From 33c829318ab87914d35cbe16d367841124a61068 Mon Sep 17 00:00:00 2001 From: bgreenwell Date: Tue, 25 Aug 2020 00:36:28 -0400 Subject: [PATCH 1/6] update article --- .Rproj.user/shared/notebooks/paths | 52 +- R/gen_pkg_bib.R | 33 + rjournal/RJwrapper.log | 1500 +++++++++++++++++ rjournal/RJwrapper.pdf | Bin 163583 -> 178421 bytes rjournal/RJwrapper.tex | 1 + rjournal/greenwell.Rmd | 275 ++- rjournal/greenwell.bib | 42 +- rjournal/greenwell.tex | 426 ++++- rjournal/greenwell_cache/latex/__packages | 2 + ...ark_ea93b04f25ab77d3e5743f87ba471dd0.RData | Bin 0 -> 3590 bytes ...hmark_ea93b04f25ab77d3e5743f87ba471dd0.rdb | Bin 0 -> 732 bytes ...hmark_ea93b04f25ab77d3e5743f87ba471dd0.rdx | Bin 0 -> 197 bytes ...act_2412fe79940d9afc3962c5dcb03a7d84.RData | Bin 0 -> 3206 bytes ...exact_2412fe79940d9afc3962c5dcb03a7d84.rdb | 0 ...exact_2412fe79940d9afc3962c5dcb03a7d84.rdx | Bin 0 -> 124 bytes ...glm_1f109ee39a9c05a547e16f1739194dbf.RData | Bin 0 -> 229 bytes 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rjournal/greenwell_cache/latex/titanic-load_7b5b154b038c175a2b2bca080ecd3ff9.rdx create mode 100644 rjournal/greenwell_files/figure-latex/titanic-benchmark-1.pdf create mode 100644 rjournal/greenwell_files/figure-latex/titanic-jack-explanations-1.pdf diff --git a/.Rproj.user/shared/notebooks/paths b/.Rproj.user/shared/notebooks/paths index 012e413..8a04e31 100644 --- a/.Rproj.user/shared/notebooks/paths +++ b/.Rproj.user/shared/notebooks/paths @@ -1,49 +1,3 @@ -/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppArmadillo/include/RcppArmadillo.h="7A2EC50" -/Users/b780620/.R/Makevars="2A873AA9" -/Users/b780620/Desktop/devel/fastshap/.Rbuildignore="9D5980A4" -/Users/b780620/Desktop/devel/fastshap/.gitattributes="1590E2AD" -/Users/b780620/Desktop/devel/fastshap/.gitignore="7E490EEC" -/Users/b780620/Desktop/devel/fastshap/.travis.yml="2720CBBB" -/Users/b780620/Desktop/devel/fastshap/DESCRIPTION="AD1517C7" -/Users/b780620/Desktop/devel/fastshap/NAMESPACE="BF528FAD" 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-/Users/b780620/Desktop/devel/vip/R/gen_friedman.R="A1AA2FC7" -/Users/b780620/Desktop/devel/vip/R/vi_permute.R="DA45DF66" +/Users/bgreenwell/Dropbox/devel/fastshap/R/gen_pkg_bib.R="E1B8FA98" +/Users/bgreenwell/Dropbox/devel/fastshap/rjournal/greenwell.Rmd="44C6DFCC" +/Users/bgreenwell/Dropbox/devel/fastshap/rjournal/greenwell.bib="6C1920D6" diff --git a/R/gen_pkg_bib.R b/R/gen_pkg_bib.R new file mode 100644 index 0000000..a41517f --- /dev/null +++ b/R/gen_pkg_bib.R @@ -0,0 +1,33 @@ +# # Grab cited packages from Rmd file +# lines <- readLines("rjournal/greenwell-boehmke.Rmd") +# z <- sapply(lines, FUN = function(x) { +# stringi::stri_extract(x, pattern = "pkg\\{[:alnum:]*\\}", regex = TRUE) +# }) +# z <- unname(sort(unique(z))) +# z <- gsub("^pkg\\{", replacement = "", x = z) +# z <- gsub("\\}$", replacement = "", x = z) + +# Remove current bib files, if they exist +files <- c("rjournal/greenwell.bib", "rjournal/packages.bib") +for (f in files) { + if (file.exists(f)) { + file.remove(f) + } +} + +# List of cited packages to include in the bibliography +pkgs <- c( + "iml", + "Rcpp" +) + +# Make sure the packages listed above are installed and up to date +required_pkgs <- setdiff(pkgs, installed.packages()[, "Package"]) +install.packages(required_pkgs) + +# Generate bibliography +knitr::write_bib(pkgs, file = "rjournal/greenwell.bib", tweak = TRUE, + width = NULL, prefix = "R-") + +# Create new bib file +file.append("rjournal/greenwell.bib", "rjournal/general.bib") diff --git a/rjournal/RJwrapper.log b/rjournal/RJwrapper.log new file mode 100644 index 0000000..7a8a653 --- /dev/null +++ b/rjournal/RJwrapper.log @@ -0,0 +1,1500 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded format=pdflatex 2017.5.23) 25 AUG 2020 00:34 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**RJwrapper.tex +(./RJwrapper.tex +LaTeX2e <2017-04-15> +Babel <3.10> and hyphenation patterns for 84 language(s) loaded. 