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Receiver Operator Characteristic Curve
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Shoh-kudo
June 19, 2019
Education
1
57
Receiver Operator Characteristic Curve
はじめての『はじめてのパターン認識』第3章
A first introduction of ROC curve (Section 3.2).
Shoh-kudo
June 19, 2019
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Transcript
ୈষϕΠζͷࣝผنଇ ड৴ऀಈ࡞ಛੑۂઢ ͡Ίͯͷύλʔϯೝࣝྠಡձ ౻ 1
065-*/& ϕΠζͷࣝผنଇ ࠞಉߦྻ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ষ ·ͱΊ ຕ ຕ ຕ ຕ
ຕ !2
ϕΠζͷࣝผنଇ ٤Ԏ͍ͯ͠Δਓ͕͍ͨʜ පؾͩͬͨʜ Cd P(Cd |x) = p(x|Cd )P(Cd )
p(x) ݈߁ͩͬͨ Ch P(Ch |x) = p(x|Ch )P(Ch ) p(x) ࣝผΫϥε = argmax p(x|Ci )P(Ci ) i x !3
ϕΠζͷࣝผنଇ ࣝผΫϥε = argmax p(x|Ci )P(Ci ) i ϵ(x) =
min [P(Ci |x)] ΫϥεΛޡΔ֬ʢසʣ i පؾͩͬͨʜ Cd P(Cd |x) = p(x|Cd )P(Cd ) p(x) ݈߁ͩͬͨ Ch P(Ch |x) = p(x|Ch )P(Ch ) p(x) !4
ϕΠζͷࣝผنଇ පؾͩͬͨʜ Cd ݈߁ͩͬͨ Ch ຊ݈߁ͩͬͨͷʹ පؾͩͱݴͬͯ͠·ͬͨ ຊපؾͩͬͨͷʹ ݈߁ͩͱݴͬͯ͠·ͬͨ ِӄੑ
ِཅੑ ࣝผڥք !5 ϵ(x) = min [P(Ci |x)] ΫϥεΛޡΔ֬ʢසʣ i පؾͰ͋Δ͜ͱΛཅੑͱ͢Δͱ
පؾͩͬͨʜ Cd ݈߁ͩͬͨ Ch ϕΠζͷࣝผنଇ ࣝผΫϥε = argmax p(x|Ci )P(Ci
) i ࠞಉߦྻ ࣝผͷੑೳΛͲ͏ධՁ͢Δ͔ !6
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ͜ΕΒͷΛͬͯࣝผੑೳΛධՁ͢Δ !7
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ِཅੑGBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷͷΛཅੑͱஅׂͨ͠߹ !8
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ਅཅੑUSVFQPTJUJWFSBUF TP P = TP TP + FN ཅੑͷͷΛਖ਼͘͠ཅੑͱஅׂͨ͠߹ !9
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ద߹QSFDJTJPO TP TP + FP ཅੑͱஅͨ͠ͷͷ͏ͪຊʹཅੑͰ͋Δׂ߹ !10
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ࠶ݱSFDBMM TP P = TP TP + FN ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨͷͷׂ߹ !11
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ਖ਼֬BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ !12
ࠞಉߦྻ Q O ߦ Q ਅཅੑ 51 ِӄੑ '/ 151
'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ''NFBTVSF 2 1 precision + 1 recall ద߹ͱ࠶ݱͷௐฏۉ !13
ࠞಉߦྻ ِཅੑGBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷͷΛཅੑͱஅׂͨ͠߹
ਅཅੑUSVFQPTJUJWFSBUF TP P = TP TP + FN ཅੑͷͷΛਖ਼͘͠ཅੑͱஅׂͨ͠߹ ద߹QSFDJTJPO TP TP + FP ཅੑͱஅͨ͠ͷͷ͏ͪ ຊʹཅੑͰ͋Δׂ߹ ࠶ݱSFDBMM TP P = TP TP + FN ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨͷͷׂ߹ ਖ਼֬BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ ''NFBTVSF 2 1 precision + 1 recall ద߹ͱ࠶ݱͷௐฏۉ !14
