Accumulated Local Effects 2021/10/30 95 R @ #TokyoR @dropout009

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RF/GBDT/NN +

Partial Dependence

Partial Dependence PD PD! 𝑥" = 𝔼 & 𝑓 𝑥" , 𝑿∖" = * & 𝑓 𝑥" , 𝒙∖" 𝑝 𝒙∖" 𝑑𝒙∖" ! 𝑓(𝑿) 𝑋!

PD ' PD! 𝑥! = 1 𝑁 . "#$ % ! 𝑓(𝑥! , 𝒙",∖! )

PD R

PD 𝑌 = 𝑋! + 𝑋" " + 𝜖 𝑋! , 𝑋" ∼ Uniform 0, 1 𝜖 ∼ 𝒩(0, 0.01") CDF

tidymodels tidymodels rand_forest() Random Forest

= 0 PD PD

= 0.99 PD PD

PD ! 𝑓 𝑋$ , 𝑋( = 𝑋$ + 𝑋( ( PD! 𝑥! = 𝔼 9 𝑓 𝑥! , 𝑋" = 𝔼 𝑥! + 𝑋" " = ! # + 𝑥! PD" 𝑥" = 𝔼[ 9 𝑓 𝑋! , 𝑥" ] = 𝔼 𝑋! + 𝑥" " = ! " + 𝑥" " PD

PD

PD 𝑋! 𝑥! 𝑋" PD! 𝑥! = 𝔼 & 𝑓 𝑥! , 𝑋" = ∫ & 𝑓 𝑥!, 𝑥" 𝑝 𝑥" 𝑑𝑥" PD 𝑋!

4 PD 𝑋! 𝑥! 𝑋" (𝑥!,# , 𝑥!,! ) (𝑥$,# , 𝑥$,! ) (𝑥%,# , 𝑥%,! ) 𝑥#,# , 𝑥#,!

𝑋! 𝑥! 𝑋" 𝑥#,# , 𝑥#,! (𝑥!,# , 𝑥!,! ) (𝑥$,# , 𝑥$,! ) - PD! 𝑥! = ! ' & 𝑓 𝑥!, 𝑥!," + ! ' & 𝑓 𝑥!, 𝑥"," + ! ' & 𝑓 𝑥! , 𝑥(," + ! ' & 𝑓 𝑥! , 𝑥'," (𝑥%,# , 𝑥%,! ) ) 𝑓(𝑥# , 𝑥!,! ) ) 𝑓(𝑥# , 𝑥!,! ) ) 𝑓(𝑥# , 𝑥%,! ) ) 𝑓(𝑥# , 𝑥#,! ) PD

Conditional Dependence (Marginal Plot)

PD 𝑋! 𝑥! 𝑋" PD! 𝑥! = 𝔼 & 𝑓 𝑥! , 𝑋" = ∫ & 𝑓 𝑥!, 𝑥" 𝑝 𝑥" 𝑑𝑥" PD 𝑋!

築 𝑋! 𝑥! 𝑋" PD! 𝑥! = 𝔼 & 𝑓 𝑥! , 𝑋" = ∫ & 𝑓 𝑥!, 𝑥" 𝑝 𝑥" 𝑑𝑥" CD! 𝑥! = 𝔼 & 𝑓 𝑥! , 𝑋" ∣ 𝑋! = 𝑥! = ∫ & 𝑓 𝑥!, 𝑥" 𝑝 𝑥" ∣ 𝑥! 𝑑𝑥"

Conditional Dependence 𝑋! 𝑥! (") 𝑋" 𝑥! (() 𝑥! (') 𝑥! (+) 𝑥! (,) 𝑥! (!)

