Receiver Operator Characteristic Curve

ୈষϕΠζͷࣝผنଇ ड৴ऀಈ࡞ಛੑۂઢ ͸͡Ίͯͷύλʔϯೝࣝྠಡձ ޻౻঑ 1

065-*/& ϕΠζͷࣝผنଇ ࠞಉߦྻ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ষ຤໰୊ ·ͱΊ ຕ ຕ ຕ ຕ
ຕ !2

ϕΠζͷࣝผنଇ පؾͩͬͨʜ Cd ݈߁ͩͬͨ Ch ຊ౰͸݈߁ͩͬͨͷʹ පؾͩͱݴͬͯ͠·ͬͨ ຊ౰͸පؾͩͬͨͷʹ ݈߁ͩͱݴͬͯ͠·ͬͨ ِӄੑ
ِཅੑ ࣝผڥք !5 ϵ(x) = min [P(Ci |x)] ΫϥεΛޡΔ֬཰ʢස౓ʣ 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 ͜ΕΒͷ஋Λ࢖ͬͯࣝผੑೳΛධՁ͢Δ !7

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ِཅੑ཰GBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹ !8

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ਅཅੑ཰USVFQPTJUJWFSBUF TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ !9

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ద߹཰QSFDJTJPO TP TP + FP ཅੑͱ൑அͨ͠΋ͷͷ͏ͪຊ౰ʹཅੑͰ͋Δׂ߹ !10

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ !11

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε ਖ਼֬౓BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ !12

'/ O ِཅੑ '1 ਅӄੑ 5' /'1 5/ ࣝผΫϥε ਅͷΫϥε '஋'NFBTVSF 2 1 precision + 1 recall ద߹཰ͱ࠶ݱ཰ͷௐ࿨ฏۉ !13

ࠞಉߦྻ ِཅੑ཰GBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹
ਅཅੑ཰USVFQPTJUJWFSBUF TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ ద߹཰QSFDJTJPO TP TP + FP ཅੑͱ൑அͨ͠΋ͷͷ͏ͪ ຊ౰ʹཅੑͰ͋Δׂ߹ ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ ਖ਼֬౓BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ '஋'NFBTVSF 2 1 precision + 1 recall ద߹཰ͱ࠶ݱ཰ͷௐ࿨ฏۉ !14

ࠞಉߦྻ ྫ୊ ద߹཰ͱ࠶ݱ཰͸ͳͥτϨʔυΦϑͷؔ܎ʹ͋Δͷ͔ʁ ໢ཏੑΛ্͛Α͏ͱཅੑͱड͚ೖΕΔ΋ͷΛ૿΍͢ͱ ِཅੑͱ൑அ͢Δ΋ͷ΋૿Ճ͠ɼ݁Ռͱͯ͠ద߹཰͕Լ͕ΔͨΊ ద߹཰QSFDJTJPO TP TP + FP
ཅੑͱ൑அͨ͠΋ͷͷ͏ͪ ຊ౰ʹཅੑͰ͋Δׂ߹ ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ !15

ِཅੑ཰GBMTFQPTJUJWFSBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹ ਅཅੑ཰USVFQPTJUJWFSBUF
TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ පؾͷਓ͔݈߁ͳਓͳͲͷαϯϓϧ਺ʹେ͖ͳภΓ͕͋Δ৔߹Ͱ΋ ཅੑӄੑͷ΋ͷ͚ͩͰ࡞ΒΕ͍ͯΔͷͰɼͦΕͧΕӄੑཅੑͷӨڹΛड͚ͳ͍ 30$ۂઢ !16

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰ ਅཅੑ཰ # " $ ࣝผڥք Cp*
Cn* ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ཅੑʢࠨʣଆͷ ֬཰෼෍ʹ஫໨͢Δ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅ !17