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The Shapley value is the average marginal contribution of a \emph{player} across all possible \emph{coalitions} in a \emph{game}. In the context of statistical/machines learning, +\begin{description} + + \item[Game:] The prediction task for a single observation $x$. + + \item[Gain:] The prediction for $x$ minus the average prediction for all training observations. + + \item[Players] The feature values of $x$ that collaborate to receive the gain (i.e., predict a certain value). + +\end{description} + +In particular, the Shapley contribution of the $i$-th feature to an instance $x$ is defined as +\begin{equation} +\nonumber +\phi_i\left(x\right) = \frac{1}{p!} \sum_{\mathcal{O} \in \pi\left(p\right)} \left[\Delta Pre^i\left(\mathcal{O}\right) \cup \left\{i\right\} - Pre^i\left(\mathcal{O}\right)\right], \quad i = 1, 2, \dots, p, +\end{equation} +where ... + +A simple example may help clarify the main ideas. + +### Fairly splitting a bar tab + +Alex, Brad, and Brandon decide to go out for drinks after work. They shared a few pitchers of beer, but nobody payed attention to how much each person drank. What's a fair way to split the tab? Suppose we knew the follow information, perhaps based on historical data: +\begin{itemize} + + \item If Alex drank alone, he'd only pay \$10. + + \item If Brad drank alone, he'd only pay \$20. + + \item If Brandon drank alone, he'd only pay \$10. + + \item If Alex and Brad drank together, they'd only pay \$25. + + \item If Alex and Brandon drank together, they'd only pay \$15. + + \item If Brad and Brandon drank together, they'd only pay \$13. + + \item If Ales, Brad, and Brandon drank together, they'd only pay \$30. + +\end{itemize} + +\begin{table}[] +\centering +\begin{tabular}{@{}llll@{}} +\toprule + & \multicolumn{3}{l}{Marginal contribution} \\ \midrule +Permutation & Alex & Brad & Brandon \\ \midrule +Alex, Brad, Brandon & \$10 & \$15 & \$5 \\ +Alex, Brandon, Brad & \$10 & \$15 & \$5 \\ +Brad, Alex, Brandon & \$5 & \$20 & \$5 \\ +Brad, Brandon, Alex & \$10 & \$20 & \$0 \\ +Brandon, Alex, Brad & \$5 & \$15 & \$10 \\ +Brandon, Brad, Alex & \$17 & \$3 & \$10 \\ \midrule +Shapley contribution: & \$9.50 & \$14.67 & \$5.83 \\ \bottomrule +\end{tabular} +\caption{Marginal contribution for each permutation of Alex, Brad, and Brandon (i.e., the order in which they arrive). The Shapley contribution is the average marginal contribution across all permutations. (Notice how each row sums to the total bill of \$30.)} +\end{table} + +So the next time the bartender asks how you want to split the tab, whip out a pencil and do the math! + + +### Estimating Shapley values via Monte Carlo simulation: SampleSHAP + +A single estimate of the contribution of $x_i$ to $f\left(x\right)$ is nothing the more than the difference between two predictions, where each prediction is based on a sort of Frankenstein instance that' are's constructed by swapping out values between the instance being explained ($x$) and an instance selected at random from the training data. To help stabilize the results, the procedure is repeated a large number, say, $R$, times, and the result averaged together. \begin{algorithm} \begin{enumerate} - \item For $i = 1, 2, \dots, j$: + \item For $j = 1, 2, \dots, R$: \begin{enumerate} - \item Permute the values of feature $X_i$ in the training data. - \item Recompute the performance metric on the permuted data $\mathcal{M}_{perm}$. - \item Record the difference from baseline using $imp\left(X_i\right) = \mathcal{M}_{perm} - \mathcal{M}_{orig}$. + \item Select a random permutation $\mathcal{O}$ of the sequence $1, 2, \dots, p$. + \item Select a random instance $w$ from the training instances $\boldsymbol{X}$. + \item Construct two new instances as follows: + \begin{itemize} + \item $b_1 = x$, but all the features in $\mathcal{O}$ that appear after feature $x_i$ get their values swapped with the corresponding values in $w$. + \item $b_2 = x$, but feature $x_j$, as well as all the features in $\mathcal{O}$ that appear after $x_j$, get their values swapped with the corresponding values in $w$. + \end{itemize} + \item $\phi_{ij}\left(x\right) = f\left(b_1\right) - f\left(b_2\right)$. \end{enumerate} - \item Return the VI scores $imp\left(X_1\right), imp\left(X_2\right), \dots, imp\left(X_j\right)$. + \item $\phi_i\left(x\right) = \sum_{j = 1} ^ R \phi_{ij}\left(x\right) / R$. \end{enumerate} -\caption{Monte