Accumulated Local Effects（ALE）で機械学習モデルを解釈する / TokyoR95

2021年10月30日に行われた、第95回R勉強会@東京（#TokyoR）での発表資料です。
https://tokyor.connpass.com/event/225967/

コードはこちらになります。
https://github.com/dropout009/tokyor95

Transcript

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

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2. TVISION INSIGHTS
   Twitter: @dropout009
   Speaker Deck: dropout009
   Blog: https://dropout009.hatenablog.com/
   

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4. RF/GBDT/NN
   +
   

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5. Partial Dependence
   

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

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

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8. PD R
   

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

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10. tidymodels
    tidymodels rand_forest()
    Random Forest
    

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11. = 0
    PD
    PD
    

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12. = 0.99
    PD
    PD
    

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13. 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
    

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14. PD
    

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

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16. 4 PD
    𝑋!
    𝑥!
    𝑋"
    (𝑥!,#
    , 𝑥!,!
    )
    (𝑥$,#
    , 𝑥$,!
    )
    (𝑥%,#
    , 𝑥%,!
    )
    𝑥#,#
    , 𝑥#,!
    

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17. 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
    

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18. Conditional Dependence
    (Marginal Plot)
    

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

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20. 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"
    

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

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22. CD
    *
    𝑓(
    𝑥"
    ($%&) + 𝑥"
    ($)
    2
    , 𝒙(,∖"
    )
    

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23. CD
    CD
    

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24. 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"
    

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25. Accumulated Local Effects
    

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

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27. 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
    

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28. ALE
    Local Effect
    Local Effect
    

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29. ALE
    ALE
    

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30. 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))
    

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31. 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
    

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32. 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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33. View Slide

34. •
    • Partial Dependence ⾒
    • Accumulated Local Effects
    ⾒
    • ALE
    

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35. • Friedman, Jerome H. "Greedy function approximation: a gradient
    boosting machine." Annals of statistics (2001): 1189-1232.
    • Hooker, Giles, and Lucas Mentch. "Please Stop Permuting Features: An
    Explanation and Alternatives." arXiv preprint arXiv:1905.03151 (2019).
    • Apley, Daniel W., and Jingyu Zhu. "Visualizing the effects of predictor
    variables in black box supervised learning models." Journal of the Royal
    Statistical Society: Series B (Statistical Methodology) 82.4 (2020): 1059-
    1086.
    • Molnar, Christoph. "Interpretable machine learning. A Guide for Making
    Black Box Models Explainable." (2019).
    https://christophm.github.io/interpretable-ml-book/.
    • Biecek, Przemyslaw and Tomasz Burzykowski. "Explanatory Model
    Analysis. Chapman and Hall/CRC (2021). https://pbiecek.github.io/ema/.
    • .
    . . (2021).
    https://is.gd/nkYPPG
    

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36. R
    

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