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰ ਅཅੑ཰ # " $ Cp* Cn*
ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ࣝผڥք ཅੑʢࠨʣଆͷ ֬཰෼෍ʹ஫໨͢Δ !18 ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰ ਅཅੑ཰ # " $ Cp* Cn*
ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ࣝผڥք ཅੑʢࠨʣଆͷ ֬཰෼෍ʹ஫໨͢Δ !19 ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑ཰ɾِཅੑ཰ (0,0) (0.1,0.7) (1.0,1.0) ཅੑʢࠨʣଆͷ ֬཰෼෍ʹ஫໨͢Δ ʢԣɼॎʣʢِཅੑ཰ɼਅཅੑ཰ʣ !20 ਅཅੑ཰ɿ੺
ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 4 ΔE = 3 !21

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 2 ΔE = 1 !22

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 0 30$͔Βݴ͑Δ͜ͱ͸Կ͔ʁ !23

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ͷॏͳΓ͕খ͍͞΄Ͳ 30$ͷԼ෦෼ͷ໘ੵ͕େ͖͘ͳ͍ͬͯΔ !24

30$ͷԼ෦෼ͷ໘ੵɿ"6$ "SFB6OEFSUIF$VSWF ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ׬ᘳʹࣝผͰ͖Δɿ AUC = 1 શࣝ͘ผͰ͖ͳ͍ɿ AUC =
0.5 0.5 ≦ AUC ≦ 1.0 !25

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ "6$Λେ͖͘͢ΔͨΊʹ͸ ྑ͍෼཭ੑͷ֬཰෼෍Λ࣋ͭಛ௃ྔͷ૊Έ߹Θͤ !26

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑ཰ɾِཅੑ཰ (0,0) (0.1,0.7) (1.0,1.0) ʢԣɼॎʣʢِཅੑ཰ɼਅཅੑ཰ʣ ͋͘·Ͱ΋ ֬཰෼෍ͷੑೳධՁ Ͳͷ஋ʹ஫໨͢Ε͹ྑ͍ʁ 30$
!27

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ͋͘·Ͱ΋ ֬཰෼෍ͷੑೳධՁ # " $ Ͳͷ஋ʹ஫໨͢Ε͹ྑ͍ʁ ͲͷࣝผڥքΛબ΂͹ྑ͍ʁ 30$ ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍
!28

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ # " $ ࣝผڥք Cp* Cn* ِཅੑ཰ ਅཅੑ཰
ཅੑͷྖҬʹ͓͚Δଛࣦ ଛࣦ ਅཅੑ཰ʹ༝དྷ͢Δଛࣦ ଛࣦ ِཅੑ཰ʹ༝དྷ͢Δଛࣦ r(C1 |x) = Σ2 k=1 L1k P(Ck |x) !29

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ӄੑྖҬʹ͓͚Δଛࣦ΋ಉ༷ʹॲཧ͢Δͱɼଛࣦ͸྆ऀͷظ଴஋ΛͱΔ͜ͱͰ 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$ʣ r = Lp*n* P(Cn* )ϵ2 + Ln*p* P(Cp* )ϵ1
30$ʹ͋Δɼਅཅੑ཰Џ ͱِཅੑ཰Џ ͰࣜΛॻ͖௚͢ͱ࣍ࣜͷΑ͏ʹͳΔɽ 1 − ϵ1 = Lp*n* P(Cn* ) Ln*p* P(Cp* ) ϵ2 + (1 − r Ln*p* P(Cp* ) ) = αϵ2 + h(r) ͜ͷࣜ͸1 $Q ͱ1 $O ͓Αͼଛࣦ-Q O ͱ-O Q ʹΑͬͯЏ ͷ܏͖͕ܾ·Γɼ ੾ยΛSͷؔ਺ͱͨ͠௚ઢͷํఔࣜͱଊ͑Δ͜ͱ͕Ͱ͖Δɽ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ Ͱ͸ɼ༩͑ΒΕͨࣜͷதͰ࠷΋ద੾ͳࣜ͸ͲͷΑ͏ͳܗʹͳΔͩΖ͏͔ʁ ৚݅͸ҎԼͷ௨ΓͰ͋Δɽ min r = max h(r) ௚ઢ͕30$ͱަ఺Λ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ৚݅ͷൣғ !32