Carlo algorithm for approximating Shapley values. \label{alg:permute}} +\caption{Approximating the $i$-th feature's contribution to $f\left(x\right)$. \label{alg:SampleSHAP}} \end{algorithm} +If there are $p$ features and $m$ instanced to be explained, this requires $2 \times R \times p \times m$ predictions (or calls to a scoring function). In practice, this can be quite computationally demanding, especially since $R$ needs to be large enough to produce good approximations to each $\phi_i\left(x\right)$. In practice, this depends on the variance of each feature in the observed training data, but typically $R >= 50--100$ will suffice (**Need reference**). + ### Special cases -TBD. +The following sections discuss two special cases where exact Shapley explanations can be computed efficiently: additive linear models, and shallow trees and tree ensembles. #### Linear models: LinearSHAP -TBD. +Cite somewhere \cite{strumbelj-2014-explaining}. + +First, lets discuss how a feature's value contributes to a prediction $f\left(X\right)$ in a simple (additive) linear model. That is, let's assume for a moment that $f$ takes the form +\begin{equation} +\nonumber + f\left(X\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p +\end{equation} + +Recall that the contribution of the $i$-th feature to the prediction $f\left(X\right)$ is the difference between $f\left(X\right)$ and the expected prediction if the $i$-th feature’s value were not known: +\begin{equation} +\nonumber +\begin{split} + \phi_i\left(X\right) &= \beta_0 + \dots + \beta_i X_i + \dots + \beta_p X_p \\ &\quad\quad - \left(\beta_0 + \dots + \beta_i \E\left(X_i\right) + \dots + \beta_p X_p\right) \\ + &= \beta_i \left(X_i - \E\left(X_i\right)\right) +\end{split}, +\end{equation} +where we estimate $\E\left(X_i\right)$ with the corresponding sample mean $\bar{X}_i$. The quantity $\phi_i\left(X\right)$ is also referred to as the \emph{situational importance of $X_i$} \citep{achen-1982-interpreting}. #### Tree-based models: TreeSHAP @@ -73,6 +165,153 @@ TBD. TBD. +### So why fastshap? + + + + +#### Efficiency + +Like many post-hoc interpretation techniques (e.g., PDPs and ICE curves), SampleSHAP can be made more efficient by generating all the data up front, and scoring it only once (or twice, in the case of SampleSHAP). For example, PDPs and ICE curves can be efficiently constructed with only a single call to a scoring function by generating all of the required data up front using a single cross-join operation (which can be done rather efficiently in SQL or Spark). The scored data can then be post-processed/aggregated and displayed as either a PDP or set of ICE curves. An example using Spark with \CRANpkg{sparklyr} \citet{R-sparklyr} can be found here: \url{https://github.com/bgreenwell/pdp/issues/97}. + +Fortunately, a similar trick can be exploited for SampleSHAP. Whether explaining a single instance with a large value of Monte Carlo reps ($R$), or explaining a large number of instances, the basic idea is to generate all the required Frankenstein instances $b_1$ and $b_2$ upfront, and stored in matrices $\boldsymbol{B}_1$ and $\boldsymbol{B}_2$, respectively. + +For example, suppose we wanted to estimate the contribution of $x_i$ for each of the $N$ rows of the available training data $\boldsymbol{X}$ using a single Monte-Carlo repetition in Algorithm~\ref{alg:SampleSHAP} (i.e., $R = 1$)\footnote{The same idea also extends to explaining new instances.}. To start, we can generate the $N$ random instances at once and store them in an $N \times p$ matrix $\boldsymbol{W}$. Rather generating $N$ random permutations $\mathcal{O}$, and constructing $b_1$ and $b_2$ one at a time, the \pkg{fastshap} package uses C++---via \CRANpkg{Rcpp} \citep{R-Rcpp}---to efficiently generate an $N \times p$ logical matrix $\boldsymbol{\mathcal{O}}$, where $\boldsymbol{\mathcal{O}}_{kl} = 1$ if feature $x_l$ appears before feature $x_i$ in the $k$-th permutation, and $0$ otherwise. This logical matrix can then be used to logically subset $\boldsymbol{X}$ and $\boldsymbol{W}$ to more efficiently construct $\boldsymbol{B}_1$ and $\boldsymbol{B}_2$ in a single swoop. The matrices (or data frames) can then be