CD * 𝑓( 𝑥" ($%&) + 𝑥" ($) 2 , 𝒙(,∖" )

CD CD

CD 𝑓 𝑋! , 𝑋" = 𝑋! + 𝑋" " 𝑋! = 𝑋" CD! 𝑥! = 𝔼 & 𝑓 𝑥!, 𝑋" ∣ 𝑋! = 𝑥! = 𝔼 𝑥! + 𝑋" " ∣ 𝑋! = 𝑥! = 𝑥! + 𝔼 𝑋" " ∣ 𝑋! = 𝑥! = 𝑥! + 𝑥! " CD" 𝑥" = 𝔼 & 𝑓 𝑋!, 𝑥" ∣ 𝑋" = 𝑥" = 𝔼 𝑋! + 𝑥" " ∣ 𝑋" = 𝑥" = 𝔼 𝑋! ∣ 𝑋" = 𝑥" + 𝑥" " = 𝑥" + 𝑥" " CD 𝑋! 𝑋"

Accumulated Local Effects

ALE CD 𝑋! 𝑥! (") 𝑋" 𝑥! (() 𝑥! (') 𝑥! (+) 𝑥! (,) 𝑥! (!) 𝑥#,# , 𝑥#,! 𝑥!,# , 𝑥!,! 𝑥$,# , 𝑥$,!

ALE Local Effect Accumulate 𝑋! 𝑥! (") 𝑋" 𝑥! (() 𝑥! (') 𝑥! (+) 𝑥! (,) 𝑥! (!) ) 𝑓 𝑥# ($), 𝑥#,! ) 𝑓 𝑥# ($) , 𝑥!,! ) 𝑓 𝑥# ($), 𝑥!,! ) 𝑓 𝑥# (%), 𝑥#,! ) 𝑓 𝑥# (%), 𝑥!,! ) 𝑓 𝑥# (%), 𝑥!,! Local Effect 1 3 ) 𝑓 𝑥# (%), 𝑥#,! − ) 𝑓 𝑥# ($), 𝑥#,! + 1 3 ) 𝑓 𝑥# (%), 𝑥!,! − ) 𝑓 𝑥# ($), 𝑥!,! + 1 3 ) 𝑓 𝑥# (%), 𝑥$,! − ) 𝑓 𝑥# ($), 𝑥$,! Local Effect

ALE Local Effect Local Effect

ALE ALE

ALE 𝑓 𝑋! , 𝑋" = 𝑋! + 𝑋" " 𝑖 𝑋! [𝑥! (12!), 𝑥! (1)) Local Effect & 𝑓 𝑥! (1), 𝑥3," − & 𝑓 𝑥! 12! , 𝑥3," = 𝑥! 1 + 𝑥3," " − 𝑥! 12! + 𝑥3," " = 𝑥! 1 − 𝑥! 12! 𝑋" 𝑋! [𝑥! (12!), 𝑥! (1))

ALE ALE Local Effect 𝑋4 𝜕 * 𝑓(𝑋", 𝑿∖") 𝜕𝑋" 𝔼 𝜕 * 𝑓(𝑋", 𝑿∖") 𝜕𝑋" ∣ 𝑋" = 𝑧" ALE" 𝑥" = ; + ! (#$%) +! 𝔼 𝜕 * 𝑓(𝑋", 𝑿∖") 𝜕𝑋" ∣ 𝑋" = 𝑧" 𝑑𝑧" 𝑋4 = 𝑧4 Local Effect Local Effect 𝑥4 (567) ALE

ALE ! 𝑓 𝑋$ + 𝑋( ( = 𝑋$ + 𝑋( ( ALE ALE! 𝑥! = 9 8 9! 𝔼 𝜕(𝑋! + 𝑋" ") 𝜕𝑋! ∣ 𝑋! = 𝑧! 𝑑𝑧! = 9 8 9! 𝔼 1 ∣ 𝑋! = 𝑧! 𝑑𝑧! = 9 8 9! 1𝑑𝑧! = 𝑥! ALE" 𝑥" = 9 8 9" 𝔼 𝜕(𝑋! + 𝑋" ") 𝜕𝑋" ∣ 𝑋" = 𝑧" 𝑑𝑧" = 9 8 9" 𝔼 2𝑋" ∣ 𝑋" = 𝑧" 𝑑𝑧" = 9 8 9" 2𝑧"𝑑𝑧" = 𝑥 " "

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• • Partial Dependence ⾒ • Accumulated Local Effects ⾒ • ALE

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