h(der)g(bliver)f(et)h(tomr)o(um.)463
4580 y Fz(5.3.3)99 b(Skalering)25 b(af)f(billederne)463
4753 y FY(Skalering)g(af)f(billederne)h(blev)g(for)o(etaget)e(ved)i
(hj\346lp)h(af)e(pr)o(ogrammet)h Fb(scale)p FY(.)f(Pr)o(ogram-)463
4868 y(met)c(tillader)g(skaleringerne)g(1:2,)f(1:4,)f(1:6)h(og)i(1:12)d
(og)j(br)o(uger)f(den)g(simple)h(algoritme)f(som)463
4982 y(er)h(beskr)o(evet)g(i)h(afsnit)f(4.2.1,)f(alts\345)h
(middelv\346r)o(dier)-6 b(.)1908 5599 y(55)p eop
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56 57 bop 1465 -129 a FY(Kapitel)20 b(5:)25 b(Implementering)p
463 -96 2975 4 v 463 192 a Fz(5.3.4)99 b(Sammenl\346gning)24
b(af)g(billeder)463 365 y FY(For)j(bl.a.)e(at)h(kunne)h(vur)o(der)o(e)e
(hvor)h(meget)h(bogstaverne)f(fylder)-6 b(,)25 b(skr)o(ev)h(vi)h(pr)o
(ogrammet)463 479 y Fb(superimpose)c FY(som)k(l\346gger)e(de)g(enkelte)
g(punkter)g(i)h(et)f(antal)g(billeder)f(sammen)i(og)g(slut-)463
594 y(teligt)f(divider)o(er)e(med)i(antallet)f(af)h(bogstaver)-6
b(.)24 b(P\345)g(den)h(m\345de)f(kan)h(man)g(f\345)f(et)h(indtryk)g(af)
463 708 y(hvor)o(dan)20 b(forskellige)h(standar)o(dbogstaver)e(ser)h
(ud,)h(se)g(\002gur)g(B.1)e(p\345)h(side)h(94.)463 969
y Fz(5.3.5)99 b(Konvertering)25 b(til)g(billedformat)463
1142 y FY(T)-5 b(il)16 b(at)g(lette)f(visualiseringen)i(af)e(de)g
(forskellige)h(tegn)g(udformede)f(vi)h(et)f(pr)o(ogram)h
Fb(chartobmp)463 1256 y FY(som)h(oms\346tter)g(en)f(tegn\002l)h(til)g
(et)f(.bmp-billede)f(ved)h(hj\346lp)h(af)e(gra\002kbiblioteket)i
(Allegr)o(o)3381 1226 y FP(1)3416 1256 y FY(.)p 463 5269
1190 4 v 553 5325 a Fy(1)583 5349 y FW(http://www)-6
b(.talula.demon.co.uk/allegr)o(o/)1908 5599 y FY(56)p
eop
%%Page: 57 59
57 58 bop 463 845 a FJ(Kapitel)44 b(6)463 1298 y FZ(Empirisk)52
b(metode)463 1745 y FY(N\345r)16 b(det)h(neurale)g(netv\346rk)g(er)f
(blevet)h(implementer)o(et,)f(er)h(der)g(behov)g(for)g(viden)g(om)h
(hvor)o(-)463 1859 y(dan)j(det)g(funger)o(er)g(i)h(samspil)g(med)f
(data)g(for)g(at)h(kunne)g(tilpasse)g(det)f(s\345)g(genkendelsesev-)463
1974 y(nen)f(bliver)g(bedst)g(mulig.)g(Denne)g(viden)g(kan)g(fr)o
(embringes)f(ved)h(at)f(eksperimenter)o(e)g(med)463 2088
y(netv\346rket,)h(men)g(disse)g(eksperimenter)g(skal)g(ske)g(gennem)h
(en)f(metode)g(der)g(giver)g(mulig-)463 2203 y(hed)j(for)g(at)f(drage)h
(konklusioner)h(p\345)f(dem,)g(samt)g(at)g(systematiser)o(e)f
(opstillingen)j(af)d(dem)463 2317 y(s\345)27 b(de)g(kan)h(udf\370r)o
(es)e(ef)o(fektivt)h(for)g(at)g(g\370r)o(e)g(det)g(praktisk)h(muligt)g
(at)f(gennemf\370r)o(e)g(fors\370-)463 2432 y(gene.)588
2547 y(Den)d FL(empiriske)g(metode)f FY(giver)g(en)h(l\370sning)h
(p\345)f(disse)g(pr)o(oblemer)f(idet)g(den)h(b\345de)e(inde-)463
2661 y(b\346r)o(er)g(m\345der)g(at)g(eftervise)g(p\345stande)g(p\345)g
(og)i(en)f(metode)f(til)i(at)e(for)o(etage)g(eksperimenter)-6
b(.)463 2776 y(F\370rst)24 b(er)g(der)f(dog)i(et)f(sp\370r)o(gsm\345l)g
(om)g(hvad)g(viden)g(er)g(\226)f(uden)h(en)g(de\002nition)h(p\345)f
(dette)f(er)463 2890 y(det)c(ikke)g(muligt)h(at)e(besvar)o(e)g(andr)o
(e)g(inter)o(essante)g(sp\370r)o(gsm\345l)h(som)h(f.eks.)e(i)h(hvor)h
(h\370j)g(grad)463 3005 y(man)h(kan)g(stole)g(p\345)f(r)o(esultaterne)g
(af)g(den)g(empiriske)h(metode.)463 3313 y FI(6.1)119
b(Positivisme)30 b(og)h(empiri)463 3517 y FY(En)c(de\002nition)i(af)e
(hvad)g(viden)g(er)-6 b(,)26 b(er)h(ikke)h(n\370dvendigvis)g(helt)g
(ligetil.)g(F)-8 b(.eks.)27 b(kan)h(man)463 3632 y(anl\346gge)j(det)f
(synspunkt)i(at)e(det)h(ikke)g(er)f(muligt)h(at)g(vide)f(noget)h(for)o
(di)f(man)h(ikke)g(kan)463 3746 y(stole)26 b(helt)g(p\345)f(sine)h
(sanser)-6 b(.)25 b(Den)g(empiriske)h(opfattelse)f(af)g(viden)g(er)g
(lige)h(modsat)f(at)g(det)463 3861 y(er)h(noget)h(som)g(kun)g(kan)g
(opn\345s)f(gennem)h(erfaringer)e(og)i(observationer)f([Kragh)h(og)f
(Pe-)463 3975 y(dersen,)19 b(1991].)d(Hertil)k(kommer)f
(veri\002kationsprincippet)h(som)g(en)g(de\002nition)g(af)e(hvilken)463
4090 y(slags)23 b(viden)f(det)h(giver)f(mening)i(at)e(besk\346ftige)g
(sig)h(med.)g(Ud)e(fra)h(dette)g(princip)h(giver)f(et)463
4205 y(udsagn)i(ikke)h(mening)h(medmindr)o(e)e(det)g(kan)g(vises)h
(sandt)f(kun)h(ud)f(fra)g(sanseerfaringer)-6 b(.)463
4319 y(Derved)24 b(afvises)f(den)i(metafysiske)f(id\351)g(om)h(en)f
(indr)o(e)g(kraft)g(der)f(styr)o(er)h(omver)o(denen)g(da)463
4434 y(den)d(ikke)g(er)f(mulig)h(at)g(p\345vise)f(med)h
(sanseerfaringer)-6 b(.)588 4548 y(Denne)18 b(\002loso\002)h(om)f(at)f
(viden)h(kun)g(kan)f(opn\345s)h(gennem)g(sanseerfaringer)f(kaldes)g
(ogs\345)463 4663 y(positivisme.)30 b(Positivisme)h(blev)f(udformet)f
(af)f(Auguste)i(Comte)g(\(1798-1857\))24 b(med)30 b(ud-)463
4777 y(gangspunkt)19 b(i)g(hans)g(lov)g(om)g(de)f(tr)o(e)f
(erkendelsesstadier)-6 b(.)17 b(F\370rste)i(stadie)f(var)f(det)h(r)o
(eligi\370se)463 4892 y(stadie)25 b(hvor)i(f\346nomener)f(blev)f
(forklar)o(et)g(ved)g(at)h(det)f(var)g(guds)h(vilje.)h(Meget)e(nyttigt)
i(til)463 5007 y(f\346nomener)19 b(man)h(ikke)f(har)g(en)g(forklaring)g
(p\345,)f(men)i(ikke)f(s\346rligt)h(god)f(til)g(at)g(skabe)g(ny)g(vi-)
463 5121 y(den.)24 b(Andet)g(stadie)g(var)g(det)g(metafysiske)h(stadie)
f(hvor)h(f\346nomener)g(blev)f(forklar)o(et)g(ved)463
5236 y(at)h(objektet)g(havde)f(en)h(\204indr)o(e)f(kraft\223.)f(T)-7
b(r)o(edje)24 b(stadie)g(er)h(det)f(positive)i(eller)e(r)o(eale)g
(stadie,)1908 5599 y(57)p eop
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58 59 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 463 192 a(under)d(hvilket)h(f\346nomenerne)g(blev)f
(forklar)o(et)f(ud)h(fra)f(observationer)h(og)h(videnskabelige)463
306 y(metoder)-6 b(.)588 421 y(Det)32 b(tr)o(edje)f(stadie)h(er)g(den)g
(sidste)g(og)h(endegyldige)f(m\345de)g(at)g(samle)g(viden)g(p\345,)g
(og)463 535 y(det)26 b(danner)g(gr)o(undlaget)g(for)g(hvor)o(dan)g
(viden)h(bliver)f(de\002ner)o(et)f(i)i(positivismen)g(og)g(em-)463
650 y(pirismen.)g(Ud)f(fra)f(dette)h(kan)g(videnskab)g(de\002nes)h(som)
g(en)g(systematisk)g(fr)o(embringelse)463 765 y(af)20
b(viden)h(og)g(erkendelse)f([Dige,)h(2002],)d(dvs.)i(at)h(videnskab)f
(bygger)h(p\345)f(at)g(kunne)i(tilegne)463 879 y(sig)f(viden)g(og)g
(formidle)g(den)f(vider)o(e)g(p\345)g(en)h(systematisk)g(og)h
(gennemskuelig)g(m\345de.)588 994 y(Man)i(kan)i(opdele)f(udsagn)g
(efter)g(om)h(de)f(er)g FL(a)g(priori)h FY(eller)f FL(a)g(posteriori)p
FY(.)h(At)f(et)g(udsagn)463 1108 y(er)31 b(a)g(priori)h(betyder)f(at)g
(det)g(kan)h(vises)g(sandt)f(eller)g(falsk)g(ud)h(fra)e(for)o
(udg\345ende)h(viden)463 1223 y(eller)20 b(viden)f(som)i(vi)f(har)g
(kendskab)f(til)i(uafh\346ngigt)f(af)f(sanseerfaringen.)g(Hvis)h
(udsagnet)463 1337 y(derimod)32 b(kun)g(kan)g(vises)h(sandt)e(ved)h
(observationer)g(siges)g(det)g(at)f(v\346r)o(e)g(a)h(posteriori.)463
1452 y(Desuden)23 b(kan)h(udsagn)f(opdeles)g(efter)g(om)h(de)f(er)g
FL(analytiske)g FY(eller)g FL(syntetiske)p FY(.)g(Analytiske)463
1567 y(udsagn)e(kan)g(veri\002cer)o(es)e(\(vises)i(sande)f(eller)h
(falske\))f(ved)g(hj\346lp)i(af)e(en)h(logisk,)g(semantisk)463
1681 y(analyse,)16 b(f.eks.)g(er)h(et)f(udsagn)h(som)g("solen)h(ikke)f
(b\345de)f(skinner)h(og)g(ikke)g(skinner"analytisk)463
1796 y(sandt)j(idet)h(det)f(ikke)h(er)f(muligt)h(at)f(noget)i(b\345de)d
(kan)i(have)f(en)h(egenskab)f(og)h(ikke)g(have)f(en)463
1910 y(egenskab)j(p\345)f(samme)h(tid,)g(uafh\346ngigt)g(af)f(hvad)g
(dette)h(noget)g(er)-6 b(.)22 b(Syntetiske)i(udsagn)e(er)463
2025 y(udsagn)j(som)g(ikke)g(kan)g(veri\002cer)o(es)f(logisk)h(ud)g
(fra)f(der)o(es)f(semantik)i(p\345)g(samme)f(m\345de)g(\226)463
2139 y(de)c(kr\346ver)g(en)h(stillingtagen)h(til)f(hvad)f(de)h(siger)g
(om)g(virkeligheden.)p 1254 2259 1393 7 v 1761 2362 a(A)f(priori)100
b(A)20 b(posteriori)p 1254 2415 1393 5 v 1304 2518 a(Analytisk)226
b(1)426 b(2)1304 2633 y(Syntetisk)246 b(3)426 b(4)p 1254
2688 1393 7 v 920 2857 a FH(T)-8 b(abel)18 b(6.1:)28
b(De)19 b(mulige)f(sammenh\346nge)h(mellem)f(forskellige)h(typer)h
(udsagn)588 3058 y FY(Ud)h(fra)g(disse)h(to)g(forskellige)h
(klassi\002kationer)g(af)e(udsagn)h(kan)g(man)g(opstille)h(\002r)o(e)e
(for)o(-)463 3173 y(skellige)e(klasser)f([Kragh)h(og)g(Pedersen,)e
(1991],)f(se)j(tabel)f(6.1.)f(Analytiske)h(a)g(priori)h(udsagn)463
3287 y(er)e(de)g(udsagn)g(som)h(kan)g(veri\002cer)o(es)e(alene)h(ud)g
(fra)f(en)i(logisk,)g(semantisk)g(analyse.)f(Analy-)463
3402 y(tiske)g(a)f(posteriori)g(udsagn)h(kan)f(p\345)g(samme)g(m\345de)
g(veri\002cer)o(es)f(alene)h(ved)g(hj\346lp)h(af)e(logisk,)463
3516 y(semantisk)24 b(analyse,)e(men)h(skulle)h(samtidig)f(afh\346nge)f
(af)g(vor)o(es)h(sanseerfaringer)e(\226)i(dette)463 3631
y(giver)e(ikke)g(meget)f(mening,)i(og)f(kategorien)g(anses)g(ogs\345)g
(normalt)g(for)f(at)h(v\346r)o(e)e(tom.)588 3745 y(De)h(syntetiske)i(a)
e(priori)h(udsagn)f(kr\346ver)g(en)h(stillingtagen)h(til)f(hvad)f(de)g
(siger)h(om)g(vir)o(-)463 3860 y(keligheden)i(og)h(s\346tter)f(\351n)g
(i)g(stand)g(til)g(at)f(udtale)h(sig)g(om)h(den,)e(men)i(kan)f
(alligevel)g(veri\002-)463 3975 y(cer)o(es)f(uafh\346ngigt)h(af)f
(erfaringer)-6 b(.)21 b(De)i(syntetiske)g(a)g(posteriori)g(udsagn)g
(s\346tter)f(ligeledes)463 4089 y(\351n)c(i)h(stand)f(til)h(at)f
(opn\345)g(viden)g(om)h(virkeligheden,)f(men)h(kr\346ver)f(samtidig)g
(observationer)463 4204 y(af)i(den)g(for)h(at)f(kunne)i(veri\002cer)o
(e)d(om)i(de)g(er)f(sande.)588 4318 y(Sp\370r)o(gsm\345let)26
b(er)g(om)i(den)f(tr)o(edje)f(kategori)g(er)h(tom)h(eller)e(ej,)h(og)h
(det)e(er)h(faktisk)g(dette)463 4433 y(sp\370r)o(gsm\345l)17
b(der)g(adskiller)g(den)g(empiriske)h(tankegang)f(fra)f(f.eks.)h
(tidliger)o(e)f(tiders)h(natur\002-)463 4547 y(loso\002.)h(If\370lge)g
(det)e(empiriske)i(synspunkt)h(er)e(den)g(tom)h(for)o(di)e(det)h(ikke)h
(er)f(muligt)h(at)f(udtale)463 4662 y(sig)23 b(om)f(virkeligheden)h
(uden)f(at)f(have)h(for)o(etaget)f(observationer)h(af)f(den)h(\226)f
(f.eks.)h(ville)g(en)463 4777 y(r)o(eligi\370s)f(veri\002kation)g(som)h
(kunne)f(for)o(etages)e(p\345)i(for)o(h\345nd)f(ikke)h(v\346r)o(e)e
(tilstr\346kkeligt.)588 4891 y(T)-5 b(ilbage)25 b(er)h(kun)g(kategori)g
(1)g(og)g(4.)f(Kategori)h(1)g(anses)g(ogs\345)g(for)g(at)f(indeholde)h
(viden,)463 5006 y(selvom)18 b(denne)f(viden)g(kan)g(opn\345s)g(alene)g
(ved)g(hj\346lp)g(af)g(analyser)f(af)h(spr)o(oget)g(\226)f(matematik)
463 5120 y(h\370r)o(er)33 b(f.eks.)f(under)h(denne)g(klasse)g(idet)g
(sandheden)g(af)f(matematiske)h(s\346tninger)h(kan)463
5235 y(afg\370r)o(es)27 b(ud)h(fra)f(de)h(gr)o(undl\346ggende)g
(de\002nitioner)-6 b(.)29 b(Hvor)f(det)g(er)g(muligt)h(i)f(f.eks.)g
(fysik,)463 5349 y(for)o(etr\346kkes)20 b(faktisk)h(viden)f(fra)g
(kategori)h(1)f(for)o(di)g(denne)g(viden)h(er)f(garanter)o(et)f(sand.)