each scored once, and the difference taken, to generate a single replication of $\phi_i\left(x\right)$ for each row of $\boldsymbol{X}$. + +Suppose instead we want to estimate the contribution of $x_i$ for a single instance $x$, but using a large value of $R$ for accuracy. We could employ the same trick, but in this case $\boldsymbol{X}$ would refer to the $R \times p$ matrix, where each row is a copy of the instance $x$. + +\pkg{fastshap} also uses efficient exact methods for the special cases described in Sections... + + +#### Parallelization + +\pkg{fastshap} is faster at computing Shapley values for a single feature for a large number of instances (or a large value of $R$ for a single instance). But what about a large number of features? Fortunately, Algorithm~\ref{alg:SampleSHAP} can be trivially parallelized across features, and this is built into \pkg{fastshap}. + + +#### A simple benchmark comparison + +This section provides a brief example comparing various implementations of Shapley values using [Kaggle’s Titanic: Machine Learning from Disaster competition](https://www.kaggle.com/c/titanic). While the true focus of the competition is to use machine learning to create a model that predicts which passengers survived the Titanic shipwreck, we’ll focus on explaining predictions from a simple logistic regression model. + +To start, we’ll load the data, which are conveniently available in the \CRANpkg{titanic} package \citep{R-titanic}, and do a little bit of cleaning. + +```{r titanic-load} +# Read in the data and clean it up a bit +titanic <- titanic::titanic_train +features <- c( + "Survived", # passenger survival indicator + "Pclass", # passenger class + "Sex", # gender + "Age", # age + "SibSp", # number of siblings/spouses aboard + "Parch", # number of parents/children aboard + "Fare", # passenger fare + "Embarked" # port of embarkation +) +titanic <- titanic[, features] +titanic$Survived <- as.factor(titanic$Survived) +titanic <- na.omit(titanic) + +# Data frame containing just the features +X <- subset(titanic, select = -Survived) +``` + +Next, we’ll use the stats::glm() to fit a logistic regression model with only main effects (i.e., no tw-way interactions, etc.). + +```{r titanic-glm} +fit <- glm(Survived ~ ., data = titanic, family = binomial) +``` +Suppose we wanted to explain the predicted survival probability for a new passenger named Jack: +```{r titanic-jack} +jack <- data.frame( + Pclass = 3, + Sex = factor("male", levels = c("female", "male")), + Age = 20, + SibSp = 0, + Parch = 0, + Fare = 15, # lower end of third-class ticket prices + Embarked = factor("S", levels = c("", "C", "Q", "S")) +) +``` +Our logistic regression model predicts that Jack’s log-odds of survival is +```{r titanic-jack-predict} +predict(fit, newdata = jack) +``` +Yikes, that’s equivalent to estimated 13.64% predicted probability of survival! With a baseline (i.e., average) survival rate of 40.62%, can we explain why the model predicts Jack to be much lower? Enter...Shapley values. + +There is a growing number of R packages that provide Shapley explanations, the two most popular arguably being \pkg{iml} and \pkg{iBreakDown}. In this example, we’ll compare those with \pkg{fastshap}. + +To start, we need to define a few things (prediction wrapper, as well as both \pkg{iml}- and \pkg{iBreakDown}-related helpers). +```{r titanic-helpers} +# Prediction wrapper to compute predicted probability of survive +pfun <- function(object, newdata) { + predict(object, newdata = newdata) +} + +# DALEX-based helper for iBreakDown +explainer <- DALEX::explain(fit, data = X, y = titanic$Survived, predict_function = pfun, verbose = FALSE) + +# Helper for iml +predictor <- iml::Predictor$new(fit, data = titanic, y = "Survived", + predict.fun = pfun) +``` +Next, we call each implementation’s Shapley-related function to compute explanations for Jack’s prediction using 100 Monte Carlo repetitions. +```{r titanic-jack-explanations, fig.width=9, fig.height=3, out.width="100%", fig.cap="TBD."