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 1 − ϵ1 = αϵ2 + h(r) ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ min
r = max h(r) ௚ઢ͕30$ͱަ఺Λ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ৚݅ͷൣғ ৚݅ ੾ยI S খଛࣦSେ ଛࣦ࠷খ͔ͭ30$ͱަ఺Λ࣋ͭ఺ (0.1,0.7) ͍͍ͩͨ ͘Β͍ʁ ਅཅੑ཰ͱِཅੑ཰ͷׂ߹͕ ఺ͷ஋ʹͳΔΑ͏ʹࣝผڥքΛఆΊΔͱ ࠷΋ޡΓΛখͨࣝ͘͞͠ผ͕ՄೳʹͳΔ !33

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ 30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ ਅͷΫϥεͱσʔλͷείΞ Q
O ᮢ஋ͷείΞ4Ҏ্ͷείΞΛ࣋ͭσʔλΛQ ͱࣝผ͢Δɽ ͜ΕΛɼᮢ஋ΛมԽͤͯ͞܁Γସ͑͢ɽ !34

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ Q O
30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ ᮢ஋4 ਅཅੑ཰ ِཅੑ཰ 㱣㱣 !35

ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ Q O
30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ !36

# " $ Cp* Cn* ِཅੑ཰ ਅཅੑ཰ ষ຤໰୊ 30$ۂઢΛඳ͍ͨͱ͜ΖɼϥϯμϜࣝผΛද͢
౓ͷ௚ઢͷԼଆʹདྷͯ͠·ͬͨɽͲ͏͢Ε͹ྑ͍͔ɽ ্ਤͷਅཅੑ཰ͱِཅੑ཰ͷେ͖͕͞ٯస͍ͯ͠Δྫͱଊ͑ΒΕΔͷͰɼ ୯७ʹཅੑͱӄੑͷΫϥεΛٯస͢Ε͹͍͍ ͋Δ͍͸ɼ྆ํͷείΞʹϚΠφεΛ͔͚Δ͔ᮢ஋ͷεΩϟϯΛٯ͔Βߦ͏ͱ͔ !37

·ͱΊ ੾ยI S খଛࣦSେ ෼ྨͷධՁ͸ࠞಉߦྻ͔ΒಘΒΕΔ ͍͔ͭ͘ͷࢦඪͰߦ͑Δ ࠞಉߦྻΛ༩͑Δ֬཰෼෍ͷධՁ͸ 30$ͷԼ෦໘ੵ "6$ ʹΑͬͯධՁ͢Δ
֬཰෼෍ͷ࠷΋ྑ͍ࣝผڥք͸ 30$ͱϕΠζͷࣝผنଇͷ ଛࣦ͕࠷খʹͳΔ఺ΛબͿ͜ͱͰಘΒΕΔ 30$͸֬཰෼෍͕Θ͔Βͳͯ͘΋ ඳ͘͜ͱ͕Ͱ͖Δ !38

ଛࣦߦྻ ଛࣦʹؔ͢Δࣜʹग़ͯ͘Δ-ʹ͍ͭͯ r(Ci |x) = ΣK k=1 Lik P(Ck |x)
,ͷ࣌ɼS͸࠷΋؆୯ͳ-Λ࢖ͬͯ࣍ͷΑ͏ʹॻ͖දΘͤΔɽ 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)) ͜ͷதͰখ͍͞ํΛऔΔ࣌͸ҎԼͷΑ͏ʹॻ͚Δ min ( p(C2 |x) p(C1 |x)) !39

Receiver Operator Characteristic Curve

Receiver Operator Characteristic Curve

More Decks by Shoh-kudo

Other Decks in Education

Featured

Transcript