1908 5599 y(58)p eop
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59 60 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 588 192 a(Den)c(empiriske)h(opfattelse)e(af)h(viden)g
(er)f(alts\345)h(at)g(det)g(best\345r)f(af)h(dels)g(analytisk)g(a)g
(priori)463 306 y(viden)i(som)h(kan)f(opn\345s)g(f.eks.)g(gennem)h
(logik)g(og)f(matematik,)g(men)g(som)h(ikke)g(s\346tter)f(\351n)g(i)463
421 y(stand)h(til)i(at)e(udtale)g(sig)h(om)g(ver)o(den,)e(og)j(dels)e
(syntetisk)i(a)e(posteriori)h(viden)g(der)f(kun)h(kan)463
535 y(opn\345s)28 b(gennem)h(sanseerfaringer)-6 b(,)26
b(om)i(ver)o(den.)f(Udsagn)g(der)g(ligger)h(uden)g(for)g(disse)g(to)463
650 y(gr)o(upper)-6 b(,)25 b(f.eks.)h(udsagn)g(det)f(ikke)h(er)g
(muligt)h(at)e(eftervise)h(ved)f(eksperimenter)h(og)g(som)463
765 y(derfor)20 b(m\345)g(ligge)i(i)e(kategori)h(3,)f(er)g
(meningsl\370se.)463 1073 y FI(6.2)119 b(Empirismens)30
b(baggrund)463 1277 y FY(Det)24 b(empiristiske)g(syn)g(p\345)f
(videnskaben)h(begyndte)f(tilbage)h(i)g(det)f(16.)f(\345r)o(hundr)o
(ede)g(hvor)463 1391 y(Galileo)27 b(Galilei)f(\(1564-1642\))c
(udf\370rte)j(kontr)o(oller)o(ede)h(iagttagelser)g(af)f(luftens)i
(tryk.)f(Det)463 1506 y(var)32 b(Galilei)h(der)g(for)f(f\370rste)h
(gang)g(systematisk)h(br)o(ugte)f(den)g(empiriske)g(metode)g(til)g(at)
463 1620 y(beskrive)24 b(sit)g(arbejde)f([Petersen,)h(1998].)d
(Form\345let)j(var)g(at)f(sikr)o(e)h(at)g(r)o(esultaterne)f(var)g
(p\345-)463 1735 y(lidelige)e(\(gr)o(unden)h(til)f(at)g(de)g(var)f
(det,)h(analyser)o(es)f(i)h(afsnit)h(6.5\).)d(P\345)i(denne)g(m\345de)g
(blev)g(fy-)463 1850 y(sikken)i(til)g(en)f(systematisk)h(videnskab,)f
(og)g(samtidig)h(var)e(gr)o(unden)i(ved)f(at)f(blive)i(lagt)f(til)463
1964 y(en)i(r)o(evolution)g(af)f(det)h(videnskabelige)g(syn.)g(Fra)f
(de)h(\002loso\002ske)h(og)g(r)o(eligi\370st)f(funder)o(ede)463
2079 y(m\345der)18 b(at)g(bedrive)g(videnskab)g(p\345)h(blev)f
(eksperimenterne)h(og)g(den)g(korr)o(ekte)f(behandling)463
2193 y(af)i(disse)h(sat)f(i)h(centr)o(um.)g(P\345)g(denne)f(m\345de)g
(begyndte)h(udviklingen)h(at)e(acceler)o(er)o(e.)588
2308 y(Gr)o(unden)28 b(til)h(denne)f(acceler)o(ering)e(skyldes)j(at)e
(den)h(empiriske)g(metode)g(var)g(s\345)f(me-)463 2422
y(get)22 b(mer)o(e)g(ef)o(fektiv)f(til)i(at)f(skabe)f(ny)i(viden,)f(og)
h(kan)f(forklar)o(es)g(med)g(to)h(ting:)f(dels)h(var)e(den)463
2537 y(r)o(eligi\370se)27 b(opfattelse)g(konservativ)h(idet)f(den)h
(cirklede)f(om)h(de)f(samme)h(begr)o(eber)d(s\345)j(der)463
2652 y(ikke)c(var)e(motivation)j(til)e(egentlig)h(ny)g(viden)f(\(snar)o
(er)o(e)e(tv\346rtimod)j(blev)f(anslag)g(til)h(dette)463
2766 y(bek\346mpet\),)c(dels)g(var)f(den)h(\002loso\002ske)i(tilgang)f
(med)f(a)g(priori)g(syntetisk)i(videnskab)e(ikke)463
2881 y(p\345lidelige,)30 b(dvs.)g(man)h(kunne)g(ikke)g(v\346r)o(e)f
(sikker)h(p\345)f(at)g(r)o(esultaterne)g(var)f(i)i(overstem-)463
2995 y(melse)18 b(med)g(virkeligheden.)g(Og)g(hvis)h(r)o(esultaterne)e
(ikke)h(er)f(p\345lidelige,)h(g\345r)f(udviklingen)463
3110 y(langsomt)22 b(for)o(di)e(hvert)g(fr)o(emskridt)g(s\345)h
(n\346rmest)g(opst\345r)f(ved)h(en)f(tilf\346ldighed.)588
3224 y(V)-9 b(ed)26 b(at)i(br)o(uge)f(eksperimenter)h(som)g(gr)o
(undlag)g(for)g(der)o(es)e(forskning)j(kunne)f(viden-)463
3339 y(skabsm\346nd)f(som)g(Blaise)e(Pascal)h(\(1623-1662\))21
b(og)27 b(Isaac)f(Newton)h(\(1642-1727\))21 b(s\345)26
b(efter)o(-)463 3454 y(h\345nden)18 b(udvikle)f(og)h(cementer)o(e)f
(den)g(empiriske)h(metode.)f(Sener)o(e)f(blev)h(metoden)h(under)o(-)463
3568 y(s\370gt)i(af)e(\002losof)o(fer)h(som)h(Geor)o(ge)f(Berkeley)g
(\(1685-1753\))14 b(og)20 b(David)f(Hume)g(\(1711-1776\))c(og)463
3683 y(den)21 b(positivistiske)h(v\346r)o(diopfattelse)d(kom)i(til.)463
3991 y FI(6.3)119 b(Bevisf\370relse)463 4195 y FY(Den)23
b(empiriske)h(metode)f(indeb\346r)o(er)e(forskellige)j(teknikker)f(til)
g(at)g(kunne)h(afg\370r)o(e)d(om)j(en)463 4310 y(hypotese)e(er)g
(rigtig)g(eller)f(ej,)h(som)h(skal)e(f\370lges)h(for)g(at)f(det)h
(videnskabelige)f(arbejde)g(inden)463 4424 y(for)32 b(et)g(felt)g
(bliver)f(anerkendt)h([Kragh)g(og)h(Pedersen,)e(1991].)f(Disse)i
(benytter)g(sig)h(i)f(vid)463 4539 y(udstr\346kning)22
b(af)e(logisk)h(ar)o(gumentation.)463 4799 y Fz(6.3.1)99
b(Deduktive)25 b(slutninger)463 4972 y FY(T)-8 b(o)35
b(gr)o(undl\346ggende)f(logiske)h(slutningsskemaer)h(er)e
FL(modus)f(ponens)i FY(og)f FL(modus)g(tollens)p FY(.)463
5086 y(Begge)22 b(g\346lder)f(helt)h(uafh\346ngigt)h(af)e(indholdet)h
(af)f(de)h(udsagn)g(der)f(arbejdes)g(med)h(\226)f(hvis)463
5201 y(pr\346misserne)g(er)f(sande,)g(vil)h(udfaldet)f(altid)g(vise)h
(sig)g(at)f(v\346r)o(e)g(korr)o(ekt.)1908 5599 y(59)p
eop
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60 61 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 588 192 a(Modus)j(tollens)h(arbejder)e(med)h
(negationer)g(og)h(kan)f(dermed)g(br)o(uges)g(til)h(at)f
FL(falsi\002cer)o(e)463 306 y FY(med,)c(dvs.)h(bevise)f(at)g(noget)i
(er)e(falsk)h(\226)f(slutningsskemaet)i(fr)o(emg\345r)d(af)h(tabel)g
(6.2.)1630 504 y(\(1\))99 b FL(Hvis)20 b FY(H,)h FL(s\345)f
FY(K)1630 618 y(\(2\))99 b FL(ikke)p FY(-K)p 1580 671
740 5 v 1630 775 a(\(3\))g FL(ikke)p FY(-H)534 954 y
FH(T)-8 b(abel)28 b(6.2:)g(Slutningsskemaet)g(modus)h(tollens.)g
FU(H)g FH(og)h FU(K)g FH(er)f(udsagn,)g(og)h(konklusionen)e(er)h(ikke-)
p FU(H)p FH(.)534 1059 y(F)-8 b(.eks.)25 b(kunne)g FU(H)h
FH(v\346r)o(e)g(en)g(hypotese)i(og)f FU(K)f FH(v\346r)o(e)g(hypotesens)
h(forudsigelse)f(af)h(et)f(eksperiment.)g(Hvis)534 1164
y(eksperimentets)19 b(udfald)f(ikke)h(er)g FU(K)p FH(,)g(vil)f
(hypotesen)i FU(H)f FH(v\346r)o(e)f(falsk.)588 1370 y
FY(Det)j(ses)h(at)g(slutningen)h(best\345r)e(af)g(to)h(pr\346misser)-6
b(.)22 b(F\370rste)f(pr\346mis)i(\(1\))d(udledes)h(af)g(den)463
1485 y(teori)j(der)e(unders\370ges.)i(Det)f(er)g(i)h(praksis)f(normalt)
h(et)g(velbegr)o(undet)f(tankeeksperiment)463 1599 y(udt\346nkt)f(af)g
(forsker)o(en.)f(Den)h(anden)g(pr\346mis)g(\(2\))f(er)h(udfaldet)e(af)i
(de)f(eksperimenter)h(for)o(-)463 1714 y(sker)o(en)d(udf\370r)o(er)f
(for)i(at)f(kunne)h(bekr\346fte/afkr\346fte)e(sin)i(hypotese.)g
(Resultatet)g(af)f(disse)h(to)463 1828 y(pr\346misser)h(\(3\))e
(sluttes)j(logisk.)811 2026 y(\(1\))98 b FL(Hvis)26 b
FY(terminering)h(ud)f(fra)f(validering)h(undg\345r)g(overtr\346ning,)
1007 2140 y FL(s\345)j FY(genkender)e(stor)o(e)h(netv\346rk)g
(tr\346net)g(med)g(validering)g(\003er)o(e)1007 2255
y(bogstaver)21 b(rigtigt)g(end)g(uden.)811 2370 y(\(2\))98
b(Genkendelsespr)o(ocenten)21 b(bliver)g FL(ikke)f FY(bedr)o(e.)p
761 2422 2378 5 v 811 2526 a(\(3\))98 b(T)-8 b(erminering)22
b(efter)e(validering)g(undg\345r)g FL(ikke)h FY(overtr\346ning.)970
2705 y FH(T)-8 b(abel)17 b(6.3:)28 b(Et)18 b(eksempel)i(p\345)f
(falsi\002cering)g(ved)g(hj\346lp)g(af)f(modus)h(tollens.)588
2911 y FY(Et)24 b(konkr)o(et)h(eksempel)f(p\345)g(br)o(ugen)h(af)f
(modus)h(tollens)h(fr)o(emg\345r)d(af)g(tabel)h(6.3)g(som)h(ud)463
3026 y(fra)g(eksperimentelle)i(data)e(i)h(\(2\))f(giver)h(en)g
(modstrid)h(til)g(den)f(opstillede)g(hypotese)h(i)f(\(1\);)463
3140 y(udfaldet)19 b(\(3\))h(viser)h(at)f(hypotesen)h(er)g(falsk)f
(\(bem\346rk)g(at)h(det)f(er)g(et)h(t\346nkt)g(eksempel\).)588
3255 y(Modus)27 b(ponens)h(arbejder)d(ikke)j(med)f(negationer)g(og)h
(kan)f(br)o(uges)g(til)h(at)f(bevise)g(ud-)463 3369 y(sagn.)j
(Slutningskemaet)h(fr)o(emg\345r)e(af)g(tabel)h(6.4.)f(Analogt)h(til)h
(modus)g(tollens)g(er)f(der)g(to)463 3484 y(pr\346misser:)22
b(\(1\))e(er)i(at)f(der)g(ud)h(fra)f(et)g(udsagn,)h(f.eks.)f(en)h
(teori,)g(H)g(kan)f(udledes)h(en)g(konse-)463 3599 y(kvens)f(K,)g(og)g
(\(2\))f(at)g(teorien)h(er)f(sand.)h(S\345)e(vil)i(konsekvensen)h(K)f
(ogs\345)g(v\346r)o(e)f(sand.)1630 3796 y(\(4\))99 b
FL(Hvis)20 b FY(H,)h FL(s\345)f FY(K)1630 3911 y(\(5\))99
b(H)p 1580 3963 740 5 v 1630 4067 a(\(6\))g(K)534 4247
y FH(T)-8 b(abel)18 b(6.4:)28 b(Slutning)17 b(ved)k(hj\346lp)e(af)g
(modus)h(ponens)f(\226)h FU(K)f FH(er)h(et)f(udsagn)h(som)g(f\370lger)f
(af)30 b FU(H)p FH(,)19 b(s\345)g(hvis)h FU(H)g FH(er)534
4352 y(sand,)e(er)h(konklusionen)e(at)i FU(K)g FH(er)g(sand.)588
4547 y FY(Et)h(konkr)o(et)h(eksempel)g(p\345)f(br)o(ug)h(af)f(modus)h
(ponens)h(fr)o(emg\345r)d(af)h(tabel)g(6.5.)588 4662
y(De)e(for)o(eg\345ende)e(eksempler)i(har)g(v\346r)o(et)f(af)g(den)h
(simpleste)h(art,)e(men)i(slutningsskema-)463 4777 y(erne)k(kan)h
(ogs\345)g(br)o(uges)g(i)g(mer)o(e)e(komplekse)j(situationer)-6
b(.)24 b(Hvis)g(f.eks.)f(der)g(ud)g(fra)g(konse-)463
4891 y(kvensen)j(K)g(af)f(en)g(hypotese)i(H)e(og)h(hypotesen)h(selv)e
(kan)h(udledes)f(en)h(r\346kke)f(for)o(uds\346t-)463
5006 y(ninger)e(F,)f(og)g(konsekvensen)h(K)g(\370nskes)g(eftervist)e
(ved)h(hj\346lp)g(af)g(H,)f(kan)h(modus)h(ponens)463
5120 y(br)o(uges)30 b(som)h(det)f(fr)o(emg\345r)e(af)h(tabel)h(6.6)f
(p\345)g(den)h(f\370lgende)g(side.)f(Hvis)i(derimod)f(sand-)463
5235 y(hedsv\346r)o(dien)c(af)f(selve)i(hypotesen)g(H)f(\370nskes)i
(fastlagt,)d(kan)i(modus)g(tollens)g(anvendes)463 5349
y(til)21 b(en)g(evt.)f(falsi\002cering.)1908 5599 y(60)p
eop
%%Page: 61 63
61 62 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 828 189 a(\(1\))98 b FL(Hvis)21 b FY(et)f(neuralt)h
(netv\346rk)f(har)h(en)f(god)h(generaliseringsevne,)1024
303 y FL(s\345)g FY(kan)g(det)f(anvendes)h(af)e(forskellige)i
(mennesker)-6 b(.)828 418 y(\(2\))98 b(Netv\346rket)20
b(har)h(en)g(god)g(generaliseringsevne.)p 778 471 2344
5 v 828 574 a(\(3\))98 b(Det)21 b(kan)g(anvendes)f(af)g(forskellige)h
(mennesker)-6 b(.)1185 754 y FH(T)e(abel)18 b(6.5:)28
b(Et)18 b(eksempel)h(p\345)h(brugen)e(af)g(modus)h(ponens.)1291
954 y FY(Modus)i(ponens)162 b(Modus)20 b(tollens)p 1241
1007 1418 5 v 1291 1111 a FL(Hvis)g FY(H)h FL(og)f FY(F,)h
FL(s\345)g FY(K)100 b FL(Hvis)20 b FY(H)h FL(og)f FY(F,)h
FL(s\345)g FY(K)1291 1225 y(H)g FL(og)f FY(F)474 b FL(ikke)p
FY(-K)p 1241 1278 V 1291 1381 a(K)649 b FL(ikke)p FY(-\(H)20
b FL(og)g FY(F\))534 1561 y FH(T)-8 b(abel)21 b(6.6:)28
b(Brug)22 b(af)g(modus)g(ponens)g(og)h(modus)f(tollens)g(p\345)g(mer)o
(e)g(komplekse)i(pr)o(oblemer)-6 b(.)22 b(Hvis)g(forud-)534
1666 y(s\346tningerne)f FU(F)h FH(antages)h(at)g(v\346r)o(e)f(sande,)h
(kan)f(modus)h(ponens)g(bruges)f(n\345r)h(blot)f(sandhedsv\346rdien)h
(af)534 1771 y FU(H)h FH(er)f(bestemt,)g(mens)h(modus)g(tollens)f
(omvendt)h(kan)f(bruges)h(til)f(at)g(vise)h(at)g FU(H)g
FH(er)f(falsk)h(n\345r)f FU(K)h FH(ikke)g(er)534 1876
y(opfyldt.)588 2146 y FY(Modus)32 b(ponens)h(og)g(modus)g(tollens)h(er)
e(s\346r)o(deles)g(vigtige)h(inden)f(for)h(logikken)h(og)463
2260 y(inden)19 b(for)g(den)g(empiriske)g(ar)o(gumentation)g(idet)g(de)
f(muligg\370r)i(systematiske)f(konklusio-)463 2375 y(ner)g(p\345)f
(sammenh\346ngen)j(mellem)e(en)g(teori)g(og)h(eksperimenterne.)e
(Slutningsskemaerne)463 2490 y(er)28 b(udledt)g(af)g(nogle)i(f\345)e
(gr)o(undl\346ggende)h(aksiomer)-6 b(,)28 b(s\345)h(hvis)g(man)g
(godtager)f(dem,)g(f\370l-)463 2604 y(ger)23 b(at)g(slutningsskemaerne)
h(altid)f(er)f(sande.)h(Hvis)g(alle)g(ens)g(antagelser)g(er)f(korr)o
(ekte,)h(og)463 2719 y(man)k(ikke)g(har)f(beg\345et)g(nogen)h(fejl)g
(under)f(udledningen)h(af)f(konsekvensen)i(fra)d(teorien,)463
2833 y(kan)i(man)g(hvis)h(man)f(benytter)g(modus)h(ponens)g(og)g(modus)
g(tollens,)f(v\346r)o(e)f(sikker)i(p\345)e(at)463 2948
y(konklusionen)d(er)d(endegyldigt)h(sand.)588 3062 y(Denne)k(korr)o
(ekthed)g(viser)h(sig)g(ved)e(n\370jer)o(e)i(eftersyn)f(imidlertid)g
(at)g(v\346r)o(e)g(af)f(udeluk-)463 3177 y(kende)19 b(teor)o(etisk)h
(karakter)e(for)i(mange)g(pr)o(oblemer)-6 b(,)18 b(hvorfor)i(den)g
(logiske)g(angr)o(ebsvinkel)463 3292 y(p\345)g(empiriske)h(pr)o
(oblemer)f(ikke)h(er)f(en)h(komplet)h(l\370sning.)463