} +# Compute explanations +set.seed(1039) # for reproducibility +ex1 <- iBreakDown::shap(explainer, B = 100, new_observation = jack) +ex2 <- iml::Shapley$new(predictor, x.interest = jack, sample.size = 100) +ex3 <- fastshap::explain(fit, X = X, pred_wrapper = pfun, nsim = 100, + newdata = jack) + +# Plot results +library(ggplot2) # for `autoplot()` function +p3 <- plot(ex1) + ggtitle("iBreakDown") +p2 <- plot(ex2) + ggtitle("iml") +p1 <- autoplot(ex3, type = "contribution") + ggtitle("fastshap") +fastshap::grid.arrange(p1, p2, p3, nrow = 1) +``` + +Each package comes loaded with it’s own bells and whistles (e.g., \pkg{iml} and \pkg{iBreakDown} have particularly fantastic visualizations). The main selling point of \pkg{fastshap} is speed! For example, all three packages (in fact, all general and practical implementations of Shapley values) use Algorithm~\ref{alg:SampleSHAP} which requires a large number of Monte Carlo repetitions to achieve accurate results. Below is a simple benchmark looking at the estimated time (in seconds) to explain Jack’s prediction as a function of the number of Monte Carlo repetitions for each implementation. (Note that this comparison does not make use of \pkg{fastshap}'s feature-wise parallelization.) + +```{r titanic-benchmark, fig.cap="Quick benchmark between three different implementations of SampleSHAP for explaining Jack's unfortunate prediction."} +nsims <- c(1, 5, 10, 25, 50, 75, seq(from = 100, to = 1000, by = 100)) +times1 <- times2 <- times3 <- numeric(length(nsims)) +set.seed(904) +for (i in seq_along(nsims)) { + message("nsim = ", nsims[i], "...") + times1[i] <- system.time({ + iBreakDown::shap(explainer, B = nsims[i], new_observation = jack) + })["elapsed"] + times2[i] <- system.time({ + iml::Shapley$new(predictor, x.interest = jack, sample.size = nsims[i]) + })["elapsed"] + times3[i] <- system.time({ + fastshap::explain(fit, X = X, newdata = jack, pred_wrapper = pfun, + nsim = nsims[i]) + })["elapsed"] +} +pal <- palette.colors(3, palette = "Okabe-Ito") # colorblind friendly palette +plot(nsims, times1, type = "b", xlab = "Number of Monte Carlo repetitions", + ylab = "Time (in seconds)", las = 1, pch = 19, col = pal[1L], + xlim = c(0, max(nsims)), ylim = c(0, max(times1, times2, times3))) +lines(nsims, times2, type = "b", pch = 19, col = pal[2L],) +lines(nsims, times3, type = "b", pch = 19, col = pal[3L],) +legend("topleft", + legend = c("iBreakDown", "iml", "fastshap"), + lty = 1, pch = 19, col = pal, inset = 0.02) +``` +The message to be taken from Figure~\ref{fig:titanic-benchmark} is that \pkg{fastshap} scales incredibly well with $N$ or $R$, as long as the corresponding \code{predict()} method does. + +Oh, and \pkg{fastshap} can produce instant (and exact) Shapley contributions for this example. + +```{r titanic-exact} +fastshap::explain(fit, newdata = jack, exact = TRUE) # ExactSHAP +fastshap::explain(fit, X = X, pred_wrapper = pfun, nsim = 10000, + newdata = jack) # SampleSHAP +predict(fit, newdata = jack, type = "terms") # ExactSHAP (base R) +``` + + ## Example: predicing sales prices TBD. diff --git a/rjournal/greenwell.bib b/rjournal/greenwell.bib index f5d9014..64eaeff 100644 --- a/rjournal/greenwell.bib +++ b/rjournal/greenwell.bib @@ -1,18 +1,28 @@ -@Manual{R, - title = {R: A Language and Environment for Statistical Computing}, - author = {{R Core Team}}, - organization = {R Foundation for Statistical Computing}, - address = {Vienna, Austria}, - year = {2012}, - note = {{ISBN} 3-900051-07-0}, - url = {http://www.R-project.org/}, +% Books ------------------------------------------------------------------------ + +@book{achen-1982-interpreting, + title = {Interpreting and Using Regression}, + author = {Achen, Christopher H.}, + isbn = {9780803900004}, + lccn = {82042675}, + series = {Interpreting and Using Regression}, + year = {1982}, + publisher = {Sage Publications} +} + + +% Articles --------------------------------------------------------------------- + +@article{strumbelj-2014-explaining, + author = {{Š}trumbelj, Erik and Kononenko, Igor}, + title = {Explaining prediction models and individual predictions with feature contributions}, + journal = {Knowledge and Information Systems}, + year = {2014}, + volume = {31}, + number = {3}, + pages = {647--665}, + url = {https://doi.org/10.1007/s10115-013-0679-x} } -@article{ihaka:1996, - Author = {Ihaka, Ross and