ࠞಉߦྻ ྫ ద߹ͱ࠶ݱͳͥτϨʔυΦϑͷؔʹ͋Δͷ͔ʁ ཏੑΛ্͛Α͏ͱཅੑͱड͚ೖΕΔͷΛ૿͢ͱ ِཅੑͱஅ͢Δͷ૿Ճ͠ɼ݁Ռͱͯ͠ద߹͕Լ͕ΔͨΊ ద߹QSFDJTJPO TP TP + FP
ཅੑͱஅͨ͠ͷͷ͏ͪ ຊʹཅੑͰ͋Δׂ߹ ࠶ݱSFDBMM TP P = TP TP + FN ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨͷͷׂ߹ !15
ِཅੑGBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷͷΛཅੑͱஅׂͨ͠߹ ਅཅੑUSVFQPTJUJWFSBUF
TP P = TP TP + FN ཅੑͷͷΛਖ਼͘͠ཅੑͱஅׂͨ͠߹ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ පؾͷਓ͔݈߁ͳਓͳͲͷαϯϓϧʹେ͖ͳภΓ͕͋Δ߹Ͱ ཅੑӄੑͷͷ͚ͩͰ࡞ΒΕ͍ͯΔͷͰɼͦΕͧΕӄੑཅੑͷӨڹΛड͚ͳ͍ 30$ۂઢ !16
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ ਅཅੑ # " $ ࣝผڥք Cp*
Cn* ཅੑͷ֬ͱӄੑͷ֬ ਅཅੑɿ ِཅੑɿ੨ ཅੑʢࠨʣଆͷ ֬ʹ͢Δ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ !17
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ ਅཅੑ # " $ Cp* Cn*
ཅੑͷ֬ͱӄੑͷ֬ ࣝผڥք ཅੑʢࠨʣଆͷ ֬ʹ͢Δ !18 ਅཅੑɿ ِཅੑɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ ਅཅੑ # " $ Cp* Cn*
ཅੑͷ֬ͱӄੑͷ֬ ࣝผڥք ཅੑʢࠨʣଆͷ ֬ʹ͢Δ !19 ਅཅੑɿ ِཅੑɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑɾِཅੑ (0,0) (0.1,0.7) (1.0,1.0) ཅੑʢࠨʣଆͷ ֬ʹ͢Δ ʢԣɼॎʣʢِཅੑɼਅཅੑʣ !20 ਅཅੑɿ
ِཅੑɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 4 ΔE = 3 !21
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 2 ΔE = 1 !22
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 0 30$͔Βݴ͑Δ͜ͱԿ͔ʁ !23
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ཅੑͷ֬ͱӄੑͷ֬ͷॏͳΓ͕খ͍͞΄Ͳ 30$ͷԼ෦ͷ໘ੵ͕େ͖͘ͳ͍ͬͯΔ !24
30$ͷԼ෦ͷ໘ੵɿ"6$ "SFB6OEFSUIF$VSWF ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ᘳʹࣝผͰ͖Δɿ AUC = 1 શࣝ͘ผͰ͖ͳ͍ɿ AUC =
0.5 0.5 ≦ AUC ≦ 1.0 !25
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ "6$Λେ͖͘͢ΔͨΊʹ ྑ͍ੑͷ֬Λ࣋ͭಛྔͷΈ߹Θͤ !26
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑɾِཅੑ (0,0) (0.1,0.7) (1.0,1.0) ʢԣɼॎʣʢِཅੑɼਅཅੑʣ ͋͘·Ͱ ֬ͷੑೳධՁ Ͳͷʹ͢Εྑ͍ʁ 30$
!27
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ͋͘·Ͱ ֬ͷੑೳධՁ # " $ Ͳͷʹ͢Εྑ͍ʁ ͲͷࣝผڥքΛબྑ͍ʁ 30$ ཅੑͷ֬ͱӄੑͷ֬
!28
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ଛࣦ͕࠷খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ # " $ ࣝผڥք Cp* Cn* ِཅੑ ਅཅੑ
ཅੑͷྖҬʹ͓͚Δଛࣦ ଛࣦ ਅཅੑʹ༝དྷ͢Δଛࣦ ଛࣦ ِཅੑʹ༝དྷ͢Δଛࣦ r(C1 |x) = Σ2 k=1 L1k P(Ck |x) !29
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ཅੑͷྖҬʹ͓͚Δଛࣦ ଛࣦ ਅཅੑʹ༝དྷ͢Δଛࣦ ଛࣦ ِཅੑʹ༝དྷ͢Δଛࣦ r(C1 |x) = Σ2
k=1 L1k P(Ck |x) r(C1 |x) = L11 P(C1 |x) r(C1 |x) = L12 P(C2 |x) ਖ਼ࣝ͘͠ผͨ͠ͷʹଛࣦʁ - Λʹͯ͠د༩Λখ͘͢͞Δ r(C1 |x) = L12 P(C1 |x) ཅੑͷྖҬʹ͓͚Δ ࠷ऴతͳଛࣦ ଛࣦ͕࠷খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ !30