3552 y Fz(6.3.2)99 b(Induktionsproblemet)463 3725 y FY(Pr)o(oblemet)20
b(ved)f(at)h(p\345vise)g(en)g(teori)g(med)g(eksperimenter)g(er)f(at)h
(der)f(altid)h(vil)h(v\346r)o(e)e(en)h(vis)463 3839 y(usikker)o(hed)26
b(om)h(r)o(esultatet)f(af)g(eksperimenterne,)g(selv)g(hvis)i(man)e(ser)
h(bort)f(fra)g(menne-)463 3954 y(skelige)c(fejl.)f(Eksperimenterne)g
(vil)g(altid)g(kun)h(med)f(en)h(vis)f(sandsynlighed)h(tale)f(for)g
(eller)463 4068 y(imod)j(en)g(teori.)f(Usikker)o(heden)g(kan)h(f.eks.)e
(dels)i(komme)g(fra)f(m\345leusikker)o(hed)g(eller)g(lig-)463
4183 y(nende,)d(men)h(ogs\345)g(fra)f(hvad)g(man)h(kunne)g(kalde)f
(statistisk)i(usikker)o(hed)e(som)h(knytter)g(sig)463
4298 y(til)g(tilf\346ldige)g(begivenheder)-6 b(.)588
4412 y(Et)26 b(eksempel)h(er)g(kast)g(med)g(en)g(terning.)h(Kastes)f
(en)g(terning)h(100)d(gange,)i(og)g(det)g(us-)463 4527
y(andsynlige)33 b(sker)f(at)g(der)g(sl\345s)h(en)f(toer)h(hver)f(af)g
(de)f(100)g(gange,)i(vil)f(den)h(umiddelbar)o(e)463 4641
y(konklusion)25 b(p\345)e(eksperimentet)g(v\346r)o(e)f(at)h(terningen)h
(igen)g(vil)f(ende)g(p\345)f(en)i(toer)f(den)g(101.)463
4756 y(gang.)30 b(Men)f(dette)g(er)g(en)h(fejlagtig)g(konklusion)i(for)
o(di)d(terningen)h(er)f(homogen)i(og)g(der)o(-)463 4870
y(for)e(besidder)f(sandsynligheden)1612 4838 y FP(1)p
1612 4852 32 4 v 1612 4899 a(6)1683 4870 y FY(for)h(at)g(sl\345)g(en)g
(toer)g(og)h(ikke)2653 4838 y FP(1)p 2653 4852 V 2653
4899 a(1)2724 4870 y FY(som)g(de)f(for)o(eg\345ende)463
4985 y(eksperimenter)21 b(har)f(vist.)588 5100 y(Denne)32
b(mulige)g(situation)h(vanskeligg\370r)f(logisk)h(deduktion)f(for)o(di)
f(udsagn)h(i)g(logik)463 5214 y(enten)f(er)f(sande)g(eller)g(falske)g
(og)h(ikke)f(f.eks.)g(84\045)f(sande,)h(og)h(skaber)f(dermed)f(et)i
(stort)463 5329 y(pr)o(oblem,)i FL(induktionspr)o(oblemet)p
FY(,)f(i)h(den)g(empiriske)h(bevisf\370r)o(else)e([Kragh)h(og)h
(Pedersen,)1908 5599 y(61)p eop
%%Page: 62 64
62 63 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 463 192 a(1991].)588 306 y(Det)i(man)h(str\346ber)g
(efter)e(n\345r)i(man)g(opstiller)g(teorier)-6 b(,)23
b(er)g(slutninger)i(fra)e(det)g(specielle)463 421 y(til)31
b(det)e(gener)o(elle;)g(induktive)i(slutninger)-6 b(.)31
b(Man)e(\370nsker)i(at)e(bevise)h(udsagn)g(p\345)g(formen)463
535 y(\204for)c(alle)g(.)12 b(.)g(.)g(g\346lder)27 b(.)12
b(.)g(.)g(\223,)26 b(hvilket)h(f.eks.)f(kan)g(g\370r)o(es)h(ved)f(at)g
(vise)g(at)g(for)o(uds\346tningerne)466 650 y FL(F)505
665 y FP(1)542 650 y FY(,)18 b FL(F)620 665 y FP(2)656
650 y FY(,)e(.)g(.)g(.)g(for)g(teorien)h(er)f(sande)g(hvorved)g(modus)h
(ponens)g(giver)f(at)g(teorien)h(ogs\345)g(er)e(sand.)463
765 y(Imidlertid)27 b(er)h(man)g(principielt)f(n\370dt)h(til)g(at)g
(afpr)o(\370ve)d(alle)j(testscenarier)f(eftersom)g(hver)463
879 y(mulighed)32 b(udg\370r)f(en)g(for)o(uds\346tning,)g(og)h(da)e
(der)g(er)h(uendeligt)g(mange,)g(er)f(dette)h(ikke)463
994 y(muligt.)18 b(En)g(universel)g(teori)g(kan)f(s\345ledes)g(ikke)h
(veri\002cer)o(es,)e(kun)i(falsi\002cer)o(es)f(med)g(modus)463
1108 y(tollens)30 b(hvor)g(der)e(blot)i(er)f(behov)g(for)g(at)g(\351n)g
(konsekvens)i(af)d(teorien)i(viser)f(sig)g(at)g(v\346r)o(e)463
1223 y(forkert.)588 1337 y(En)19 b(mulig)h(l\370sning)h(p\345)e(pr)o
(oblemet)f(er)h(at)g(indf\370r)o(e)g(sandsynligheder)g(i)h(de)f
(logiske)h(slut-)463 1452 y(ninger)j(i)g(stedet)f(for)g(kun)h(at)f
(arbejde)g(med)g(v\346r)o(dierne)f(sand)h(og)h(falsk.)f(F)-8
b(.eks.)23 b(skulle)g(man)463 1567 y(s\345)29 b(kunne)g(konkluder)o(e)g
(at)g(en)g(teori)g(var)f(99\045)g(sand)g(hvis)i(der)e(var)h(99\045)e
(sandsynlighed)463 1681 y(for)21 b(at)g(for)o(uds\346tningerne)h(var)f
(sande,)f(se)i(tabel)f(6.7.)f(Det)h(er)g(imidlertid)g(ikke)h(lykkedes)f
(at)463 1796 y(opstille)f(en)g(s\345dan)f(logik)h([Kragh)f(og)h
(Pedersen,)f(1991],)e(idet)i(de)g(hidtidige)h(fors\370g)g(er)f(endt)463
1910 y(med)i(at)g(have)f(u\370nskede)h(egenskaber)-6
b(,)21 b(som)g(f.eks.)g(at)g(sandsynligheden)g(for)g(alle)g(udsagn)463
2025 y(p\345)h(formen)h(\204for)f(alle)g(.)12 b(.)g(.)g(g\346lder)23
b(.)12 b(.)g(.)g(\223)23 b(er)g(lig)g(med)f(nul,)h(eller)g(at)f
(observationer)h(der)f(tyde-)463 2139 y(ligvis)c(ikke)f(har)g(med)g
(teorien)g(at)g(g\370r)o(e,)f(kan)h(bidrage)f(til)h(at)g(af-)f(eller)g
(bekr\346fte)h(den)f([H\374ttel,)463 2254 y(2001].)1590
2442 y(\(7\))102 b FL(P)r FG(\()q FY(H)p FS(j)q FY(F)r
FG(\))25 b(=)f FY(0)q(,)q(99)1590 2557 y(\(8\))99 b(F)p
1540 2610 820 5 v 1590 2713 a(\(9\))j FL(P)r FG(\()q
FY(H)p FG(\))26 b(=)d FY(0)q(,)q(99)534 2893 y FH(T)-8
b(abel)20 b(6.7:)27 b(Et)21 b(eksempel)h(p\345)f(en)g(s\345kaldt)g(pr)o
(obabilistisk)f(slutning)f(\226)j(hypotesen)f FU(H)h
FH(givet)f(foruds\346tnin-)534 2998 y(gen)c FU(F)g FH(har)h(en)f
(sandsynlighed,)i(P)r Fn(\()q FU(H)r Fv(j)q FU(F)p Fn(\))q
FH(.)d(Hvis)i FU(F)f FH(er)g(sand,)g(vil)h(sandsynligheden)f(for)h
FU(H)f FH(v\346r)o(e)g(sandsyn-)534 3103 y(ligheden)i(0,99.)588
3315 y FY(Induktionspr)o(oblemet)i(har)f(bl.a.)g(som)h(f\370lge)g(for)f
(dette)g(pr)o(ojekt)h(at)f(det)g(ikke)h(er)f(muligt)463
3430 y(at)i(opstille)h(en)f(gener)o(el)f(teori)h(der)g(beviser)f(at)h
(det)g(konstr)o(uer)o(ede)f(netv\346rk)h(er)g(det)g(univer)o(-)463
3545 y(selt)g(bedste)f(ud)h(fra)f(de)g(gennemf\370rte)h(test.)g(Det)g
(vil)g(dog)g(v\346r)o(e)e(muligt)j(at)f(benytte)f(modus)463
3659 y(tollens)i(til)e(at)h(opstille)g(slutningsskemaer)h(der)d
(afkr\346fter)g(nogle)j(teorier)-6 b(,)20 b(og)i(ligeledes)g(vil)463
3774 y(det)h(v\346r)o(e)g(muligt)i(at)e(br)o(uge)h(modus)g(ponens)h
(til)f(at)g(bekr\346fte)e(speci\002kke)i(hypoteser)h(i)f(te-)463
3888 y(stene)29 b(\226)g(kravet)f(er)h(blot)h(at)f(antallet)f(af)h(for)
o(uds\346tninger)h(skal)f(v\346r)o(e)f(overkommeligt)i(at)463
4003 y(teste.)588 4117 y(Da)i(induktionspr)o(oblemet)i(umuligg\370r)g
(logiske)g(beviser)f(for)g(gener)o(elle)f(hypoteser)-6
b(,)463 4232 y(men)18 b(naturvidenskaben)g(alligevel)f(er)g(fuld)h(af)f
(den)g(slags)h(teorier)g(\226)f(opstilling)j(af)d(s\345danne)463
4347 y(teorier)31 b(kan)g(faktisk)h(ses)f(som)h(selve)f
FL(m\345let)g FY(\226)g(m\345)g(der)g(v\346r)o(e)f(andr)o(e)g
(muligheder)h(for)g(at)463 4461 y(p\345vise)20 b(dem.)463
4718 y Fz(6.3.3)99 b(Slutning)25 b(til)h(bedste)e(forklaring)463
4891 y FY(Karl)30 b(Popper)g(\(1902-1994\))c(mente)k(at)g(muligheden)h
(for)f(falsi\002cering)h(var)e(tilstr\346kkelig)463 5006
y(idet)g(udv\346lgelsen)g(af)g(en)g(teori)h(forl\370ber)f(efter)f(en)h
(n\346rmest)h(darwinistisk)g(fr)o(emgangs-)463 5120 y(m\345de)20
b([Thornton,)i(2002].)588 5235 y(P\345)17 b(et)h(givet)f(tidspunkt)i
(vil)f(der)f(v\346r)o(e)f(en)i(vis)g(m\346ngde)g(teorier)-6
b(.)17 b(Disse)i(skal)f(naturligvis)463 5349 y(v\346r)o(e)i(fri)g(for)g
(selvmodsigelser)i(og)f(desuden)f(v\346r)o(e)g(indr)o(ettet)g
(s\345ledes)g(at)g(det)h(er)f(muligt)h(at)1908 5599 y(62)p
eop
%%Page: 63 65
63 64 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 463 192 a(opstille)c(p\345stande)f(der)f(kan)i(afpr)o
(\370ves)e(eksperimentelt,)h(ellers)h(m\345)f(de)g(forkastes.)g(Her)o
(efter)463 306 y(udv\346lges)26 b(en)g(lovende)g(teori)h(efter)e(en)h
(moderne)g(vider)o(eudvikling)g(af)f FL(Occams)g(ragekniv)463
421 y FY([H\374ttel,)j(2001])e(efter)h(den)h(franciskanske)g(munk)h(W)
-5 b(illiam)29 b(of)f(Occam)g(\(1285-1349\))23 b(som)463
535 y(g\345r)d(ud)h(p\345)f(at)g(den)h(simpleste)g(mulige)g(teori)g(er)
g(den)f(bedste:)567 706 y(1.)41 b(En)27 b(hypotese)i(der)e(forklar)o
(er)f(noget)i(er)g(bedr)o(e)e(end)h(en)h(der)f(blot)h(kan)g(beskrives)g
(af)671 820 y(korr)o(elationer)-6 b(.)24 b(Det)i(vil)g(sige)g(at)f
(selvom)h(en)g(matematisk)f(kvantisering)h(er)g(vigtigt,)671
935 y(s\345)20 b(er)g(det)h(dog)g(vigtiger)o(e)f(at)g(hypotesen)i
(indeholder)f(en)f(forklaring.)567 1110 y(2.)41 b(En)26
b(gener)o(el)g(hypotese)h(er)f(bedr)o(e)f(end)i(en)f(speci\002k,)h
(hvilket)g(til)g(dels)f(h\346nger)h(sam-)671 1225 y(men)18
b(med)g(n\346ste)h(princip)f(\226)g(jo)h(f\346rr)o(e)d(teorier)i(der)g
(er)g(n\370dvendige,)g(jo)h(mindr)o(e)f(kom-)671 1339
y(plicer)o(et.)567 1515 y(3.)41 b(En)20 b(simpel)g(hypotese)h(er)f
(bedr)o(e)e(end)i(en)h(komplicer)o(et.)e(Gr)o(unden)i(til)g(dette)e
(skal)i(m\345-)671 1630 y(ske)26 b(ses)g(i)g(den)f(sammenh\346ng)j(at)d
(en)h(teori)g(i)g(forvejen)g(er)f(en)h(forsimplet)h(model)f(af)671
1744 y(virkeligheden)f(\226)f(jo)i(mer)o(e)e(simpelt)h(den)g(kan)g
(formuler)o(es,)f(jo)h(mer)o(e)f(vil)h(den)g(s\345)g(be-)671
1859 y(v\346ge)20 b(sig)h(i)g(r)o(etning)g(af)f(det)g(gr)o
(undl\346ggende)h(og)h(dermed)e(mest)h(gener)o(elle.)567
2034 y(4.)41 b(En)27 b(hypotese)h(der)e(h\346nger)i(sammen)g(og)g(er)e
(integr)o(er)o(et)g(med)i(eksister)o(ende)e(viden)671
2149 y(er)18 b(bedr)o(e)g(end)h(en)g(isoler)o(et;)f(tr)o(oen)h(p\345)g
(de)f(andr)o(e)g(teorier)h(smitter)g(n\346rmest)g(af)g(p\345)f(den)671
2264 y(nye)j(teori.)588 2434 y(N\345r)31 b(en)g(teori)h(er)g(udvalgt,)f
(skal)h(den)f(uds\346ttes)h(for)g(et)f(antal)g(eksperimenter)-6
b(.)32 b(Hvis)463 2548 y(nogle)22 b(af)d(disse)i(viser)g(sig)g(at)g
(v\346r)o(e)e(i)i(modstrid)g(med)g(teorien,)g(kan)f(denne)h
(falsi\002cer)o(es)f(ved)463 2663 y(hj\346lp)k(af)e(modus)i(tollens,)g
(og)f(m\345)g(derfor)f(forkastes.)h(Ellers)f(vil)i(teorien)f(g\345)g
(styrket)g(ud)g(af)463 2777 y(pr)o(\370velsen)c([Kragh)g(og)h
(Pedersen,)f(1991])f(idet)h(den)g(i)h(endnu)g(h\370jer)o(e)f(grad)g
(end)g(f\370r)g(har)h(vist)463 2892 y(sig)h(at)g(v\346r)o(e)e
FL(konsistent)p FY(.)588 3007 y(P\345)24 b(denne)i(m\345de)e(er)h(det)g
(muligt)h(at)f(afvise)g(en)g(r\346kke)g(teorier)g(og)h(l\370bende)f
(slutte)g(sig)463 3121 y(til)g(bedste)f(forklaring)g(som)h(blot)g(er)f
(den)g(teori)h(der)f(hidtil)h(har)f(klar)o(et)f(sig)i(bedst.)f(Man)g
(ar)o(-)463 3236 y(bejder)k(ikke)h(p\345)f(at)h(bevise)f(teorierne,)h
(men)g(p\345)f(at)g(vise)h(at)g(de)f(er)g(konsistente)i(\226)f(herved)
463 3350 y(forsvinder)24 b(induktionspr)o(oblemet.)h(Da)f
(v\346rkt\370jet)g(er)g(falsi\002kation,)g(er)g(teorier)f(der)h(ikke)
463 3465 y(kan)e(falsi\002cer)o(es,)f(ikke)h(br)o(ugbar)o(e)f(og)i(er)f
(if\370lge)g(Popper)g(ogs\345)h(blot)f(meninger)h([Thornton,)463
3579 y(2002].)18 b(I)j(centr)o(um)g(for)g(metoden)g(st\345r)f(alts\345)
g(stadig)h(de)f(empiriske)h(fors\370g.)463 3885 y FI(6.4)119
b(Empiriske)29 b(fors\370g)463 4089 y FY(If\370lge)24
b(den)g(empiriske)g(metode)g(skal)g(de)g(iagttagelser)f(der)h(er)o
(hverves)f(ved)g(eksperimen-)463 4204 y(ter)i(beskrives)f(og)i(or)o
(dnes.)e(T)-8 b(o)26 b(ting)g(er)e(kritiske)h(ved)g(alle)f(empiriske)i
(fors\370g)f(idet)g(de)f(kan)463 4318 y(p\345virke)j(r)o(esultaterne)f
(af)g(eksperimenterne:)i(tilr)o(ettel\346ggelsen)f(af)g(fors\370gene)g
(f\370r)g(selve)463 4433 y(pr)o(ocessen)22 b(er)g(g\345et)g(i)h(gang,)g
(og)g(den)f(efterf\370lgende)g(fortolkning)i(af)d(de)h(opn\345ede)g(r)o
(esulta-)463 4547 y(ter)-6 b(.)21 b(V)-9 b(ed)20 b(den)h(empiriske)h
(arbejdsgang)f(m\345)g(alts\345)g(forst\345s)g(en)h(sammenh\346ng)g
(mellem)g(op-)463 4662 y(stilling)k(af)f(hypoteser)-6
b(,)25 b(korr)o(ekt)f(afvikling)h(af)g(testene)f(samt)i(fortolkning)g
(af)e(r)o(esultaterne)463 4777 y([Kragh)c(og)i(Pedersen,)e(1991,)e(s.)j
(53-54].)588 4891 y(N\345r)28 b(der)g(udf\370r)o(es)g(empiriske)h
(fors\370g)g(for)g(at)f(p\345vise)h(eller)f(afvise)g(hypoteser)-6
b(,)29 b(er)f(det)463 5006 y(sj\346ldent)d(at)e(pr)o(oblemomr\345det)g
(er)h(totalt)g(isoler)o(et.)f(Det)h(er)g(derfor)f(n\370dvendigt)h(at)f
(kunne)463 5120 y(afgr\346nse)h(det)h(omr\345de)f(man)h(for)o(etager)e
(eksperimenterne)i(inden)g(for)-6 b(,)25 b(s\345)f(de)h(bliver)f(tek-)
463 5235 y(nisk)17 b(og)g(tidsm\346ssigt)g(mulige)g(at)f(gennemf\370r)o
(e,)f(og)i(samtidig)f(vur)o(der)o(e)e(hvor)o(dan)i(afgr\346ns-)463
5349 y(ningen)22 b(har)e(en)h(ind\003ydelse)g(p\345)f(r)o(esultatet.)