Gentleman, Robert}, - Journal = {Journal of Computational and Graphical Statistics}, - Number = 3, - Pages = {299--314}, - Title = {R: A Language for Data Analysis and Graphics}, - Volume = 5, - Year = 1996} + +% Other ------------------------------------------------------------------------ diff --git a/rjournal/greenwell.tex b/rjournal/greenwell.tex index 832ec70..0897b0c 100644 --- a/rjournal/greenwell.tex +++ b/rjournal/greenwell.tex @@ -18,47 +18,145 @@ \subsection{Introduction}\label{introduction}} \hypertarget{background}{% \subsection{Background}\label{background}} -This section may contain a figure such as Figure \ref{fig:Rlogo}. +So what's a Shapley value? The Shapley value is the average marginal +contribution of a \emph{player} across all possible \emph{coalitions} in +a \emph{game}. In the context of statistical/machines learning, -\begin{Schunk} -\begin{figure}[htbp] +\begin{description} -{\centering \includegraphics[width=2in]{/Users/b780620/Desktop/devel/fastshap/man/figures/logo} + \item[Game:] The prediction task for a single observation $x$. + + \item[Gain:] The prediction for $x$ minus the average prediction for all training observations. + + \item[Players] The feature values of $x$ that collaborate to receive the gain (i.e., predict a certain value). + +\end{description} -} +In particular, the Shapley contribution of the \(i\)-th feature to an +instance \(x\) is defined as \begin{equation} +\nonumber +\phi_i\left(x\right) = \frac{1}{p!} \sum_{\mathcal{O} \in \pi\left(p\right)} \left[\Delta Pre^i\left(\mathcal{O}\right) \cup \left\{i\right\} - Pre^i\left(\mathcal{O}\right)\right], \quad i = 1, 2, \dots, p, +\end{equation} where \ldots{} -\caption[The logo of fastshap]{The logo of fastshap.}\label{fig:Rlogo} -\end{figure} -\end{Schunk} +A simple example may help clarify the main ideas. -\hypertarget{estimating-shapley-values-in-practice}{% -\subsubsection{Estimating Shapley values in -practice}\label{estimating-shapley-values-in-practice}} +\hypertarget{fairly-splitting-a-bar-tab}{% +\subsubsection{Fairly splitting a bar +tab}\label{fairly-splitting-a-bar-tab}} -TBD. +Alex, Brad, and Brandon decide to go out for drinks after work. They +shared a few pitchers of beer, but nobody payed attention to how much +each person drank. What's a fair way to split the tab? Suppose we knew +the follow information, perhaps based on historical data: + +\begin{itemize} + + \item If Alex drank alone, he'd only pay \$10. + + \item If Brad drank alone, he'd only pay \$20. + + \item If Brandon drank alone, he'd only pay \$10. + + \item If Alex and Brad drank together, they'd only pay \$25. + + \item If Alex and Brandon drank together, they'd only pay \$15. + + \item If Brad and Brandon drank together, they'd only pay \$13. + + \item If Ales, Brad, and Brandon drank together, they'd only pay \$30. + +\end{itemize} + +\begin{table}[] +\centering +\begin{tabular}{@{}llll@{}} +\toprule + & \multicolumn{3}{l}{Marginal contribution} \\ \midrule +Permutation & Alex & Brad & Brandon \\ \midrule +Alex, Brad, Brandon & \$10 & \$15 & \$5 \\ +Alex, Brandon, Brad & \$10 & \$15 & \$5 \\ +Brad, Alex, Brandon & \$5 & \$20 & \$5 \\ +Brad, Brandon, Alex & \$10 & \$20 & \$0 \\ +Brandon, Alex, Brad & \$5 & \$15 & \$10 \\ +Brandon, Brad, Alex & \$17 & \$3 & \$10 \\ \midrule +Shapley contribution: & \$9.50 & \$14.67 & \$5.83 \\ \bottomrule +\end{tabular} +\caption{Marginal contribution for each permutation of Alex, Brad, and Brandon (i.e., the order in which they arrive). The Shapley contribution is the average marginal contribution across all permutations. (Notice how each row sums to the total bill of \$30.)