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ӄੑྖҬʹ͓͚Δଛࣦಉ༷ʹॲཧ͢Δͱɼଛࣦ྆ऀͷظΛͱΔ͜ͱͰ r = E[r(x)] = ∫ R1 +R2 min[L12
P(C2 |x), L21 P(C1 |x)]p(x)dx = ∫ R1 L12 p(x|C2 )P(C2 )dx + ∫ R2 L21 p(x|C1 )P(C1 )dx ΫϥεΛཅੑʢQ ʣɼΫϥεΛӄੑʢO ʣͱ͢Ε r = ∫ p* Lp*n* p(x|Cn* )P(Cn* )dx + ∫ n* Ln*p* p(x|Cp* )P(Cp* )dx = Lp*n* P(Cn* )∫ p* p(x|Cn* )dx + Ln*p* P(Cp* )∫ n* p(x|Cp* )dx = Lp*n* P(Cn* )ϵ2 + Ln*p* P(Cp* )ϵ1 ୈछͷޡΓ ୈछͷޡΓ ଛࣦ͕࠷খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ !31
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ r = Lp*n* P(Cn* )ϵ2 + Ln*p* P(Cp* )ϵ1
30$ʹ͋ΔɼਅཅੑЏ ͱِཅੑЏ ͰࣜΛॻ͖͢ͱ࣍ࣜͷΑ͏ʹͳΔɽ 1 − ϵ1 = Lp*n* P(Cn* ) Ln*p* P(Cp* ) ϵ2 + (1 − r Ln*p* P(Cp* ) ) = αϵ2 + h(r) ͜ͷࣜ1 $Q ͱ1 $O ͓Αͼଛࣦ-Q O ͱ-O Q ʹΑͬͯЏ ͷ͖͕ܾ·Γɼ ยΛSͷؔͱͨ͠ઢͷํఔࣜͱଊ͑Δ͜ͱ͕Ͱ͖Δɽ ଛࣦ͕࠷খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ Ͱɼ༩͑ΒΕͨࣜͷதͰ࠷దͳࣜͲͷΑ͏ͳܗʹͳΔͩΖ͏͔ʁ ݅ҎԼͷ௨ΓͰ͋Δɽ min r = max h(r) ઢ͕30$ͱަΛ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ݅ͷൣғ !32
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 1 − ϵ1 = αϵ2 + h(r) ଛࣦ͕࠷খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ min
r = max h(r) ઢ͕30$ͱަΛ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ݅ͷൣғ ݅ ยI S খଛࣦSେ ଛࣦ࠷খ͔ͭ30$ͱަΛ࣋ͭ (0.1,0.7) ͍͍ͩͨ ͘Β͍ʁ ਅཅੑͱِཅੑͷׂ߹͕ ͷʹͳΔΑ͏ʹࣝผڥքΛఆΊΔͱ ࠷ޡΓΛখͨࣝ͘͞͠ผ͕ՄೳʹͳΔ !33
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ 30$Ϋϥεͷ͕Θ͔Βͳͯ͘ඳ͘͜ͱ͕Ͱ͖Δɽ ਅͷΫϥεͱσʔλͷείΞ Q
O ᮢͷείΞ4Ҏ্ͷείΞΛ࣋ͭσʔλΛQ ͱࣝผ͢Δɽ ͜ΕΛɼᮢΛมԽͤͯ͞܁Γସ͑͢ɽ !34
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ Q O
30$Ϋϥεͷ͕Θ͔Βͳͯ͘ඳ͘͜ͱ͕Ͱ͖Δɽ ᮢ4 ਅཅੑ ِཅੑ 㱣 㱣 !35
ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ Q O
30$Ϋϥεͷ͕Θ͔Βͳͯ͘ඳ͘͜ͱ͕Ͱ͖Δɽ !36
# " $ Cp* Cn* ِཅੑ ਅཅੑ ষ 30$ۂઢΛඳ͍ͨͱ͜ΖɼϥϯμϜࣝผΛද͢
ͷઢͷԼଆʹདྷͯ͠·ͬͨɽͲ͏͢Εྑ͍͔ɽ ্ਤͷਅཅੑͱِཅੑͷେ͖͕͞ٯస͍ͯ͠Δྫͱଊ͑ΒΕΔͷͰɼ ୯७ʹཅੑͱӄੑͷΫϥεΛٯస͢Ε͍͍ ͋Δ͍ɼ྆ํͷείΞʹϚΠφεΛ͔͚Δ͔ᮢͷεΩϟϯΛٯ͔Βߦ͏ͱ͔ !37
·ͱΊ ยI S খଛࣦSେ ྨͷධՁࠞಉߦྻ͔ΒಘΒΕΔ ͍͔ͭ͘ͷࢦඪͰߦ͑Δ ࠞಉߦྻΛ༩͑Δ֬ͷධՁ 30$ͷԼ෦໘ੵ "6$ ʹΑͬͯධՁ͢Δ
֬ͷ࠷ྑ͍ࣝผڥք 30$ͱϕΠζͷࣝผنଇͷ ଛࣦ͕࠷খʹͳΔΛબͿ͜ͱͰಘΒΕΔ 30$͕֬Θ͔Βͳͯ͘ ඳ͘͜ͱ͕Ͱ͖Δ !38
ଛࣦߦྻ ଛࣦʹؔ͢Δࣜʹग़ͯ͘Δ-ʹ͍ͭͯ r(Ci |x) = ΣK k=1 Lik P(Ck |x)
,ͷ࣌ɼS࠷؆୯ͳ-Λͬͯ࣍ͷΑ͏ʹॻ͖දΘͤΔɽ Lkj p(Ck |x) = (1 − Ikj )p(Ck |x) = {( 1 1 1 1) − ( 1 0 0 1)} ( p(C1 |x) p(C2 |x)) = ( 0 1 1 0) ( p(C1 |x) p(C2 |x)) = ( p(C2 |x) p(C1 |x)) ͜ͷதͰখ͍͞ํΛऔΔ࣌ҎԼͷΑ͏ʹॻ͚Δ min ( p(C2 |x) p(C1 |x)) !39