1908 5599 y(63)p eop
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64 65 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 588 192 a(En)16 b(almindelig)i(m\345de)e(at)g(for)o
(etage)g(afgr\346nsninger)h(p\345)f(er)g(ved)g(at)h(g\370r)o(e)f(sig)h
(nogle)h(anta-)463 306 y(gelser)i(om)g(hvor)o(dan)g(de)f(forskellige)i
(dele)e(i)h(eksperimentet)g(h\346nger)h(sig,)f(f.eks.)f(at)h(antage)463
421 y(at)d(de)g(er)g(uafh\346ngige.)h(P\345)f(gr)o(und)h(af)e(den)i
(m\346ngde)g(tid)f(det)g(tager)g(at)g(udf\370r)o(e)f(optr\346ning)j(af)
463 535 y(et)d(neuralt)h(netv\346rk)f(og)h(de)f(begr\346nsede)g(r)o
(esour)o(cer)f(der)h(er)g(til)h(r\345dighed,)e(vil)i(eksempelvis)463
650 y(de)f(test)i(som)f(bliver)g(for)o(etaget)f(i)h(forbindelse)f(med)h
(dette)f(pr)o(ojekt)h(ogs\345)h(blevet)e(opdelt)h(efter)463
765 y(en)k(r\346kke)g(antagelser)-6 b(.)19 b(Da)i(videnskab)g(er)f
(systematisk)i(og)f(gennemskuelig)h(tilegnelse)g(af)463
879 y(viden)k(m\345)g(disse)g(antagelser)g(imidlertid)g(overvejes)g(og)
h(evt.)f(dokumenter)o(es)g(s\345)f(eventu-)463 994 y(elle)e(forbehold)h
(det)f(m\345tte)g(medf\370r)o(e)g(for)g(r)o(esultatet,)f(kan)i
(inddrages)f(i)h(konklusionen)i(p\345)463 1108 y(eksperimentet.)588
1223 y(B\345de)18 b(eksperimentets)i(udformning)h(og)g(fortolkningen)g
(af)e(det)h(bliver)f(pr\346get)h(af)f(den)463 1337 y(forsker)31
b(der)g(udf\370r)o(er)g(eksperimentet.)g(Dels)h(har)f(forskellige)h
(personer)g(forskellige)g(til-)463 1452 y(gangsvinkler)g(og)g(metoder)
-6 b(,)31 b(f.eks.)g(p\345)g(gr)o(und)h(af)f(baggr)o(und,)h
(uddannelse,)e(personlige)463 1567 y(idealer)24 b(og)i(lignende,)g
(dels)f(vil)h(forsker)o(en)f(blive)g(p\345virket)g(at)g(vedkommende)h
(er)f(ved)f(at)463 1681 y(afpr)o(\370ve)18 b(en)i(bestemt)g(teori)g(og)
g(dermed)f(i)h(nogen)h(grad)e(tilpasse)h(r)o(esultatet)f(efter)g(sin)h
(over)o(-)463 1796 y(bevisning)28 b(om)h(hvad)d(udfaldet)h(vil)g(blive)
h(\226)f(man)g(kigger)h(efter)f(bestemt)g(\204tegn\223.)g(Det)g(er)463
1910 y(derfor)20 b(vigtigt)h(at)f(observat\370r)o(en)g(er)g(s\345)g
(objektiv)i(som)f(muligt,)h(og)f(af)f(samme)g(gr)o(und)h(for)o(e-)463
2025 y(tr\346kkes)i(r)o(esultater)e(som)i(afh\346nger)f(mindst)h
(muligt)g(af)f(kontakt)h(med)f(observat\370r)o(en;)g(de)463
2139 y(skal)f(helst)g(kunne)g(kvanti\002cer)o(es)f(til)i(tal.)588
2254 y(I)j(forbindelse)g(med)f(dette)h(pr)o(ojekt)g(er)g(dette)f(et)h
(mindr)o(e)g(pr)o(oblem)g(da)f(r)o(esultaterne)g(af)463
2369 y(fors\370gene)18 b(alle)f(kan)h(opg\370r)o(es)g(i)g(tal)f
(uafh\346ngigt)h(af)f(observat\370r)o(en.)g(Desuden)h(er)f(pr)o(ogram-)
463 2483 y(merne)k(deterministiske)g(s\345)g(alle)f(k\370rsler)h(kan)f
(efterpr)o(\370ves.)588 2598 y(Fortolkningen)h(af)e(eksperimentet)h
(indeb\346r)o(er)e(en)i(analyse)f(af)g(dataene)f(og)i(har)f(til)h(for)o
(-)463 2712 y(m\345l)31 b(at)f(kunne)h(konkluder)o(e)f(om)i
(eksperimentet)e(har)h(eftervist)e(eller)i(modvist)g(teorien.)463
2827 y(Fortolkningen)i(er)d(dermed)f(i)i(endnu)g(h\370jer)o(e)f(grad)g
(end)g(dataene)g(forbundet)g(med)g(teo-)463 2941 y(rien)23
b(hvilket)h(kan)f(r)o(esulter)o(er)e(i)i(fejlagtige)g(konklusioner)i
(hvis)f(dataene)e(og)h(de)g(opstillede)463 3056 y(hypoteser)j(i)f
(virkeligheden)h(ikke)f(h\346nger)h(sammen.)g(Eksempelvis)f(kan)g
(fortolkningen)463 3170 y(indeb\346r)o(e)20 b(en)i(frasortering)f(af)g
(visse)h(dele)f(af)f(dataene)g(hvis)j(de)e(sk\370nnes)h(at)f(v\346r)o
(e)g(forstyr)o(-)463 3285 y(r)o(et)f(af)g(en)h(fejlkilde)f(der)g
(virker)h(uafh\346ngigt)g(af)f(teorien.)588 3400 y(Konklusionen)h(r)o
(epr\346senter)o(er)16 b(det)i(endelige)h(r)o(esultat)f(af)f
(eksperimentet,)i(men)g(for)f(at)463 3514 y(undg\345)28
b(fejl)g(og)g(sikr)o(e)g(gennemskueligheden)h(skal)f(b\345de)f(data,)f
(analyse)i(og)h(fortolkning)463 3629 y(dokumenter)o(es)22
b(s\345dan)g(at)h(en)f(vilk\345rlig)h(person)g(teor)o(etisk)f(kan)h
(gentage)f(pr)o(ocessen)h(og)g(f\345)463 3743 y(samme)d(r)o(esultater)
-6 b(.)18 b(P\345)h(denne)g(m\345de)g(sikr)o(es)h(ogs\345)g(den)f
(tillid)h(der)f(er)g(n\370dvendig)h(at)f(det)g(er)463
3858 y(muligt)j(at)e(br)o(uge)h(hinandens)g(r)o(esultater)-6
b(.)588 3972 y(T)h(il)18 b(dokumentationen)g(af)f(r)o(esultaterne)f(er)
h(der)f(endeligt)i(opbygget)g(et)f(stort)h(system)g(af)463
4087 y(blade,)24 b(b\370ger)i(og)g(konfer)o(encer)e(med)h(en)h(standar)
o(diser)o(et)d(formidlingsform,)j(den)f(viden-)463 4202
y(skabelige)20 b(artikel,)g(der)f(var)o(etager)f(delingen)j(af)e
(viden.)h(Desuden)g(var)o(etager)e(disse)i(publi-)463
4316 y(kationskanaler)f(en)g(del)f(af)g(det)h(sociale)g(pr)o(es)f(for)g
(korr)o(ekt)h(empirisk)g(fr)o(emgangsm\345de)f(ved)463
4431 y(at)i(stille)i(krav)e(til)h(det)f(de)g(publicer)o(er)-6
b(.)463 4739 y FI(6.5)119 b(P\345lidelighed)29 b(og)i(anvendelighed)463
4943 y FY(T)-5 b(il)21 b(den)g(empiriske)g(metode)g(som)g(et)g(hele)g
(h\370r)o(er)f(alts\345:)567 5141 y(1.)41 b(En)27 b(v\346r)o
(diopfattelse)e(der)i(till\346gger)g(betydning)h(til)f(det)g(der)f(kan)
h(erfar)o(es)e(gennem)671 5255 y(sanserne,)20 b(m\345les)h(ved)f(kontr)
o(oller)o(ede)g(fors\370g)h(og)g(ikke)g(mindst)h(efter)o(g\370r)o(es.)
1908 5599 y(64)p eop
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65 66 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 567 192 a(2.)41 b(Et)31 b(s\346t)g(logiske)h(r)o
(egler)f(der)f(accepter)o(es)g(som)i(sande)f(og)h(derved)e(utvetydigt)h
(kan)671 306 y(br)o(uges)21 b(til)g(at)f(r\346sonner)o(e)g(sig)i(fr)o
(em)e(til)h(konklusioner)h(p\345)e(eksperimenterne.)567
478 y(3.)41 b(En)26 b(erkendelse)f(af)g(at)h(disse)g(r)o(egler)f(dog)h
(ikke)g(er)f(nok)i(til)g(at)e(kunne)i(udtale)e(sig)h(om)671
593 y(gener)o(elle)c(sammenh\346nge,)i(og)f(en)g(pragmatisk)g(udvej)g
(i)h(form)f(af)f(en)h(for)o(el\370big)g(ac-)671 707 y(cept)d(af)h(at)g
(en)g(teori)h(er)e(br)o(ugbar)h(n\345r)g(der)g(er)g(blevet)f(gjort)j
(et)e(vist)g(antal)g(fors\370g)h(p\345)e(at)671 822 y(falsi\002cer)o(e)
f(den)h(uden)h(held.)567 993 y(4.)41 b(En)17 b(r\346kke)g
(udv\346lgelseskriterier)g(for)g(hvor)o(dan)g(teorier)g(for)o
(etr\346kkes;)f(f.eks.)g(teorier)671 1108 y(der)i(kan)h(indpasses)g
(godt)g(med)g(andr)o(e)f(teorier)h(inden)g(for)g(det)g(givne)g(felt,)g
(og)g(teorier)671 1223 y(der)h(er)g(simple)h(i)g(der)o(es)f(opbygning)i
(og)f(gener)o(elle)f(i)h(stedet)f(for)h(meget)g(speci\002kke.)567
1394 y(5.)41 b(Nogle)17 b(normer)h(for)f(hvor)o(dan)g(eksperimenterne)g
(skal)g(tilr)o(ettel\346gges,)g(udf\370r)o(es,)f(for)o(-)671
1509 y(tolkes)i(og)h(dokumenter)o(es)f(med)g(henblik)h(p\345)e(at)h
(sikr)o(e)f(at)h(de)g(kan)g(efter)o(g\370r)o(es)e(og)j(er)e(s\345)671
1623 y(alment)k(gyldige)h(som)g(mulige)g(i)g(stedet)f(for)g(f.eks.)g
(udelukkende)g(at)g(afh\346nge)g(af)g(en)671 1738 y(bestemt)28
b(forskers)f(tilgangsvinkel.)i(Samt)e(et)h(distributionssystem)i(der)d
(muligg\370r)671 1853 y(vidensdelingen.)588 2027 y(Betydningen)g(af)f
(den)h(empiriske)h(metode)f(er)f(steget)h(eksplosivt)h(op)g(gennem)f
(histo-)463 2142 y(rien)19 b(i)g(takt)f(med)h(den)g(teknologiske)h
(udvikling)f(fra)f(Galileis)h(f\370rste)g(fors\370g)g(til)g
(situationen)463 2256 y(i)25 b(dag)g(hvor)g(alle)g(teknologiske)h
(landvindinger)f(g\370r)o(es)g(ved)f(hj\346lp)i(af)e(metoden)h(idet)g
(den)463 2371 y(er)e(en)g(integr)o(er)o(et)f(del)h(af)g(den)g
(arbejdsmetode)g(der)f(undervises)i(i)f(p\345)g(f.eks.)g(universiteter)
-6 b(.)463 2485 y(Uden)31 b(den)g(viden)g(som)h(den)f(empiriske)h(fr)o
(emgangsm\345de)e(har)h(skabt,)f(ville)i(konstr)o(uk-)463
2600 y(tion)22 b(af)e(de)g(mange)h(teknologier)h(der)e(anvendes)g(i)h
(dag)f(ikke)h(v\346r)o(e)f(muligt,)i(og)f(det)f(danske)463
2714 y(samfund)d(ville)g(m\345ske)g(stadig)f(v\346r)o(e)g(et)h(landbr)o
(ugssamfund)g(i)g(stil)g(med)g(1500-tallets)d(Dan-)463
2829 y(mark.)588 2943 y(Gr)o(unden)26 b(til)g(metodens)g(succes)g
(p\345)f(det)g(naturvidenskabelige)g(omr\345de)g(kan)h(m\345ske)463
3058 y(forklar)o(es)c(med)i(p\345lideligheden)f(af)g(de)g(enkelte)g(r)o
(esultater)-6 b(,)22 b(dvs.)i(at)f(de)g(er)g(i)h(over)o(enstem-)463
3173 y(melse)e(med)f(virkeligheden.)g(Afsnittet)h(om)g(induktionspr)o
(oblemet)h(viste)e(at)g(r)o(esultaterne)463 3287 y(ikke)j
(n\370dvendigvis)g(er)f(korr)o(ekte)g(i)h(det)f(gener)o(elle)f
(tilf\346lde,)h(men)h(samtidig)g(g\370r)g(den)f(po-)463
3402 y(sitivistiske)k(v\346r)o(diopfattelse)e(der)g(l\346gger)h
(v\346gt)g(p\345)g(f\346nomerne)g(skal)g(kunne)h(m\345les,)e(at)463
3516 y(teorierne)20 b(udformes)h(s\345)f(det)g(er)g(muligt)i(at)e(vise)
g(forskellige)h(specialtilf\346lde)f(af)f(teorierne.)463
3631 y(Dermed)g(opbygges)g(en)g(vis)g(p\345lidelighed)g(inden)g(for)g
(de)f(omr\345der)g(man)h(efterpr)o(\370ver)e(\226)i(og)463
3745 y(selvom)29 b(de)e(for)o(esl\345ede)g(teorier)h(viser)g(sig)g(at)g
(v\346r)o(e)f(forkerte,)g(vil)i(de)e(nye)i(teorier)e(typisk)463
3860 y(vise)21 b(sig)g(at)f(v\346r)o(e)g(i)h(meget)g(god)g(over)o
(enstemmelse)g(med)f(de)h(gamle)f(p\345)g(de)h(omr\345der)f(hvor)463
3975 y(de)g(tidliger)o(e)g(teorier)h(viste)g(sig)g(at)f(v\346r)o(e)g
(konsistente)i(med)e(m\345lingerne.)588 4089 y(Et)k(kendt)h(eksempel)g
(er)g(Newtons)h(mekanik)f(som)h(kan)f(br)o(uges)h(til)f(pr\346cist)g
(at)f(for)o(ud-)463 4204 y(sige)29 b(f.eks.)f(hvor)h(lang)f(tid)h(det)f
(tager)g(for)g(en)h(genstand)g(at)f(tilbagel\346gge)g(en)h
(str\346kning)463 4318 y(under)24 b(p\345virkning)i(af)e(en)h(anden)f
(genstand)h(\226)f(det)h(viste)g(sig)g(sener)o(e)f(at)g(r)o
(esultaterne)g(var)463 4433 y(forkerte)e(ved)g(meget)g(stor)o(e)g
(hastigheder)-6 b(,)22 b(hvorfor)h(teorien)g(m\345tte)f(erstattes)g
(med)g(Einste-)463 4547 y(ins)28 b(specielle)e(r)o(elativitetsteori.)h
(Men)f(denne)h(giver)g(n\370jagtigt)h(samme)f(r)o(esultat)f(for)h
(sm\345)463 4662 y(hastigheder)-6 b(.)588 4777 y(Det)16
b(ses)h(at)f(p\345lideligheden)h(afh\346nger)f(af)g(to)h(ting:)g
(p\345lideligheden)f(af)g(kun)h(at)f(for)o(etage)463
4891 y(\204stikpr)o(\370ver\223)23 b(n\345r)h(en)h(teori)g(testes)g
(inden)g(for)f(et)h(omr\345de,)f(og)h(p\345lideligheden)f(af)g(de)g
(en-)463 5006 y(kelte)h(fors\370g.)g(Sidstn\346vnte)g(s\370ger)g
(normerne)h(for)f(det)f(eksperimentelle)h(arbejde)f(at)h(\370ge)463
5120 y(idet)20 b(de)f(skal)h(sikr)o(e)f(at)g(r)o(esultaterne)g(er)g
(uafh\346ngige)h(af)f(den)h(person)g(der)f(udf\346r)o(diger)f(eks-)463
5235 y(perimentet)i(\226)g(r)o(ent)g(praktisk)h(set)f(fors\370ges)h
(desuden)f(altid)g(at)h(minimer)o(e)f(andr)o(e)f(fejlkilder)463
5349 y(der)h(kunne)h(p\345virke)f(p\345lideligheden,)g(f.eks.)h
(usikker)o(hed)f(p\345)g(m\345leinstr)o(umenterne.)1908
5599 y(65)p eop
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66 67 bop 1446 -129 a FY(Kapitel)20 b(6:)25 b(Empirisk)c(metode)p
463 -96 2975 4 v 588 192 a(At)k(stikpr)o(\370vemetoden)h(er)g
(p\345lidelig,)f(er)h(ikke)g(umiddelbart)f(indlysende,)h(men)h(ikke)463
306 y(desto)g(mindr)o(e)g(som)h(r)o(egel)f(et)g(faktum)g(der)f(er)h
(blevet)g(erfar)o(et)e(\226)i(tilsyneladende)g(opf\370r)o(er)463
421 y(ver)o(den)e(sig)i(tilr)o(egneligt.)g(Desuden,)f(hvis)i
(eksperimenterne)e(involver)o(er)g(s\345)g(komplekse)463
535 y(sammenh\346nge,)e(f.eks.)f(p\345)f(det)h(medicinske)h(omr\345de,)
f(at)g(der)f(mistanke)i(om)g(at)f(der)g(ikke)463 650
y(udf\370r)o(es)29 b(tilstr\346kkeligt)h(mange,)g(er)f(en)h(del)f(af)g
(objektivitetskravet)h(at)f(konklusionerne)463 765 y(ogs\345)21
b(skal)g(v\346r)o(e)e(tilsvar)o(ende)h(forsigtige.)588
879 y(P\345)32 b(omr\345der)g(hvor)h(det)f(er)g(muligt)i(at)e(udf\346r)
o(dige)f(hypoteser)i(der)f(kan)g(testes)h(gen-)463 994
y(nem)23 b(kontr)o(oller)o(ede)f(eksperimenter)-6 b(,)21
b(vil)i(den)f(empiriske)h(metode)g(\226)f(under)g(antagelse)g(af)463
1108 y(at)31 b(eksperimenterne)g(udf\370r)o(es)f(p\345lideligt)g(\226)h
(derfor)f(v\346r)o(e)g(p\345lidelig)h(som)h(videnskabe-)463
1223 y(lig)24 b(metode,)g(ogs\345)g(selvom)g(det)g(ikke)g(er)f(muligt)i
(at)e(opn\345)h(en)g(garanti)g(for)f(at)h(de)f(gener)o(elle)463
1337 y(teorier)d(er)h(gyldige)g(i)g(hele)f(det)h(omr\345de)f(de)g
(postuler)o(es)h(inden)g(for)-6 b(.)463 1598 y Fz(6.5.1)99
b(Anvendelse)24 b(i)h(dette)g(projekt)463 1771 y FY(V)-9
b(ed)20 b(at)g(anvende)g(den)g(empiriske)h(metode)f(med)h
(vekselvirkning)g(mellem)g(teoridannelse)463 1885 y(og)34
b(fors\370g)f(til)h(at)e(skabe)h(viden)g(bliver)g(det)g(muligt)h(at)e
(forbedr)o(e)g(det)h(neurale)f(netv\346rk)463 2000 y(vi)e(har)g(konstr)
o(uer)o(et.)h(Desuden)f(kan)h(metoden)f(hj\346lpe)h(med)f(til)h(at)f
(vur)o(der)o(e)e(hvor)j(godt)463 2114 y(det)g(endelige)g(netv\346rk)g
(er)-6 b(.)30 b(Der)h(synes)h(ikke)f(at)g(v\346r)o(e)f(andr)o(e)g
(muligheder)h(for)g(at)g(opn\345)463 2229 y(tilsvar)o(ende)20
b(r)o(esultater)f(i)i(l\370bet)g(af)f(kort)h(tid.)588
2343 y(Det)i(ville)h(f.eks.)f(ikke)h(v\346r)o(e)e(muligt)j(at)e
(forbedr)o(e)f(netv\346rket)h(gennem)h(samtaler)f(med)463
2458 y(forskellige)e(br)o(uger)o(e)g(der)f(har)h(f\345et)f(hver)h(sit)g
(netv\346rk,)g(dels)g(for)o(di)f(det)h(ville)g(medf\370r)o(e)f(pr)o(o-)
463 2573 y(blemer)25 b(med)h(om)g(udsagnene)f(var)g(p\345lidelige,)g
(dels)g(for)o(di)g(det)g(ikke)h(er)f(praktisk)g(muligt)463
2687 y(at)e(skabe)f(tilstr\346kkelig)i(viden)f(om)h(netv\346rkets)f
(virkem\345de)f(gennem)h(den)g(metode)g(til)g(at)463
2802 y(det)d(er)g(muligt)i(at)e(stille)h(p\345)f(de)g(mulige)h
(parametr)o(e.)e(Der)h(er)g(br)o(ug)h(for)f(en)h(opdeling)g(af)f(pr)o
(o-)463 2916 y(blemet)h(gennem)h(hypoteser)f(om)h(hvor)o(dan)f
(netv\346rket)g(funger)o(er)-6 b(,)19 b(for)i(at)g(det)g(er)f(muligt)i
(at)463 3031 y(opn\345)f(tilstr\346kkeligt)h(detaljer)o(et)d(viden.)588
3145 y(Den)h(empiriske)g(metode)f(giver)h(derimod)g(en)f(ef)o(fektiv)g
(og)h(p\345lidelig)g(l\370sning.)h(T)-5 b(il)20 b(gen-)463
3260 y(g\346ld)27 b(er)f(det)h(ikke)g(sikkert)g(den)f(ville)h(v\346r)o
(e)f(n\346r)h(s\345)g(ef)o(fektiv)e(til)i(at)g(vur)o(der)o(e)d(om)k
(det)e(en-)463 3375 y(delige)21 b(netv\346rk)h(implementer)o(et)f
(p\345)g(en)g(h\345ndholdt)h(ville)g(v\346r)o(e)e(tilfr)o(edsstillende)
h(nok)i(til)463 3489 y(br)o(ugerne,)16 b(ogs\345)g(selvom)h(metoden)f
(kan)g(give)g(pr\346cise)g(tal)g(for)f(f.eks.)h(genkendelsesevnen.)588
3604 y(De)k(belyste)h(metoder)g(til)g(at)f(unders\370ge)h(hypoteser)g
(er)f(f\370lgende:)588 3785 y FS(\017)41 b FY(Slutning)21
b(vha.)f(modus)i(tollens,)f(falsi\002cering.)588 3966
y FS(\017)41 b FY(Slutning)21 b(vha.)f(modus)i(ponens,)f(logisk)h
(slutning.)588 4147 y FS(\017)41 b FY(Poppers)20 b(metode,)h(med)f
(slutning)j(til)e(bedste)f(forklaring.)463 4328 y(Den)g(type)f
(hypoteser)h(der)f(\370nskes)h(afklar)o(et)e(i)h(forbindelse)g(med)h
(eksperimenterne)f(i)h(dette)463 4442 y(pr)o(ojekt,)30
b(vil)h(typisk)g(v\346r)o(e)f(meget)g(speci\002kke,)h(hvorfor)f(modus)i
(ponens)f(kan)f(komme)i(i)463 4557 y(anvendelse.)20 b(Samtidig)h(er)f
(der)g(dog)h(behov)h(for)e(ogs\345)h(at)g(kunne)g(forsvar)o(e)f(mer)o
(e)g(gener)o(elle)463 4671 y(hypoteser)-6 b(,)16 b(f.eks.)g(for)g(at)g
(kunne)h(begr)o(unde)g(de)e(afgr\346nsninger)i(der)f(er)g(for)o(etaget)
f(i)h(testene.)463 4786 y(Hertil)23 b(er)g(princippet)h(om)g(at)f
(benytte)g(den)g(bedste)g(forklaring)h(velegnet.)f(Modus)g(tollens)463
4900 y(kan)e(br)o(uges)g(s\345)f(snart)h(et)f(eksperiment)h(viser)g
(sig)g(at)g(f\345)e(et)i(uventet)g(r)o(esultat.)1908
5599 y(66)p eop
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67 68 bop 463 845 a FJ(Kapitel)44 b(7)463 1298 y FZ(T)-23
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(blevet)h(gennem-)463 1859 y(g\345et)29 b(i)h(kapitel)g(4.)f(En)h(test)
g(kunne)h(f.eks.)e(v\346r)o(e)g(at)g(vi)h(s\370ger)g(afklar)o(et)e(om)j
(en)f(centr)o(ering)463 1974 y(af)22 b(bogstaverne)h(er)f(en)i(god)f
(id\351)f(eller)h(ej.)g(Ud)f(fra)g(disse)h(r)o(esultater)f
(sammens\346tter)h(vi)g(det)463 2088 y(netv\346rk)e(som)g(ud)g(fra)f
(vor)o(es)g(test)h(giver)f(den)h(h\370jeste)g(genkendelsespr)o(ocent.)