} +\end{table} + +So the next time the bartender asks how you want to split the tab, whip +out a pencil and do the math! + +\hypertarget{estimating-shapley-values-via-monte-carlo-simulation-sampleshap}{% +\subsubsection{Estimating Shapley values via Monte Carlo simulation: +SampleSHAP}\label{estimating-shapley-values-via-monte-carlo-simulation-sampleshap}} + +A single estimate of the contribution of \(x_i\) to \(f\left(x\right)\) +is nothing the more than the difference between two predictions, where +each prediction is based on a sort of Frankenstein instance that' are's +constructed by swapping out values between the instance being explained +(\(x\)) and an instance selected at random from the training data. To +help stabilize the results, the procedure is repeated a large number, +say, \(R\), times, and the result averaged together. \begin{algorithm} \begin{enumerate} - \item For $i = 1, 2, \dots, j$: + \item For $j = 1, 2, \dots, R$: \begin{enumerate} - \item Permute the values of feature $X_i$ in the training data. - \item Recompute the performance metric on the permuted data $\mathcal{M}_{perm}$. - \item Record the difference from baseline using $imp\left(X_i\right) = \mathcal{M}_{perm} - \mathcal{M}_{orig}$. + \item Select a random permutation $\mathcal{O}$ of the sequence $1, 2, \dots, p$. + \item Select a random instance $w$ from the training instances $\boldsymbol{X}$. + \item Construct two new instances as follows: + \begin{itemize} + \item $b_1 = x$, but all the features in $\mathcal{O}$ that appear after feature $x_i$ get their values swapped with the corresponding values in $w$. + \item $b_2 = x$, but feature $x_j$, as well as all the features in $\mathcal{O}$ that appear after $x_j$, get their values swapped with the corresponding values in $w$. + \end{itemize} + \item $\phi_{ij}\left(x\right) = f\left(b_1\right) - f\left(b_2\right)$. \end{enumerate} - \item Return the VI scores $imp\left(X_1\right), imp\left(X_2\right), \dots, imp\left(X_j\right)$. + \item $\phi_i\left(x\right) = \sum_{j = 1} ^ R \phi_{ij}\left(x\right) / R$. \end{enumerate} -\caption{Monte Carlo algorithm for approximating Shapley values. \label{alg:permute}} +\caption{Approximating the $i$-th feature's contribution to $f\left(x\right)$. \label{alg:SampleSHAP}} \end{algorithm} +If there are \(p\) features and \(m\) instanced to be explained, this +requires \(2 \times R \times p \times m\) predictions (or calls to a +scoring function). In practice, this can be quite computationally +demanding, especially since \(R\) needs to be large enough to produce +good approximations to each \(\phi_i\left(x\right)\). In practice, this +depends on the variance of each feature in the observed training data, +but typically \(R >= 50--100\) will suffice (\textbf{Need reference}). + \hypertarget{special-cases}{% \subsubsection{Special cases}\label{special-cases}} -TBD. +The following sections discuss two special cases where exact Shapley +explanations can be computed efficiently: additive linear models, and +shallow trees and tree ensembles. \hypertarget{linear-models-linearshap}{% \paragraph{Linear models: LinearSHAP}\label{linear-models-linearshap}} -TBD. +Cite somewhere \cite{strumbelj-2014-explaining}. + +First, lets discuss how a feature's value contributes to a prediction +\(f\left(X\right)\) in a simple (additive) linear model. That is, let's +assume for a moment that \(f\) takes the form \begin{equation} +\nonumber + f\left(X\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p +\end{equation} + +Recall that the contribution of the \(i\)-th feature to the prediction +\(f\left(X\right)\) is the difference between \(f\left(X\right)\) and +the expected prediction if the \(i\)-th feature's value were not known: +\begin{equation} +\nonumber +\begin{split} + \phi_i\left(X\right) &= \beta_0 + \dots + \beta_i X_i + \dots + \beta_p X_p \\ &\quad\quad - \left(\beta_0 + \dots + \beta_i \E\left(X_i\right) + \dots + \beta_p X_p\right) \\ + &= \beta_i \left(X_i - \E\left(X_i\right)\right) +\end{split}, +\end{equation} where we estimate \(\E\left(X_i\right)\) with the +corresponding sample mean \(\bar{X}_i\). The quantity +\(\phi_i\left(X\right)\) is also referred to as the +\emph{situational importance of $X_i$} \citep{achen-1982-interpreting}. \hypertarget{tree-based-models-treeshap}{% \paragraph{Tree-based models: @@ -71,6 +169,294 @@ \subsection{Shapley values in R}\label{shapley-values-in-r}} TBD. +\hypertarget{so-why-fastshap}{% +\subsubsection{So why fastshap?