588 2203 y(Efter)31 b(at)h(have)g(sammensat)h(dette)f(endelige)g
(netv\346rk)h(udf\370r)o(er)e(vi)h(en)h(fuldst\346ndig)463
2317 y(krydstest)g(og)h(en)f(optr\346ning)h(med)f(forskellige)g
(startv\346r)o(dier)e(for)i(v\346gtene)g(for)g(at)g(be-)463
2432 y(d\370mme)21 b(netv\346rkets)g(endelige)f(genkendelsesevne.)588
2547 y(Med)i(dette)h(netv\346rk)g(s\370ger)h(vi)g(afklar)o(et)d(om)j
(en)g(eftertr\346ning)f(af)g(netv\346rket)g(p\345)g(bog-)463
2661 y(staver)e(fra)g(en)g(isoler)o(et)h(testperson)g(vil)g(forbedr)o
(e)e(netv\346rkets)h(evne)h(til)g(at)f(genkende)h(\003er)o(e)463
2776 y(af)e(vedkommendes)h(bogstaver)-6 b(.)588 2890
y(Her)o(efter)23 b(r)o(edeg\370r)g(vi)i(for)g(om)g(det)g(netv\346rk)g
(vi)g(har)f(valgt)h(ud,)f(er)h(praktisk)g(muligt)g(at)463
3005 y(implementer)o(e)20 b(p\345)g(en)h(h\345ndholdt)h(computer)-6
b(.)588 3119 y(T)h(il)17 b(sidst)h(s\370ger)f(vi)g(afklar)o(et)e(om)i
(vor)o(es)g(testafsnit)g(har)g(levet)f(op)h(til)h(den)f(empiriske)g
(teori)463 3234 y(der)i(er)g(beskr)o(evet)g(i)g(kapitel)h(6,)e(og)j
(hvilke)f(pr)o(oblemer)f(der)f(kan)i(v\346r)o(e)f(med)g(den)g(metode)h
(vi)463 3349 y(benytter)-6 b(.)463 3657 y FI(7.1)119
b(Empiriske)29 b(overvejelser)463 3861 y FY(V)-9 b(or)o(es)32
b(beslutning)i(om)f(at)f(benytte)h(den)f(empiriske)h(metode)g(til)g(at)
f(efterpr)o(\370ve)f(teorien)463 3975 y(medf\370r)o(er)20
b(nogle)i(krav)f(til)g(planl\346gning,)i(udf\370r)o(else,)d
(fortolkning)i(og)g(dokumentation)h(af)463 4090 y(testene.)h(V)-5
b(i)25 b(har)f(derfor)f(opstillet)i(en)f(r\346kke)g(krav)g(til)h(vor)o
(es)e(arbejdsgang)h(og)h(dokumen-)463 4205 y(tation:)588
4402 y FS(\017)41 b FY(Inden)21 b(en)g(test)h(udf\370r)o(es)e(skal)h
(der)g(opstilles)h(en)g(velbegr)o(undet)f(hypotese.)h(I)f(hypote-)671
4517 y(sen)26 b(skal)h(der)e(indg\345)i(hvad)f(testen)g(skal)h(vise)f
(og)h(hvilken)g(tankepr)o(oces)f(der)g(ligger)671 4631
y(bag.)588 4812 y FS(\017)41 b FY(Alle)27 b(udf\370rte)g(test)h(skal)g
(dokumenter)o(es)g(p\345)f(en)h(s\345dan)f(m\345de)g(at)h(det)f(er)h
(muligt)g(at)671 4927 y(efterpr)o(\370ve)18 b(r)o(esultaterne.)588
5108 y FS(\017)41 b FY(Efter)21 b(udf\370r)o(elsen)h(af)h(en)g(test)g
(skal)g(alle)f(v\346sentlige)h(r)o(esultater)f(fr)o(emg\345,)f(og)j
(gyldig-)671 5222 y(hedsomr\345det)c(for)g(hypotesen)i(vur)o(der)o(es.)
1908 5599 y(67)p eop
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68 69 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 588 192 a FS(\017)41 b FY(I)24 b(fortolkningen)j(af)d
(en)h(test)g(skal)g(ar)o(gumentation)g(v\346r)o(e)e(klar)i(s\345)f(der)
h(skelnes)g(mel-)671 306 y(lem)32 b(ar)o(gumenter)f(der)g(br)o(uger)h
(logiske)h(slutninger)-6 b(,)33 b(og)g(ar)o(gumenter)e(der)g(g\345r)h
(p\345)671 421 y(slutning)22 b(til)f(bedste)f(forklaring.)463
729 y FI(7.2)119 b(Praktiske)29 b(begr\346nsninger)463
933 y FY(De)21 b(mange)f(test)h(er)f(meget)g(ber)o(egnings-)h(og)g
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1048 y(gang)32 b(til)g(en)f(super)o(computer)-6 b(,)31
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1162 y(maskiner)19 b(med)g(256)e(MB)h(ram)h(til)g(test.)g(For)g(at)f
(koor)o(diner)o(e)h(de)f(enkelte)h(maskiners)g(arbejde)463
1277 y(konstr)o(uer)o(ede)j(vi)h(et)g(databasesystem)e(som)j(k\370rte)f
(fra)e(basisuddannelsens)i(centrale)f(ser)o(-)463 1391
y(ver)-6 b(.)24 b(Maskinerne)h(\002k)h(fra)e(denne)h(tildelt)h(en)f
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1506 y(tene)c(gemt)g(og)g(log\002len)h(opdater)o(et.)588
1620 y(For)j(at)f(modvirke)h(overtr\346ning)h(som)g(omtalt)f(i)g
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y(ringss\346t)f(som)g(termineringskriterium)h(hvor)e(netv\346rket)g
(fors\370gte)h(sig)g(med)f(maksimalt)463 1850 y(50)d(iterationer)g
(efter)g(det)g(hidtidige)h(minimum)h(for)f(valideringsfejlen)f(blev)h
(opn\345et.)f(Des-)463 1964 y(uden)29 b(blev)g(krydstest)h(benyttet)g
(til)f(at)g(bestemme)h(genkendelsespr)o(ocenten.)f(Desv\346rr)o(e)463
2079 y(tillod)35 b(tiden)f(ikke)g(at)f(vi)h(udf\370rte)f(en)i
(fuldst\346ndig)f(krydstest,)f(s\345)h(i)g(stedet)g(blev)g(kryd-)463
2193 y(stesten)d(udf\370rt)f(med)g(to)h(personer)g(hvor)f(henholdsvis)i
(Anders')e(og)h(Laus)f(tr\346ningsek-)463 2308 y(sempler)19
b(blev)g(udeladt)f(fra)g(optr\346ning.)i(Den)f(endelige)g
(genkendelsespr)o(ocenten)g(blev)g(s\345)463 2422 y(estimer)o(et)28
b(ved)h(at)f(teste)h(med)i FL(x)r FY('s)e(tr\346ningseksempler)g(p\345)
g(det)f(netv\346rk)h(hvor)j FL(x)r FY('s)d(tr\346-)463
2537 y(ningseksempler)23 b(blev)g(udeladt,)e(og)i(til)g(sidst)g(ber)o
(egne)e(gennemsnittet)j(af)d(disse)i(genken-)463 2652
y(delsespr)o(ocenter)-6 b(.)588 2766 y(For)29 b(at)g(sikr)o(e)g(at)g
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h(hvilke)463 2881 y(parametr)o(e)17 b(der)h(passer)g(bedst)h(til)g(et)g
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(es)463 2995 y(\003er)o(e)24 b(test)h(med)f(forskellige)h(startv\346r)o
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(antallet)463 3110 y(af)h(test)h(kraftigt)g(og)h(var)e(s\345ledes)g
(ikke)i(praktisk)f(muligt,)g(undtagen)g(for)g(det)g(afsluttende)463
3224 y(estimat.)h(For)g(at)g(g\370r)o(e)f(testene)h(sammenlignelige)h
(holdes)g(startv\346r)o(dierne)d(for)i(v\346gtene)463
3339 y(ellers)21 b(ens.)588 3454 y(Da)e(nogle)i(af)f(parametr)o(ene)e
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(parame-)463 3568 y(tersammens\346tninger)d(umulig.)g(Derfor)g(bliver)f
(vi)g(n\370dt)h(til)g(at)f(begr\346nse)g(os)h(til)g(en)g(endelig)463
3683 y(m\346ngde)i(parametr)o(e.)d(En)j(del)f(af)g(denne)g
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3797 y(r)o(etisk)i(\226)f(f.eks.)g(giver)h(det)g(ikke)g(mening)h(at)e
(s\346tte)h(inertikonstanten)i(til)e(v\346r)o(dier)e(uden)i(for)463
3912 y(intervallet)26 b FG([)q FY(0;)14 b(1)q FG([)25
b FY(\226)g(og)h(ud)g(fra)e(dette)h(udv\346lges)g(nogle)h(rimelige)g
(v\346r)o(dier)e(p\345)h(baggr)o(und)463 4026 y(af)20
b(gr)o(uppens)h(erfaringer)f(eller)g(forslag)h(i)f(litteratur)o(en.)588
4141 y(Selv)28 b(med)i(nogle)g(f\345)e(udvalgte)h(v\346r)o(dier)f(er)h
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4370 y(de)f(forskellige)h(parametr)o(e)e(ikke)i(er)g(korr)o(eler)o(ede)
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4485 y(de)h(andr)o(e)g(parametr)o(e)e(fastholdt.)463
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b(opstiller)h(f\370lgende)e(hypoteser:)588 5349 y FS(\017)41
b FY(Centr)o(ering)20 b(forbedr)o(er)f(genkendelsespr)o(ocenten.)1908
5599 y(68)p eop
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69 70 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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(e)f(optr\346ning)671 306 y(med)20 b(en)h(lignende)g(genkendelsespr)o
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(netv\346rk.)588 668 y FS(\017)41 b FY(Konjuger)o(et)17
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(gradientnedstigning)i(med)e(en)671 783 y(lignende)k(genkendelsespr)o
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1145 y FS(\017)41 b FY(En)20 b(h\370j)i(ekstern)f(skaleringsgrad)f
(forringer)h(genkendelsespr)o(ocenten.)588 1326 y FS(\017)41
b FY(Intern)26 b(skalering)g(forbedr)o(er)d(genkendelsespr)o(ocenten)j
(i)g(for)o(hold)g(til)g(ekstern)f(ska-)671 1440 y(lering.)588
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1736 y(metoder)-6 b(.)588 1917 y FS(\017)41 b FY(St\370rr)o(elsen)19
b(af)h(valideringss\346ttet)h(har)f(ind\003ydelse)i(p\345)e
(genkendelsesevnen)588 2098 y(De)i(enkelte)h(test)f(er)g(opbygget)h
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(og)463 2212 y(til)d(sidst)f(fortolkes)h(r)o(esultaterne.)e(Denne)h
(opbygning)i(sker)e(for)g(at)g(skabe)g(et)g(overblik)g(over)463
2327 y(de)h(enkelte)h(r)o(esultater)e(og)j(konsekvenserne)f(heraf.)463
2587 y Fz(7.3.1)99 b(Ordforklaring)p 474 2686 2952 7
v 524 2783 a FU(Forkortelse)e(Forklaring)p 474 2833 2952
5 v 524 2930 a(Centr)-6 b(.)274 b(Hvis)18 b(\204til\223,)g(er)f
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(inden)g(bil-)997 3035 y(ledet)g(br)o(uges)g(som)g(input.)524
3140 y(Skal.)314 b(1:)r FH(x)22 b FU(angiver)d(graden)h(af)g(den)f
(eksterne)g(skalering)f(der)h(er)g(for)o(etaget)g(som)h(en)g(ek-)997
3245 y(stern)e(forbehandling.)524 3350 y(Neur)-6 b(.)287
b(Antal)19 b(neur)o(oner)f(i)h(det)f(skjulte)h(lag.)524
3455 y(Akt.fkt.)220 b(Aktiveringsfunktionen)18 b(i)g(neur)o(onerne,)g
(enten)i(tanh)g(eller)d(sigmoid.)524 3560 y(T)-7 b(r\346ning)197
b(T)-7 b(r\346ningsmetoden,)17 b(enten)j(konjuger)o(et)d(gradient)i
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b(genkendte)h(bogstaver)f(fra)h(det)g(tr\346ningss\346t)g(som)f
(netv\346rket)h(ikke)997 3770 y(er)h(blevet)h(optr\346net)f(p\345.)524
3875 y(Gr)-6 b(.)18 b(neds.)179 b(T)-7 b(r\346ning)18
b(med)g(gradientnedstigning)g(med)g(inerti.)524 3980
y(Konj.)h(gr)-6 b(.)195 b(T)-7 b(r\346ning)18 b(med)g(skaler)o(et)f
(konjuger)o(et)h(gradient.)524 4085 y(V)-10 b(.str)k(.)311
b(Pr)o(ocentdelen)18 b(af)h(tr\346ningss\346ttet)f(der)g(er)g(anvendt)i
(som)e(valideringss\346t.)525 4190 y FA(h)433 b FU(T)-7
b(r\346ningskonstanten)19 b(til)f(gradientnedstigning.)525
4295 y FA(m)428 b FU(Inertikonstanten)20 b(der)d(tr\346nes)i(med.)524
4400 y(Ant.)g(iter)-6 b(.)185 b(Antallet)18 b(af)g(genneml\370b)g
(tr\346ningsalgoritmen)e(for)o(etager)g(over)h(hele)h(tr\346nings-)997
4505 y(s\346ttet)h(inden)f(terminering.)524 4610 y(Seed)316
b(Den)19 b(v\346r)o(di)e(gener)o(erator)o(en)f(af)j(tilf\346ldige)e
(tal)i(startes)f(med.)524 4715 y(T)-7 b(estperson)115
b(Den)18 b(person)g(hvis)h(bogstaver)f(isoler)o(es)e(under)i(selve)f
(optr\346ningen)i(og)f(efterf\370l-)997 4820 y(gende)g(br)o(uges)g(til)
g(at)i(bed\370mme)d(netv\346rkets)h(genkendelsespr)o(ocent.)p
474 4872 2952 7 v 1908 5599 a FY(69)p eop
%%Page: 70 72
70 71 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 463 192 a Fz(7.3.2)99 b(Opstilling)26
b(af)e(netv\346rk)463 365 y FY(Netv\346rket)18 b(opbygges)i(som)g(et)f
(fuldt)f(forbundet)h(fr)o(emadf\370r)o(ende)d(netv\346rk)j
(best\345ende)f(af)463 479 y(et)28 b(inputlag,)g(et)f(skjult)i(lag)f
(og)g(et)g(outputlag.)g(Antallet)f(af)g(neur)o(oner)h(i)g(inputlaget)g
(kan)g(i)463 594 y(vor)o(es)23 b(opstilling)j(antage)d(4)h(st\370rr)o
(elser:)f(2880)f(uden)i(ekstern)g(skalering,)f(720)g(med)h(1:2)e(ek-)
463 708 y(stern)j(skalering,)f(180)f(med)h(1:4)f(ekstern)i(skalering)g
(eller)f(80)f(med)h(1:6)g(ekstern)g(skalering.)463 823
y(Skaleringsgraden)c(er)g(angivet)h(i)f(r)o(esultattabellerne.)f
(Antallet)h(af)g(skjulte)i(neur)o(oner)e(vari-)463 938
y(er)o(er)15 b(mellem)j(50-200)c(og)j(er)f(ligeledes)h(angivet)g(under)
f(hver)h(hypotese.)g(Antallet)g(af)f(output)463 1052
y(neur)o(oner)24 b(er)g(altid)g(29.)f(V)-9 b(ed)24 b(intern)g(og)h(r)o
(egionsskalering)g(benyttes)g(dog)f(en)h(anden)f(arki-)463
1167 y(tektur)-6 b(,)20 b(som)i(beskr)o(evet)d(i)i(afsnit)g(4.3.2.)588
1281 y(Hvis)31 b(intet)g(andet)g(er)f(angivet,)h(benyttes)g(de)f
(f\370lgende)h(v\346r)o(dier)f(for)g(de)h(forskellige)463
1396 y(parametr)o(e:)p 1082 1517 1736 7 v 1132 1614 a
FU(Parameter)116 b(V\346r)o(di)p 1082 1664 1736 5 v 1132
1761 a(Centr)-6 b(.)258 b(T)l(il)1132 1866 y(Akt.fkt.)204
b(tanh)1132 1971 y(T)-7 b(r\346ning)181 b(Gradientnedstigning)1133
2076 y FA(h)417 b FU(0,3)1133 2181 y FA(m)412 b FU(0,1)1132
2286 y(V)-10 b(.str)k(.)295 b(10\045)1132 2391 y(Seed)300
b(1)1132 2496 y(T)-7 b(estperson)99 b(Gennemsnit)19 b(mellem)f(Lau)h
(og)f(Anders)p 1082 2548 1736 7 v 463 2778 a Fz(7.3.3)99
b(Hypotese:)25 b(Centrering)g(forbedrer)f(genkendelsesprocenten)463
2951 y FY(V)-5 b(i)21 b(\370nsker)g(efterpr)o(\370vet)e(om)j(centr)o
(ering)e(forbedr)o(er)f(genkendelsespr)o(ocenten.)588
3066 y(Baggr)o(unden)g(for)h(at)f(opstille)i(hypotesen)f(er)g(at)f(en)h
(centr)o(ering)f(vil)h(ensarte)g(billederne,)463 3180
y(hvilket)i(for)g(netv\346rket)g(formodentlig)g(vil)h(virke)e(som)i(et)
f(st\370rr)o(e)f(antal)g(tr\346ningseksemp-)463 3295
y(ler)29 b(for)g(de)g(v\346gte)g(der)g(ligger)g(omkring)h
(tr\346ningseksemplets)h(centr)o(um.)f(Hvis)f(der)g(var)463
3410 y(tilstr\346kkeligt)j(mange)e(tr\346ningseksempler)-6
b(,)32 b(bur)o(de)d(centr)o(eringen)h(v\346r)o(e)g(un\370dvendig,)463
3524 y(men)21 b(dette)f(er)h(n\346ppe)f(tilf\346ldet.)588
3639 y(Da)e(en)h(for)o(\370gelse)g(af)f(tr\346ningss\346ttet)i(gener)o
(elt)e(vil)h(modvirke)g(overtr\346ning)h(som)g(er)e(et)463
3753 y(pr)o(oblem)k(for)h(alle)f(de)g(opstillede)h(netv\346rk)g(for)o
(di)e(antallet)i(af)f(v\346gte)g(er)g(stort)h(i)g(for)o(hold)g(til)463
3868 y(antallet)18 b(af)g(tr\346ningseksempler)-6 b(,)19
b(bur)o(de)e(tendensen)i(ogs\345)g(v\346r)o(e)e(ens)i(for)f(alle)g
(netv\346rk.)g(V)-5 b(i)463 3982 y(tester)20 b(derfor)g(kun)i(centr)o
(ering)e(med)g(et)h(enkelt)g(netv\346rk.)463 4223 y FT(Resultater)p
1137 4298 1627 7 v 1187 4401 a FY(Centr)-6 b(.)99 b(Skal.)f(Neur)-6
b(.)99 b(T)-7 b(r\346ning)100 b(Genk.)p 1137 4454 1627
5 v 1237 4557 a(Fra)186 b(1:2)196 b(50)166 b(Konj.)22
b(gr)108 b(54,7\045)1250 4672 y(T)-5 b(il)200 b(1:2)c(50)166
b(Konj.)22 b(gr)108 b(76,0\045)p 1137 4727 1627 7 v 463
4945 a FT(Fortolkning)463 5118 y FY(Netv\346rkets)20
b(genkendelsespr)o(ocent)h(er)f(alts\345)g(forbedr)o(et)f(ved)h(br)o
(ug)g(af)g(centr)o(ering)g(i)h(oven-)463 5233 y(st\345ende)30
b(testscenarie.)g(Ud)g(fra)g(testen)h(kan)f(man)h(ikke)g(med)g(sikker)o
(hed)f(konkluder)o(e)g(at)463 5347 y(forbedringen)19
b(i)h(genkendelsespr)o(ocenten)f(skyldes)h(at)f(centr)o(eringen)g(er)g
(\346kvivalent)g(med)1908 5599 y(70)p eop
%%Page: 71 73
71 72 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 463 192 a(en)22 b(for)o(\370gelse)g(af)f
(tr\346ningseksemplerne,)i(men)g(dette)e(synes)i(at)f(v\346r)o(e)f(den)
h(bedste)g(forkla-)463 306 y(ring.)588 421 y(V)-9 b(ed)20
b(at)i(slutte)f(til)h(denne)g(f\345s)f(at)g(centr)o(ering)g(ogs\345)h
(vil)g(forbedr)o(e)e(genkendelsespr)o(ocen-)463 535 y(ten)h(p\345)f
(andr)o(e)f(netv\346rk)i(under)g(antagelse)f(af)g(at)g(disse)h(ogs\345)
g(bliver)f(overtr\346net.)463 796 y Fz(7.3.4)99 b(Hypotese:)34
b(T)-11 b(anh)32 b(som)i(aktiveringsfunktion)f(fremfor)f(sigmoid)762
929 y(giver)e(hurtigere)g(optr\346ning)g(med)f(en)g(lignende)h
(genkendelses-)762 1063 y(procent)463 1236 y FY(Det)20
b(\370nskes)g(afgjort)g(hvilken)h(aktiveringsfunktion)g(der)e(giver)g
(den)h(hurtigste)g(optr\346ning)463 1350 y(af)25 b(netv\346rket.)h
(Samtidig)g(\370nskes)h(efterpr)o(\370vet)e(om)i(valget)e(af)h
(aktiveringsfunktion)h(kun)463 1465 y(giver)21 b(en)f(mindr)o(e)h
(forskel)g(i)f(genkendelsespr)o(ocenten.)588 1580 y(Motivationen)g(for)
f(hypotesen)h(kommer)h(fra)d(litteratur)o(en,)h(bl.a.)f([Bishop,)i
(1995;)d(Sarle,)463 1694 y(1999],)f(hvor)i(det)g(fr)o(emg\345r)f(at)h
(br)o(ug)h(af)e(tanh)i(som)g(aktiveringsfunktion)h(erfaringsm\346ssigt)
463 1809 y(giver)31 b(hurtiger)o(e)g(konver)o(gens)h(end)f(sigmoid.)h
(Hvis)g(tanh)f(viser)g(sig)h(at)f(v\346r)o(e)g(hurtiger)o(e)463
1923 y(end)18 b(sigmoid,)i(bliver)e(det)g(muligt)i(at)e(teste)g(\003er)
o(e)g(netv\346rk)g(p\345)g(den)h(tid)f(der)g(er)g(til)h(r\345dighed,)
463 2038 y(hvilket)i(indir)o(ekte)f(kan)h(hj\346lpe)g(med)g(til)g(at)f
(\002nde)h(et)g(bedr)o(e)e(netv\346rk.)588 2152 y(Under)h(testen)h
(blev)h(m\345lv\346r)o(dierne)d(for)j(sigmoid)g(naturligvis)g(\346ndr)o
(et)e(til)i(0)q(,)q(1)e(og)i(0)q(,)q(9)e(i)463 2267 y(stedet)g(for)i
FS(\000)q FY(0)q(,)q(9)d(og)i(0)q(,)q(9)f(som)i(benyttes)f(ved)f(tanh.)
463 2508 y FT(Resultater)p 478 2582 2944 7 v 528 2686
a FY(T)-8 b(estperson)100 b(Ant.)21 b(iter)-6 b(.)99
b(Centr)-6 b(.)99 b(Skal.)g(Neur)-6 b(.)108 b(Akt.fkt.)i(T)-7
b(r\346ning)100 b(Genk.)p 478 2739 2944 5 v 655 2842
a(Lau)324 b(237)246 b(Fra)186 b(1:2)196 b(50)228 b(tanh)166
b(Konj.)22 b(gr)-6 b(.)101 b(63,3\045)655 2957 y(Lau)324
b(335)246 b(Fra)186 b(1:2)196 b(50)161 b(sigmoid)100
b(Konj.)22 b(gr)-6 b(.)101 b(64,0\045)591 3071 y(Anders)259
b(220)246 b(Fra)186 b(1:2)196 b(50)228 b(tanh)166 b(Konj.)22
b(gr)-6 b(.)101 b(46,1\045)591 3186 y(Anders)259 b(388)246
b(Fra)186 b(1:2)196 b(50)161 b(sigmoid)100 b(Konj.)22
b(gr)-6 b(.)101 b(46,2\045)p 478 3241 2944 7 v 463 3459
a FT(Fortolkning)463 3632 y FY(Det)19 b(ses)g(at)g(tanh)g(omtr)o(ent)g
(giver)g(den)f(samme)h(genkendelsespr)o(ocent)g(som)h(sigmoid,)g(men)
463 3747 y(at)g(sigmoid)h(skal)f(br)o(uge)h(henh.)f(41\045)f(og)i
(76\045)e(\003er)o(e)g(iterationer)-6 b(.)20 b(Alts\345)g(kan)g(vi)g
(konkluder)o(e)463 3861 y(at)d(hypotesen)g(har)g(vist)g(sig)h(at)e
(v\346r)o(e)g(rigtig)i(for)f(disse)g(speci\002kke)g(tilf\346lde.)f(At)h
(tanh)g(i)g(begge)463 3976 y(tilf\346lde)26 b(giver)g(en)h(lidt)f
(laver)o(e)f(genkendelsespr)o(ocent,)h(kan)h(tilskrives)g
(tilf\346ldigheder)e(\226)463 4090 y(det)17 b(ville)g(v\346r)o(e)f
(n\370dvendigt)h(med)g(\003er)o(e)f(test)h(for)g(at)g(afg\370r)o(e)e
(dette,)i(men)g(da)f(forskellen)i(er)e(s\345)463 4205
y(lille,)i(har)g(vi)f(ikke)i(anset)e(dette)h(for)f(ber)o(ettiget;)g(r)o
(esultatet)g(er)g(klart)h(med)f(hensyn)i(til)f(antallet)463
4320 y(af)i(iterationer)-6 b(.)588 4434 y(Da)24 b(vi)g(ikke)g(kan)h
(falsi\002cer)o(e)d(hypotesen)j(som)h(i)e(\370vrigt)g(underst\370ttes)h
(af)e(erfaringerne)463 4549 y(fra)d(litteratur)o(en,)f(v\346lger)h(vi)h
(derfor)e(at)i(konkluder)o(e)f(dens)g(for)o(el\370bige)g(korr)o(ekthed)
g(og)h(f\370l-)463 4663 y(ger)f(den)h(ved)f(at)h(br)o(uge)f(tanh)h(i)g
(det)f(f\370lgende)h(for)f(at)h(kunne)g(udf\370r)o(e)f(\003est)h
(mulige)g(test.)463 4924 y Fz(7.3.5)99 b(Hypotese:)35
b(V)-11 b(alget)34 b(af)h Fp(h)k Fz(og)d Fp(m)g Fz(har)e(stor)h
(betydning)g(for)g(et)f(givet)762 5057 y(netv\346rk)463
5230 y FY(Det)18 b(\370nskes)h(fastsl\345et)f(om)h(v\346r)o(dierne)e
(af)g(tr\346ningskonstanten)j(og)f(inertikonstanten)h(ved)463
5345 y(gradientnedstigning)g(har)g(nogen)g(st\370rr)o(e)f(betydning)h
(for)f(antallet)h(af)e(iterationer)i(og)g(st\370r)o(-)1908
5599 y(71)p eop
%%Page: 72 74
72 73 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 463 192 a(r)o(elsen)18 b(af)g(genkendelsespr)o
(ocenten)h(for)f(det)g(f\346r)o(dige)f(netv\346rk.)i(Desuden)g
(\370nskes)g(en)g(god)463 306 y(v\346r)o(di)h(for)g(konstanterne)i
(bestemt.)588 421 y(If\370lge)d(den)f(teor)o(etiske)h(opfattelse)g(af)f
(hvor)o(dan)h(tr\346ningen)h(forl\370ber)e(b\370r)h(de)g(optimale)463
535 y(v\346r)o(dier)27 b(af)i FN(h)k FY(og)d FN(m)h FY(v\346r)o(e)c
(afh\346ngige)i(af)f(hvor)o(dan)g(fejlr)o(ummet)i(ser)e(ud,)h(og)g(da)f
(enhver)463 650 y(\346ndring)18 b(i)f(netv\346rkets)h(parametr)o(e)d
(giver)i(et)h(nyt)f(fejlr)o(um,)h(forventer)f(vi)g(ikke)h(at)f(r)o
(esultatet)463 765 y(fra)30 b(denne)h(test)g(vil)h(stemme)g
(n\370jagtigt)g(over)o(ens)e(med)h(r)o(esultatet)g(fra)f(andr)o(e)g
(test.)h(Hvis)463 879 y(parametr)o(ene)23 b(ikke)j(\346ndr)o(es)e(for)h
(drastisk)g(\226)g(f.eks.)f(ved)h(at)g(skifte)g(aktiveringsfunktion)h
(\226)463 994 y(b\370r)19 b(de)g(lokale)h(minima)g(i)g(fejlr)o(ummet)g
(dog)g(v\346r)o(e)e(af)h(nogenlunde)h(af)f(samme)h(st\370rr)o(else)f
(og)463 1108 y(udstr\346kning,)29 b(s\345)f(parameterv\346r)o(dierne)e
(fundet)i(i)g(denne)h(test)f(stadig)g(er)g(fornuftige)h(til)463
1223 y(andr)o(e)19 b(test.)588 1337 y(Meningsfulde)h(v\346r)o(dier)g
(for)h(inertikonstanten)h(ligger)g(i)f(intervallet)g(0)j
FS(\024)h FN(m)h FM(<)e FY(1.)d(Hvis)463 1452 y(inertikonstanten)i(er)f
(st\370rr)o(e)f(end)h(eller)g(lig)g(med)g(1,)g(vil)g(optr\346ningen)h
(aldrig)e(konver)o(ger)o(e,)463 1567 y(og)g(hvis)g(den)f(ligger)h
(under)f(0,)g(vil)h(man)f(modarbejde)g(optr\346ningen)h(og)g(g\345)g
(v\346k)f(fra)f(mini-)463 1681 y(mum.)28 b(Ligeledes)g(skal)h
FN(h)i FY(v\346r)o(e)c(st\370rr)o(e)g(end)h(0.)f(V)-5
b(i)29 b(har)e(yderliger)o(e)g(erfar)o(et)f(at)h(fornuftige)463
1796 y(v\346r)o(dier)19 b(ligger)i(omkring)h(0)q(,)q(1.)463
2037 y FT(Resultater)c(med)j FN(h)p 887 2133 2126 7 v
937 2237 a FY(Ant.)f(iter)-6 b(.)100 b(Skal.)e(Neur)-6
b(.)123 b(T)-7 b(r\346ning)152 b FN(h)133 b(m)101 b FY(Genk.)p
887 2289 2126 5 v 1014 2393 a(2623)211 b(1:2)196 b(50)160
b(Gr)-6 b(.)21 b(nedst.)99 b(0,1)k(0)i(79,3\045)1014
2507 y(1338)211 b(1:2)196 b(50)160 b(Gr)-6 b(.)21 b(nedst.)99
b(0,3)k(0)i(76,8\045)p 887 2563 2126 7 v 463 2781 a FT(Fortolkning)463
2954 y FY(Det)23 b(ses)g(at)f(v\346r)o(dien)g(af)h FN(h)j
FY(har)d(stor)g(betydning)g(for)g(b\345de)e(antallet)i(af)e
(iterationer)i(og)g(gen-)463 3068 y(kendelsespr)o(ocenten)f(og)g(at)g
(0)q(,)q(1)f(gav)h(bedst)f(genkendelsespr)o(ocent.)h(Dette)g(kan)g
(forklar)o(es)463 3183 y(med)h(at)g(jo)i(mindr)o(e)e
(v\346gtopdateringer)g(der)f(for)o(etages)h(pr)-6 b(.)22
b(iteration,)i(jo)g(dyber)o(e)e(i)i(det)f(lo-)463 3297
y(kale)j(minima)h(kan)f(algoritmen)h(n\345.)e(Det)i(ses)f(desuden)g(at)
g(algoritmen)g(ved)g(v\346r)o(dien)f(0)q(,)q(1)463 3412
y(skulle)c(br)o(uge)g(omkring)h(dobbelt)f(s\345)f(mange)h(iterationer)f
(for)h(at)f(n\345)h(dette)f(minimum.)588 3527 y(Ovenst\345ende)26
b(r)o(esultater)g(antyder)g(at)h(en)g(mindr)o(e)g(tr\346ningskonstant)i
(altid)e(vil)g(give)463 3641 y(en)20 b(h\370jer)o(e)f(genkendelsespr)o
(ocent,)h(men)g(man)f(kan)h(for)o(estille)f(sig)i(at)e(optr\346ning)h
(med)g(sm\345)463 3756 y(v\346r)o(dier)k(af)g(tr\346ningskonstanten)j
(vil)e(have)g(en)g(tendens)g(til)g(at)g(havne)g(i)g(selv)g(sm\345)g
(lokale)463 3870 y(minima.)d(Dette)e(er)h(dog)g(ikke)g(en)h(konklusion)
h(der)d(kan)h(uddrages)f(af)g(ovenst\345ende)h(r)o(esul-)463
3985 y(tater)-6 b(.)463 4226 y FT(Resultater)18 b(med)j
FN(m)p 861 4322 2179 7 v 910 4426 a FY(Ant.)g(iter)-6
b(.)99 b(Skal.)g(Neur)-6 b(.)122 b(T)-7 b(r\346ning)152
b FN(h)160 b(m)127 b FY(Genk.)p 861 4478 2179 5 v 987
4582 a(2008)211 b(1:2)196 b(50)161 b(Gr)-6 b(.)20 b(nedst.)99
b(0,1)g(0,1)h(78,7\045)1008 4696 y(645)232 b(1:2)196
b(50)161 b(Gr)-6 b(.)20 b(nedst.)99 b(0,1)g(0,5)h(78,0\045)1008
4811 y(380)232 b(1:2)196 b(50)161 b(Gr)-6 b(.)20 b(nedst.)99
b(0,1)g(0,9)h(77,7\045)p 861 4864 V 1008 4967 a(606)232
b(1:2)196 b(50)161 b(Gr)-6 b(.)20 b(nedst.)99 b(0,3)g(0,1)h(78,6\045)
1008 5082 y(275)232 b(1:2)196 b(50)161 b(Gr)-6 b(.)20
b(nedst.)99 b(0,3)g(0,5)h(75,4\045)1029 5196 y(53)253
b(1:2)196 b(50)161 b(Gr)-6 b(.)20 b(nedst.)99 b(0,3)g(0,9)h(19,7\045)p
861 5252 2179 7 v 1908 5599 a(72)p eop
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1052 y(som)22 b(endepunkt.)588 1167 y(For)17 b(begge)g(v\346r)o(dier)f
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74 75 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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765 y(ent)29 b(kan)f(ogs\345)h(forklar)o(e)e(hvorfor)h
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1108 y(person.)h(LeCun)g(et)g(al.)g([1998])e(angiver)i(et)g(lignende)g
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3050 y(kan)f(dette)f(interval)g(ikke)h(speci\002cer)o(es)f(yderliger)o
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5599 y(74)p eop
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76 77 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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b(81,9\045)1172 2761 y(1:4)175 b(200)282 b(83,3\045)357
b(82,2\045)1172 2875 y(1:6)175 b(200)282 b(77,7\045)357
b(78,7\045)p 1085 2931 1730 7 v 463 3149 a FT(Fortolkning)463
3322 y FY(Det)22 b(ses)f(at)g(hypotesen)i(er)e(falsi\002cer)o(et)f(da)g
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(genkendelsespr)o(ocent)h(kan)g(m\345ske)g(skyl-)463
3551 y(des)17 b(at)g(de)g(\003er)o(e)g(v\346gte)h(giver)f(st\370rr)o(e)
g(risiko)h(for)f(overtr\346ning.)h(Da)g(intern)g(skalering)f(under)463
3665 y(alle)23 b(omst\346ndigheder)i(giver)e(langsommer)o(e)h
(tr\346ning)g(for)o(di)f(de)g(mindr)o(e)h(eksempelst\370r)o(-)463
3780 y(r)o(elser)i(ved)h(ekstern)g(skalering)h(neds\346tter)f(pr)o
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4124 y(for)16 b(intern)g(skalering)g(idet)f(genkendelsespr)o(ocenten)h
(falder)e(med)i(st\370rr)o(e)f(skalering)h(selvom)463
4238 y(1:4)k(skalering)h(i)f(to)i(af)d(tilf\346ldene)i(faktisk)g(gav)f
(bedr)o(e)f(genkendelsespr)o(ocent)i(end)f(1:2.)463 4498
y Fz(7.3.10)99 b(Hypotese:)28 b(En)f(regionsskalering)h(giver)g(bedre)f
(genkendelses-)812 4632 y(procent)e(end)f(andre)g(skaleringsmetoder)463
4805 y FY(\330nsket)c(er)f(at)g(efterpr)o(\370ve)f(om)i(en)f(skalering)
h(der)f(er)g(tilpasset)g(den)g(typiske)h(informations-)463
4920 y(for)o(deling)g(i)g(et)h(tr\346ningseksempel,)g(forbedr)o(er)d
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5034 y(andr)o(e)e(skaleringsmetoder)-6 b(.)588 5149 y(Motivationen)22
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o(oblemet)g(med)g(over)o(-)1908 5599 y(76)p eop
%%Page: 77 79
77 78 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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b(,)18 b(kan)463 306 y(disse)30 b(ikke)f(dir)o(ekte)g(sammenlignes)h
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(dem)i(efter)f(sammenlignelige)i(antal)e(af)g(v\346gte.)463
662 y FT(Resultater)p 794 736 2312 7 v 958 840 a FY(Skal.)213
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b(.)99 b(Genkend.)p 794 892 2312 5 v 844 996 a(Ekstern)21
b(1:4)209 b(10.450)334 b(Lau)288 b(50)231 b(89,3\045)872
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b(50)231 b(87,0\045)p 794 1278 V 844 1381 a(Ekstern)21
b(1:4)209 b(20.900)334 b(Lau)267 b(100)210 b(88,6\045)872
1496 y(Intern)21 b(1:4)237 b(23.780)334 b(Lau)267 b(100)210
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b(1:4)209 b(41.800)334 b(Lau)267 b(200)210 b(90,2\045)872
1881 y(Intern)21 b(1:4)237 b(44.680)334 b(Lau)267 b(200)210
b(93,5\045)917 1996 y(Region)284 b(54.680)334 b(Lau)267
b(200)210 b(90,6\045)p 794 2051 2312 7 v 463 2269 a FT(Fortolkning)463
2442 y FY(Det)28 b(ses)f(at)g(hypotesen)i(er)e(falsi\002cer)o(et)f(da)h
(r)o(egionsskalering)h(i)f(ingen)i(af)d(tilf\346ldene)h(gav)463
2557 y(den)19 b(bedste)g(genkendelsespr)o(ocent,)f(men)i(altid)f
(ligger)g(mellem)h(intern)g(og)f(ekstern)h(skale-)463
2671 y(ring.)j(Resultaterne)g(kan)g(tolkes)h(som)f(at)g(noget)h(af)e
(den)h(omkringliggende)h(information)g(i)463 2786 y(nogle)h
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(m\345de)f(at)h(for)o(etage)e(r)o(egionsska-)463 2900
y(lering)i(p\345)g(er)f(alts\345)h(ikke)g(den)g(optimale)g(m\345de)g
(at)f(skaler)o(e)g(p\345,)g(og)i(vi)f(v\346lger)g(derfor)f(ikke)463
3015 y(at)d(forts\346tte)h(med)f(denne.)463 3275 y Fz(7.3.11)99
b(Hypotese:)35 b(St\370rrelsen)f(af)f(valideringss\346ttet)i(har)f
(ind\003ydelse)812 3409 y(p\345)24 b(genkendelsesevnen)463
3582 y FY(\330nsket)h(er)f(at)g(afg\370r)o(e)e(om)j(andelen)f(af)g
(data)f(der)g(br)o(uges)i(til)g(valideringss\346t,)f(har)g(ind\003y-)
463 3696 y(delse)c(p\345)h(genkendelsesevnen.)588 3811
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(br)o(uges)h(til)g(at)f(estimer)o(e)g(ge-)463 3926 y
(neraliseringsevnen)i(undervejs)h(med)e(henblik)i(p\345)f(at)f
(afg\370r)o(e)g(hvorn\345r)h(tr\346ningen)h(skal)463
4040 y(stoppe.)i(Hvis)g(valideringss\346ttet)g(ikke)h(er)e(omfattende)h
(nok,)h(vil)f(estimatet)g(ikke)g(v\346r)o(e)463 4155
y(korr)o(ekt,)20 b(og)h(tr\346ningen)h(kan)f(derfor)e(blive)i(stoppet)g
(enten)g(for)g(tidligt)g(eller)f(for)h(sent.)588 4269
y(Men)d(n\345r)g(st\370rr)o(elsen)h(af)e(valideringss\346ttet)i(for)o
(\370ges,)f(formindskes)h(st\370rr)o(elsen)g(af)f(tr\346-)463
4384 y(ningss\346ttet)i(samtidig.)f(Derved)f(for)o(\370ges)h
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4498 y(r)o(else)h(af)g(validerings\346ttet)g(er)h(derfor)e(en)i
(afvejning)g(af)f(de)g(to)i(f\346nomener)-6 b(.)463 4740
y FT(Resultater)p 1361 4814 1179 7 v 1410 4917 a FY(V)-11
b(.str)-6 b(.)100 b(Skal.)e(Neur)-6 b(.)99 b(Genk.)p
1361 4970 1179 5 v 1424 5073 a(10\045)148 b(1:2)196 b(50)162
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b(77,7\045)p 1361 5358 1179 7 v 1908 5599 a(77)p eop
%%Page: 78 80
78 79 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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(esultater)g(idet)h(genkendelses-)463 479 y(evnen)31
b(er)g(skiftet)g(med)f(st\370rr)o(elsen)h(af)f(valideringss\346ttet.)h
(Forskellene)h(i)f(v\346r)o(dierne)e(er)463 594 y(dog)21
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(tilf\346ldige)g(udsving)g(\226)g(men)g(v\346r)o(dierne)e(f\370l-)463
708 y(ger)h(dog)h(den)g(for)o(udsete)f(for)o(deling)h(med)f(et)h
(toppunkt)g(ved)g(20\045.)463 1017 y FI(7.4)119 b(Opstilling)29
b(af)g(de)h(endelige)f(test)463 1221 y FY(Ud)20 b(fra)g(de)g
(ovenst\345ende)g(test)h(kan)g(vi)g(sammens\346tte)g(et)g(netv\346rk)f
(best\345ende)g(af:)588 1402 y FS(\017)41 b FY(Centr)o(ering)20
b(efter)g(massemidtpunkt.)588 1583 y FS(\017)41 b FY
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b FY(Gradient)20 b(nedstigning)i(som)g(tr\346ningsalgoritme.)588
1945 y FS(\017)41 b FY(200)19 b(neur)o(oner)h(i)h(det)f(skjulte)i(lag.)
588 2126 y FS(\017)41 b FY(Ekstern)20 b(1:2)g(skalering)h(\(720)e
(inputneur)o(oner\).)588 2307 y(V)-9 b(or)o(es)28 b(indledende)g(test)h
(synes)g(at)g(vise)f(at)h(\003er)o(e)f(skjulte)i(neur)o(oner)e(og)h
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(genkendelsespr)o(ocent,)g(men)g(der)g(er)g(imidlertid)g(et)g(pr)o
(oblem)g(med)g(at)463 2536 y(tr\346ningstiden)f(for)f(et)g(st\370rr)o
(e)f(antal)h(neur)o(oner)f(og)i(en)f(1:1)f(skalering)h(vil)g(v\346r)o
(e)f(uhensigts-)463 2650 y(m\346ssig)28 b(stor)-6 b(.)26
b(Uhensigtsm\346ssig)i(for)o(di)e(antallet)g(af)g(test)h(p\345)f(det)g
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b(skulle)h(begr\346nses)g(til)g(ganske)g(f\345.)f(Endvider)o(e)f(vil)i
(vi)g(sener)o(e)e(kigge)j(p\345)e(om)h(net\346rket)463
2879 y(kan)h(eftertr\346nes)e(til)i(en)g(enkelt)g(br)o(uger)-6
b(.)22 b(Dette)g(skal)h(i)g(s\345fald)e(ske)i(p\345)f(en)h
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(v\346r)o(e)e(krav)g(om)i(en)f(r)o(elativ)f(kort)i(tr\346ningstid.)f(V)
-5 b(i)22 b(har)f(derfor)463 3109 y(valgt)g(ikke)g(at)f(teste)g
(st\370rr)o(e)h(netv\346rk)f(end)h(dem)f(der)g(er)h(beskr)o(evet)e(i)i
(de)f(indledende)h(test.)588 3223 y(Et)i(netv\346rk)g(med)g(de)g
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3567 y(for)26 b(dette)g(netv\346rks)g(genkendelsesevne)h(og)f(om)h(en)g
(fuldst\346ndig)f(krydstest)g(vil)h(\346ndr)o(e)463 3681
y(p\345)20 b(r)o(esultatet.)463 3942 y Fz(7.4.1)99 b(Resultater)25
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7 v 1165 4144 a FY(Seed)99 b(T)-8 b(estperson)100 b(Skal.)f(Neur)-6
b(.)98 b(Genk.)p 1115 4197 1670 5 v 1231 4300 a(1)293
b(Lau)262 b(1:2)175 b(200)141 b(92,3\045)1231 4415 y(2)293
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1670 7 v 463 4803 a FT(Fortolkning)463 4976 y FY(Det)21
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(genkendelsespr)o(ocenter)-6 b(.)20 b(Dette)463 5090
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5599 y(78)p eop
%%Page: 79 81
79 80 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
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b(76,2\045)p 1252 1890 V 1302 1993 a(Gen.snt.)775 b(81,5\045)p
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y FY(Det)20 b(ses)h(at)f(n\345r)g(forskellige)g(personer)g(udelades,)f
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(skrivem\345de.)f(M\345ske)g(afviger)g(disse)g(perso-)463
3008 y(ners)e(h\345ndskrift)g(markant)g(fra)f(normen)i(eller)e
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3123 y(r)o(esten)e(af)f(personerne.)h(De)h(fem)f(andr)o(e)f(personers)h
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y(tr\346net)h(p\345,)f(har)h(for)f(disse)h(tr)o(e)f(personer)h(i)g
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4154 y(vende)k(overtr\346ningspr)o(oblemet)g(til)h(en)g(for)o(del)e
(gennem)i(en)f(specialisering)h(af)e(netv\346rket)463
4269 y(til)j(den)g(enkelte)f(br)o(uger)-6 b(.)463 4572
y FI(7.5)119 b(Eftertr\346ning)463 4777 y FY(En)20 b(s\345dan)f
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(til)g(en)g(br)o(uger)g(som)h(s\345)f(ef-)463 5235 y(terf\370lgende)j
(kan)h(forbedr)o(e)f(genkendelsespr)o(ocenten)h(ved)f(at)h(lade)f(det)h
(eftertr\346ne.)e(Om)463 5349 y(dette)d(kan)h(lade)f(sig)h(g\370r)o(e)f
(\370nskes)i(efterpr)o(\370vet)d(i)i(dette)f(afsnit.)1908
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80 81 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 588 192 a(Det)20 b(ville)g(v\346r)o(e)f(for)o
(delagtigt)g(for)h(en)g(br)o(uger)g(at)g(en)g(eftertr\346ning)g(kunne)h
(forl\370be)f(hur)o(-)463 306 y(tigt)d(og)f(med)g(f\345)g(indskr)o
(evne)f(tegn.)i(For)f(at)g(kunne)h(f\345)e(en)h(id\351)g(om)h(hvor)f
(stort)h(et)f(tr\346ningss\346t)463 421 y(der)27 b(beh\370ves,)h(og)g
(hvor)g(mange)g(iterationer)g(det)f(kr\346ver)g(at)h(for)o(h\370je)g
(genkendelsespr)o(o-)463 535 y(centen,)d(testes)g(dette.)f(T)-8
b(esten)25 b(blev)g(udf\370rt)f(p\345)g(det)h(netv\346rk)f(der)h(blev)f
(krydstestet)h(i)g(for)o(-)463 650 y(rige)18 b(afsnit.)f(Dog)i(var)e
(der)g(kun)h(data)f(tilg\346ngelige)i(til)f(at)f(eftertr\346ne)g(p\345)
g(en)h(enkelt)g(person,)463 765 y(nemlig)k(Lau;)e(af)g(denne)h(gr)o
(und)h(blev)f(der)f(eftertr\346net)g(p\345)h(det)f(netv\346rk)h(der)g
(ikke)g(var)g(ble-)463 879 y(vet)g(optr\346net)g(med)g
(tr\346ningseksempler)h(fra)f(Lau.)f(F\370rst)i(blev)f(40)f(billeder)g
(af)h(hvert)g(bog-)463 994 y(stav)f(isoler)o(et)h(og)g(udelukkende)g
(br)o(ugt)g(til)g(at)f(estimer)o(e)g(genkendelsespr)o(ocenten)h(med.)
588 1108 y(En)g(test)g(af)g(forskellige)g(st\370rr)o(elser)g
(datas\346t)f(fr)o(emg\345r)g(af)g(\002gur)i(7.1)e(hvor)h(tallet)g
(angiver)463 1223 y(antallet)k(af)f(tr\346ningseksempler)h(af)g(hvert)f
(bogstav)-9 b(,)25 b(og)h(for)e(hver)h(af)f(disse)h(datas\346t)f(blev)
463 1337 y(20\045)c(br)o(ugt)h(til)g(at)f(stoppe)h(vha.)f(validering.)
612 3334 y @beginspecial 50 @llx 50 @lly 410 @urx 302
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 @endspecial 534 3522 a FH(Figur)15 b(7.1:)28 b(Genkendelsespr)o(ocent)
15 b(som)h(funktion)f(af)g(antallet)g(af)h(iterationer)f(med)h
(forskellige)g(st\370rr)o(elser)534 3627 y(af)i(tr\346ningss\346ttet.)f
(Genkendelsespr)o(ocenten)h(stiger)h(kraftigt)g(i)f(l\370bet)h(af)f(de)
h(f\370rste)g(10)g(iterationer)-6 b(.)588 3839 y FY(Det)26
b(ses)g(at)g(jo)h(st\370rr)o(e)e(datas\346t)h(der)f(bliver)h
(eftertr\346net)f(p\345,)g(jo)i(bedr)o(e)e(bliver)h(genken-)463
3953 y(delsespr)o(ocenten.)18 b(Desuden)g(ses)g(det)g(at)g
(st\370rstedelen)g(af)g(forbedringen)g(sker)g(i)h(l\370bet)f(af)g(de)
463 4068 y(f\370rste)i(10)g(iterationer)-6 b(.)588 4182
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%%Page: 81 83
81 82 bop 1690 -129 a FY(Kapitel)20 b(7:)25 b(T)-8 b(est)p
463 -96 2975 4 v 612 1982 a @beginspecial 50 @llx 50
@lly 410 @urx 302 @ury 3212 @rwi @setspecial
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 @endspecial 534 2170 a FH(Figur)20 b(7.2:)27 b(Netv\346rket)21
b(optr\346net)g(med)g(l\370bende)f(indsamling)g(af)h
(tr\346ningseksempler)-6 b(.)20 b(I)g(kumulativ)g(ef-)534
2275 y(tertr\346ning)g(er)i(de)g(f\370rste)f(10)h(iterationer)f(med)h
(et)g(datas\346t)f(p\345)h(5)g(af)f(hvert)i(bogstav)l(,)e(her)o(efter)h
(10,)g(20,)f(40)534 2380 y(og)h(120)h(af)f(hvert)h(bogstav)l(.)f(I)g
(successiv)h(eftertr\346ning)e(er)h(de)g(f\370rste)g(10)h(iterationer)f
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(10)g(iterationer)-6 b(,)18 b(osv)l(.)588 2755 y FY(Den)j(mest)g
(praktiske)g(metode)g(at)f(eftertr\346ne)g(p\345)h(er)f(den)h(metode)g
(der)f(kun)i(br)o(uger)f(de)463 2870 y(5)h(senest)h(skr)o(evne)f
(bogstaver)h(da)f(man)g(med)h(denne)f(metode)h(ikke)g(beh\370ver)f(at)h
(opbevar)o(e)463 2984 y(st\370rr)o(e)d(datas\346t)g(end)g(disse)h(29)16
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b(.)463 3292 y FI(7.6)119 b(Praktiske)29 b(overvejelser)463
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3611 y(m\346ssige)21 b(krav)g(der)f(er)g(blevet)g(opstillet)i(afsnit)e
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(hukommelse,)i(heller)f(ikke)g(midlertidigt.)588 4269
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b(.)23 b(Desuden)h(blev)g(det)g(maksimale)g(hukommelsesforbr)o(ug)1908
5599 y(81)p eop
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2433 y(en)h(god)g(mar)o(gin)f(ligger)h(under)g(kravet.)588
2547 y(Hukommelsesforbr)o(uget)31 b(er)e(heller)i(ikke)f(uden)g(for)g
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3120 y(tene)i(i)g(variable)e(med)i(mindr)o(e)f(pr\346cision.)588
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5349 y(konstr)o(uer)o(e)26 b(et)g(netv\346rk)g(med)f(en)h
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(testperso-)1908 5599 y(82)p eop
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421 y(viden)f(om)g(eftertr\346ning.)f(Konklusionerne)j(har)d(alts\345)g
(i)h(et)g(vist)g(omfang)g(vist)g(sig)g(p\345lide-)463
535 y(lige.)588 650 y(De)k(to)g(hovedpr)o(oblemer)g(i)g(forbindelse)g
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994 y(muligt)33 b(at)f(p\345vise,)f(eller)h(tilsvar)o(ende)f(hvor)i
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1108 y(at)26 b(falsi\002cer)o(e)e(teorien.)i(V)-5 b(i)27
b(er)e(blevet)h(tvunget)g(ud)g(i)g(begge)g(pr)o(oblemer)f(p\345)h(gr)o
(und)g(af)f(den)463 1223 y(for)o(holdsvis)19 b(lange)g(tr\346ningstid)h
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(til)h(r\345-)463 1337 y(dighed.)588 1452 y(Med)f(hensyn)i(til)f
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2025 y(ligg\370r)o(e)e(den)f(empiriske)h(metode.)588
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3170 y(begr\346nset.)1908 5599 y(83)p eop
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3234 y(ind\003ydelse)18 b(p\345)f(udf\370rselstiden)h(og)g
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(forbehandling)i(af)e(billederne)g(er)h(at)f(cen-)463
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3921 y(for)o(eg\345r)25 b(iterativt)h(da)g(pr)o(oblemet)g(ikke)h(kan)g
(l\370ses)g(analytisk,)f(hvilket)i(medf\370r)o(er)d(at)h(man)463
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(i)g(den)463 4494 y(modsatte)29 b(r)o(etning)h(af)f(gradienten)g(til)i
FL(E)q FY(,)f(eller)f(konjuger)o(et)g(gradient)g(der)g(udnytter)h(an-)
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4723 y(nedstigning.)d(F\346lles)e(for)g(de)g(forskellige)h(varianter)e
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4838 y(a\003edede)20 b(af)h(fejlfunktionen)j FL(E)f FY(som)f(kan)g
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(positivistisk)i(opfattelse)d(af)h(et)f(kun)1908 5599
y(84)p eop
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85 86 bop 1549 -129 a FY(Kapitel)20 b(8:)25 b(Konklusion)p
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421 y(metode)g(sig)h(dog)f(alligevel)g(at)g(v\346r)o(e)f(p\345lidelig.)
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535 y(ning)26 b(til)e(bedste)h(forklaring)f(har)g(vist)h(sig)g(at)g
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650 y(pr)o(ojekts)c(pr)o(oblemstillinger)-6 b(.)588 765
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879 y(blevet)24 b(opstillet)h(som)g(et)g(kvalitativt)f(udvalg)f(da)h
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1108 y(test)e(har)g(vi)g(kunnet)g(drage)f(nogle)i(konklusioner)-6
b(.)28 b(Af)f(de)f(to)i(aktiveringsfunktioner)f(har)463
1223 y FL(tanh)18 b FY(vist)i(sig)g(hurtigst.)f(Konjuger)o(et)h
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1452 y(T)-7 b(r\346nings-)27 b(og)h(inertikonstant)g(har)f(vist)h(sig)g
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1567 y(0,1.)19 b(Centr)o(ering)h(viste)h(sig)g(at)g(have)f(en)h
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1681 y(mens)25 b(en)f(h\370jer)o(e)g(skalering)g(end)g(1:2)e(viste)j
(sig)f(at)g(medf\370r)o(e)f(en)h(d\345rliger)o(e)e(genkendelses-)463
1796 y(evne.)588 1910 y(For)c(at)g(f\345)g(et)g(endeligt,)h
(p\345lideligt)f(r)o(esultat)g(har)g(et)g(netv\346rk)h(med)f(de)g
(f\370r)g(fundne)h(para-)463 2025 y(metr)o(e)k(gennemg\345et)g(en)h
(fuldst\346ndig)g(krydstest.)f(Resultatet)h(var)f(81,7\045)e
(gennemsnitlig)463 2139 y(genkendelse)27 b(p\345)f(ukendte)h(tegn)h
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(kan)g(ogs\345)g(modvir)o(-)463 2369 y(kes)e(gennem)g(en)g
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2712 y(99\045)20 b(p\345)g(en)h(person.)g(En)g(s\345dan)g
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3531 y FY(T)-5 b(il)23 b(hvilke)f(opgaver)f(kan)h(kunstige)h(neurale)e
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3646 y(kunne)32 b(v\346r)o(e)e(n\345r)g(et)h(neurale)f(netv\346rk)h
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3760 y(ninger)-6 b(,)28 b(f.eks.)e(mennesker)-6 b(.)28
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(kom-)463 3875 y(plicer)o(et)i(neuralt)h(netv\346rk,)f(er)h(det)g(dog)g
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3989 y(kunstigt)20 b(netv\346rk)e(skulle)i(v\346r)o(e)d(bedr)o(e)g(til)
j(meget)e(komplekse)i(opgaver)e(som)h(det)g(at)f(diag-)463
4104 y(nosticer)o(e)h(patienter)f(p\345)h(linje)h(med)f(en)g(praktiser)
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4219 y(delige)25 b(samtaler)g(\226)h(til)g(geng\346ld)g(kan)f(et)h
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4333 y(computer)19 b(i)h(mods\346tning)h(til)e(et)g(menneske.)h(Fejlpr)
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4448 y(sammenlignes)j(med)e(andr)o(e)g(kunstige)i(modeller)-6
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h(i)h(at)e(de)h(er)g(et)g(meget)g(gener)o(elt)g(l\370snings-)463
4677 y(v\346rkt\370j)d(som)h(i)f(teorien)g(kan)g(anvendes)f(p\345)g
(alt)h(hvad)f(der)g(kan)h(opstilles)h(et)f(datas\346t)f(for)-6
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5021 y(taene)h(og)h(m\345lv\346r)o(dierne.)e(Det)i(d\346kker)f(over)h
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5135 y(kan)21 b(v\346r)o(e)e(sv\346rt)i(tiln\346rmelige,)g
(s\345som[Sarle,)f(2001]:)588 5333 y FS(\017)41 b FY
(Ansigtsgenkendelse)1908 5599 y(85)p eop
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86 87 bop 1549 -129 a FY(Kapitel)20 b(8:)25 b(Konklusion)p
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(hvor)o(dan)g(det)h(n\345r)g(til)g(sine)h(konklusioner)-6
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(mange)e(tr\346ningseksempler)i(er)e(pr)o(oblematisk)463
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2163 y(idet)22 b(den)h(teor)o(etiske)f(ramme)h(ikke)g(er)f
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2277 y(del)e(kan)g(opstilles)h(p\345)f(for)o(h\345nd)f(uden)h
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2392 y(empirien)e(som)h(v\346rkt\370j.)f(Neurale)f(netv\346rk)h
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2506 y(ind)30 b(i)f(teorien)h(bag)f(for)g(at)g(kunne)h(anvende)f(dem)g
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2621 y(begr\346nser)20 b(den)g(gener)o(elle)f(anvendelighed.)h(Den)g
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2735 y(i)g(afsnit)g(3.1)e(kr\346ver)h(derimod)h(ikke)g(n\346r)g(s\345)f
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b(Implementering)23 b(p\345)i(h\345ndholdt)463 3169 y
FY(Hvis)31 b(vor)o(es)e(tegngenkendelsessystem)j(skulle)f(udgives)f
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3398 y(ger)o(en)21 b(at)f(f\345)h(genkendt)g(et)g(skr)o(evet)g(tegn.)g
(V)-9 b(or)o(es)20 b(forslag)h(er)g(beskr)o(evet)f(i)i(afsnit)f(3.2.)f
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463 3627 y(kunne)21 b(man)g(for)o(estille)g(sig)g(at)f(den)h(blev)f
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y Fz(8.1.3)99 b(L\370bende)24 b(eftertr\346ning)463 4620
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463 4734 y(h\345ndholdt)f(computer)g(ikke)g(den)g(bedste)f(maskine)h
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4849 y(ningen)22 b(er)f(meget)g(ber)o(egningskr\346vende.)f(Hvorvidt)h
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y(86)p eop
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87 88 bop 1549 -129 a FY(Kapitel)20 b(8:)25 b(Konklusion)p
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2783 y(de)22 b(h\370jeste)h(outputv\346r)o(dier)-6 b(,)21
b(s\345)g(vil)i(det)f(korr)o(ekte)f(bogstav)i(sandsynligvis)g(v\346r)o
(e)e(at)h(\002nde)463 2897 y(blandt)i(disse)g(3,)g(og)h(man)f(kan)g
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(et)g(af)g(de)463 3012 y(bogstaver)29 b(der)f(svar)o(er)f(til)i(disse)g
(outputneur)o(oner)-6 b(.)29 b(P\345)g(denne)g(m\345de)f(slipper)h(br)o
(uger)o(en)463 3126 y(for)21 b(at)f(slette)h(bogstavet)f(og)i(skrive)e
(et)h(nyt.)588 3241 y(Hvis)g(man)g(kombiner)o(er)g(dette)g(med)g
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3355 y(at)e(optr\346ne)h(p\345)f(de)h(bogstaver)f(som)i(f\370r)e(kunne)
h(v\346r)o(e)f(genkendt)h(forkert.)463 3597 y FT(Raf)o(\002neret)d
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3884 y(hj\346lp)24 b(i)f(form)g(af)f(de)g(tegn)i(der)e(aller)o(ede)f
(er)h(blevet)h(genkendt.)g(Hvis)g(en)g(br)o(uger)g(f.eks.)f(har)463
3999 y(skr)o(evet)g(\204m\370d\223,)g(er)g(der)g(stor)i(sandsynlighed)f
(for)g(at)f(det)h(n\346ste)g(bogstav)g(er)f(et)h(\204e\223.)e(Der)o(-)
463 4113 y(ved)f(kan)h(det)f(blive)h(letter)o(e)f(at)g(afg\370r)o(e)f
(tvivlstilf\346lde.)588 4228 y(En)33 b(mulighed)h(ville)g(v\346r)o(e)e
(at)h(s\370r)o(ge)g(for)g(at)g(outputtet)h(fra)e(netv\346rket)i(dir)o
(ekte)e(kan)463 4342 y(fortolkes)22 b(som)h(sandsynligheder)f(efter)f
(principperne)h(beskr)o(evet)e(i)i([Bishop,)g(1995].)e(Den)463
4457 y(samlede)k(sandsynlighed)i(for)f(at)f(det)g(bogstav)h(der)g(er)f
(ved)g(at)h(blive)g(genkendt,)g(er)f(et)h(be-)463 4572
y(stemt)c(bogstav)-9 b(,)21 b(kan)g(s\345)f(\002ndes)h(ved)f(at)h
(gange)g(sandsynligheden)g(givet)g(konteksten)h(med)463
4686 y(sandsynligheden)g(fra)d(netv\346rket.)588 4801
y(Sandsynlighedstabeller)d(for)h(konteksterne)h(kunne)g(i)f(en)g
(implementering)h(opbygges)463 4915 y(l\370bende)j(eller)f(\002ndes)h
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5144 y(stav)36 b(der)f(skulle)h(genkendes,)g(men)h(man)f(bliver)f
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5259 y(konteksten)24 b(da)f(denne)g(vokser)g(eksponentielt)i(med)e
(antallet)f(af)h(bogstaver)-6 b(.)22 b(V)-9 b(ed)23 b(\351t)g(bog-)1908
5599 y(87)p eop
%%Page: 88 90
88 89 bop 1549 -129 a FY(Kapitel)20 b(8:)25 b(Konklusion)p
463 -96 2975 4 v 463 192 a(stavs)g(kontekst)i(skal)f(den)f
(h\345ndholdte)h(potentielt)g(have)f(en)g(sandsynlighedstabel)h(med)463
306 y(29)547 276 y FP(2)606 306 y FG(=)d FY(841)18 b(indgange)h(hvor)g
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(tabel)463 421 y(med)i(29)732 391 y FP(4)791 421 y FG(=)j
FY(707.281)18 b(indgange)j(\226)f(i)h(praksis)g(kan)g(dette)f(antal)h
(dog)g(begr\346nses)g(betydeligt)463 535 y(da)f(visse)h(kombinationer)h
(aldrig)e(optr\346der)-6 b(.)588 650 y(Pr)o(oblemet)26
b(med)h(at)f(br)o(uge)h(kontekstafh\346ngig)h(information)f(til)h(at)e
(forbedr)o(e)f(g\346ttet)463 765 y(er)i(at)g(det)g(for)o(eg\345r)f(med)
h(den)h(antagelse)f(at)g(konteksten)h(faktisk)g(har)f(en)h(stor)f
(betydning)463 879 y(for)32 b(hvor)o(dan)g(det)g(n\346ste)g(bogstav)h
(bliver)-6 b(.)31 b(V)-9 b(ed)32 b(almindelig)h(tekstskrivning)g(kan)g
(dette)463 994 y(sagtens)17 b(t\346nkes)f(at)g(v\346r)o(e)g(en)g(god)h
(antagelse,)f(men)g(netop)h(p\345)f(en)g(h\345ndholdt)h(er)f(der)g
(m\345ske)463 1108 y(st\370rr)o(e)25 b(sandsynlighed)i(for)e(at)h(br)o
(uger)o(en)f(benytter)h(sig)g(af)f(forskellige)i(forkortelser)e(eller)
463 1223 y(sammenskrivninger)k(hvis)g(teksterne)f(kun)g(er)g(til)g
(notebr)o(ug.)h(Desuden)f(g\345r)g(det)f(ikke)h(at)463
1337 y(s\346tte)f(sandsynligheden)h(for)f(usandsynlige)g(bogstaver)g
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1452 y(v\346r)o(e)20 b(muligt)h(at)g(skrive)f(disse)h(bogstaver)-6
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1681 y(ders\370ge)21 b(br)o(ugernes)g(vaner)-6 b(.)21
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(f\345)g(gavn)g(af)463 1796 y(konteksten)28 b(p\345)f(en)g(indlysende)g
(m\345de)g(i)g(mods\346tning)i(til)e(f.eks.)g(den)g(eksister)o(ende)f
(l\370s-)463 1910 y(ning.)1908 5599 y(88)p eop
%%Page: 89 91
89 90 bop 463 893 a FZ(Litteratur)463 1340 y FY(Shigeo)31
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FY(.)61 b(Kluwer)31 b(Academic)e(Publishers)546 1454
y(Gr)o(oup,)20 b(1995.)463 1635 y(Peter)g(Bickfor)o(d.)29
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(bickf)o(ord_w)o(ait.)546 1865 y(htm)p FY(.)463 2046
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2160 y(Pr)o(ess,)g(1995.)463 2341 y(S\370r)o(en)c(Br)o(unak)h(og)g
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2637 y(Steve)h(Bush.)30 b FL(Brighthand)18 b(-)j(nothing)e(but)i
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FY(.)463 2932 y(Morten)21 b(Dige.)30 b FL(Intr)o(oduktion)19
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Fa(http://www.teo.au.dk/html/rv-)t(sem/dig)o(e/vid)o(enskab)o(steor)o
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(S2001/)546 3638 y(DeEnkelteKursusgange/gang2-)t(\045noter.pd)o(f)p
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4114 y(Helge)c(Kragh)g(og)h(Stig)f(Andur)g(Pedersen.)j
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4229 y(1991.)463 4410 y(David)21 b(C.)g(Lay)-9 b(.)30
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FY(.)30 b(Addison-W)-8 b(esley)f(,)21 b(anden)g(udgave,)546
4525 y(2000.)463 4706 y(Y)-8 b(ann)20 b(LeCun,)f(L\351on)g(Bottou,)h(Y)
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5001 y(Per)k(Printz)g(Madsen.)39 b FL(Neurale)25 b(Netv\346rk)e
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5599 y(89)p eop
%%Page: 90 92
90 91 bop 1773 -129 a FY(Litteratur)p 463 -96 2975 4
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1193 y Fa(ftp://ftp.sas.com/pub/neural/FAQ.ht)o(ml)p
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%%Page: 91 93
91 92 bop 463 845 a FJ(Bilag)43 b(A)463 1298 y FZ(Konjugeret)52
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92 93 bop 1219 -129 a FY(Kapitel)20 b(A:)g(Konjuger)o(et)h(gradient)g
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  pixels 0 number_pixels getinterval
} bind def

/DirectClassImage
{
  %
  % Display a DirectClass image.
  %
  systemdict /colorimage known
  {
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { DirectClassPacket } false 3 colorimage
  }
  {
    %
    % No colorimage operator;  convert to grayscale.
    %
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { GrayDirectClassPacket } image
  } ifelse
} bind def

/GrayDirectClassPacket
{
  %
  % Get a DirectClass packet;  convert to grayscale.
  %
  % Parameters:
  %   red
  %   green
  %   blue
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/GrayPseudoClassPacket
{
  %
  % Get a PseudoClass packet;  convert to grayscale.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassPacket
{
  %
  % Get a PseudoClass packet.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  compression 0 eq
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassImage
{
  %
  % Display a PseudoClass image.
  %
  % Parameters:
  %   class: 0-PseudoClass or 1-Grayscale.
  %
  currentfile buffer readline pop
  token pop /class exch def pop
  class 0 gt
  {
    currentfile buffer readline pop
    token pop /depth exch def pop
    /grays columns 8 add depth sub depth mul 8 idiv string def
    columns rows depth
    [
      columns 0 0
      rows neg 0 rows
    ]
    { currentfile grays readhexstring pop } image
  }
  {
    %
    % Parameters:
    %   colors: number of colors in the colormap.
    %   colormap: red, green, blue color packets.
    %
    currentfile buffer readline pop
    token pop /colors exch def pop
    /colors colors 3 mul def
    /colormap colors string def
    currentfile colormap readhexstring pop pop
    systemdict /colorimage known
    {
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { PseudoClassPacket } false 3 colorimage
    }
    {
      %
      % No colorimage operator;  convert to grayscale.
      %
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { GrayPseudoClassPacket } image
    } ifelse
  } ifelse
} bind def

/DisplayImage
{
  %
  % Display a DirectClass or PseudoClass image.
  %
  % Parameters:
  %   x & y translation.
  %   x & y scale.
  %   label pointsize.
  %   image label.
  %   image columns & rows.
  %   class: 0-DirectClass or 1-PseudoClass.
  %   compression: 0-none or 1-RunlengthEncoded.
  %   hex color packets.
  %
  gsave
  /buffer 512 string def
  /byte 1 string def
  /color_packet 3 string def
  /pixels 768 string def

  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  x y translate
  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  currentfile buffer readline pop
  token pop /pointsize exch def pop
  /Times-Roman findfont pointsize scalefont setfont
  x y scale
  currentfile buffer readline pop
  token pop /columns exch def
  token pop /rows exch def pop
  currentfile buffer readline pop
  token pop /class exch def pop
  currentfile buffer readline pop
  token pop /compression exch def pop
  class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
  grestore
} bind def
%%EndProlog
%%Page:  1 1
%%PageBoundingBox: 0 0 48 60
userdict begin
DisplayImage
0 0
48 60
12.000000
48 60
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end
%%PageTrailer
%%Trailer
%%EOF

%%EndDocument
 @endspecial 278 w @beginspecial 0 @llx 0 @lly 48 @urx
60 @ury 214 @rwi @setspecial
%%BeginDocument: superset/super-a-center.bmp.eps
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: (ImageMagick)
%%Title: (super-a-center.char.bmp.eps)
%%CreationDate: (Sat May 25 14:41:16 2002)
%%BoundingBox: 0 0 48 60
%%DocumentData: Clean7Bit
%%LanguageLevel: 1
%%Pages: 1
%%EndComments

%%BeginDefaults
%%EndDefaults

%%BeginProlog
%
% Display a color image.  The image is displayed in color on
% Postscript viewers or printers that support color, otherwise
% it is displayed as grayscale.
%
/DirectClassPacket
{
  %
  % Get a DirectClass packet.
  %
  % Parameters:
  %   red.
  %   green.
  %   blue.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  compression 0 eq
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/DirectClassImage
{
  %
  % Display a DirectClass image.
  %
  systemdict /colorimage known
  {
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { DirectClassPacket } false 3 colorimage
  }
  {
    %
    % No colorimage operator;  convert to grayscale.
    %
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { GrayDirectClassPacket } image
  } ifelse
} bind def

/GrayDirectClassPacket
{
  %
  % Get a DirectClass packet;  convert to grayscale.
  %
  % Parameters:
  %   red
  %   green
  %   blue
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/GrayPseudoClassPacket
{
  %
  % Get a PseudoClass packet;  convert to grayscale.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 eq
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassPacket
{
  %
  % Get a PseudoClass packet.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  compression 0 eq
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassImage
{
  %
  % Display a PseudoClass image.
  %
  % Parameters:
  %   class: 0-PseudoClass or 1-Grayscale.
  %
  currentfile buffer readline pop
  token pop /class exch def pop
  class 0 gt
  {
    currentfile buffer readline pop
    token pop /depth exch def pop
    /grays columns 8 add depth sub depth mul 8 idiv string def
    columns rows depth
    [
      columns 0 0
      rows neg 0 rows
    ]
    { currentfile grays readhexstring pop } image
  }
  {
    %
    % Parameters:
    %   colors: number of colors in the colormap.
    %   colormap: red, green, blue color packets.
    %
    currentfile buffer readline pop
    token pop /colors exch def pop
    /colors colors 3 mul def
    /colormap colors string def
    currentfile colormap readhexstring pop pop
    systemdict /colorimage known
    {
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { PseudoClassPacket } false 3 colorimage
    }
    {
      %
      % No colorimage operator;  convert to grayscale.
      %
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { GrayPseudoClassPacket } image
    } ifelse
  } ifelse
} bind def

/DisplayImage
{
  %
  % Display a DirectClass or PseudoClass image.
  %
  % Parameters:
  %   x & y translation.
  %   x & y scale.
  %   label pointsize.
  %   image label.
  %   image columns & rows.
  %   class: 0-DirectClass or 1-PseudoClass.
  %   compression: 0-none or 1-RunlengthEncoded.
  %   hex color packets.
  %
  gsave
  /buffer 512 string def
  /byte 1 string def
  /color_packet 3 string def
  /pixels 768 string def

  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  x y translate
  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  currentfile buffer readline pop
  token pop /pointsize exch def pop
  /Times-Roman findfont pointsize scalefont setfont
  x y scale
  currentfile buffer readline pop
  token pop /columns exch def
  token pop /rows exch def pop
  currentfile buffer readline pop
  token pop /class exch def pop
  currentfile buffer readline pop
  token pop /compression exch def pop
  class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
  grestore
} bind def
%%EndProlog
%%Page:  1 1
%%PageBoundingBox: 0 0 48 60
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