}\label{so-why-fastshap}} + +\hypertarget{efficiency}{% +\paragraph{Efficiency}\label{efficiency}} + +Like many post-hoc interpretation techniques (e.g., PDPs and ICE +curves), SampleSHAP can be made more efficient by generating all the +data up front, and scoring it only once (or twice, in the case of +SampleSHAP). For example, PDPs and ICE curves can be efficiently +constructed with only a single call to a scoring function by generating +all of the required data up front using a single cross-join operation +(which can be done rather efficiently in SQL or Spark). The scored data +can then be post-processed/aggregated and displayed as either a PDP or +set of ICE curves. An example using Spark with \CRANpkg{sparklyr} +\citet{R-sparklyr} can be found here: +\url{https://github.com/bgreenwell/pdp/issues/97}. + +Fortunately, a similar trick can be exploited for SampleSHAP. Whether +explaining a single instance with a large value of Monte Carlo reps +(\(R\)), or explaining a large number of instances, the basic idea is to +generate all the required Frankenstein instances \(b_1\) and \(b_2\) +upfront, and stored in matrices \(\boldsymbol{B}_1\) and +\(\boldsymbol{B}_2\), respectively. + +For example, suppose we wanted to estimate the contribution of \(x_i\) +for each of the \(N\) rows of the available training data +\(\boldsymbol{X}\) using a single Monte-Carlo repetition in +Algorithm\textasciitilde{}\ref{alg:SampleSHAP} (i.e., +\(R = 1\))\footnote{The same idea also extends to explaining new instances.}. +To start, we can generate the \(N\) random instances at once and store +them in an \(N \times p\) matrix \(\boldsymbol{W}\). Rather generating +\(N\) random permutations \(\mathcal{O}\), and constructing \(b_1\) and +\(b_2\) one at a time, the \pkg{fastshap} package uses C++---via +\CRANpkg{Rcpp} \citep{R-Rcpp}---to efficiently generate an +\(N \times p\) logical matrix \(\boldsymbol{\mathcal{O}}\), where +\(\boldsymbol{\mathcal{O}}_{kl} = 1\) if feature \(x_l\) appears before +feature \(x_i\) in the \(k\)-th permutation, and \(0\) otherwise. This +logical matrix can then be used to logically subset \(\boldsymbol{X}\) +and \(\boldsymbol{W}\) to more efficiently construct +\(\boldsymbol{B}_1\) and \(\boldsymbol{B}_2\) in a single swoop. The +matrices (or data frames) can then be each scored once, and the +difference taken, to generate a single replication of +\(\phi_i\left(x\right)\) for each row of \(\boldsymbol{X}\). + +Suppose instead we want to estimate the contribution of \(x_i\) for a +single instance \(x\), but using a large value of \(R\) for accuracy. We +could employ the same trick, but in this case \(\boldsymbol{X}\) would +refer to the \(R \times p\) matrix, where each row is a copy of the +instance \(x\). + +\pkg{fastshap} also uses efficient exact methods for the special cases +described in Sections\ldots{} + +\hypertarget{parallelization}{% +\paragraph{Parallelization}\label{parallelization}} + +\pkg{fastshap} is faster at computing Shapley values for a single +feature for a large number of instances (or a large value of \(R\) for a +single instance). But what about a large number of features? +Fortunately, Algorithm\textasciitilde{}\ref{alg:SampleSHAP} can be +trivially parallelized across features, and this is built into +\pkg{fastshap}. + +\hypertarget{a-simple-benchmark-comparison}{% +\paragraph{A simple benchmark +comparison}\label{a-simple-benchmark-comparison}} + +This section provides a brief example comparing various implementations +of Shapley values using \href{https://www.kaggle.com/c/titanic}{Kaggle's +Titanic: Machine Learning from Disaster competition}. While the true +focus of the competition is to use machine learning to create a model +that predicts which passengers survived the Titanic shipwreck, we'll +focus on explaining predictions from a simple logistic regression model. + +To start, we'll load the data, which are conveniently available in the +\CRANpkg{titanic} package \citep{R-titanic}, and do a little bit of +cleaning. + +\begin{Schunk} +\begin{Sinput} +# Read in the data and clean it up a bit +titanic <- titanic::titanic_train +features <- c( + "Survived", # passenger survival indicator + "Pclass", # passenger class + "Sex", # gender + "Age", # age + "SibSp", # number of siblings/spouses aboard + "Parch", # number of parents/children aboard + "Fare", # passenger fare + "Embarked" # port of embarkation +) +titanic <- titanic[, features] +titanic$Survived <- as.factor(titanic$Survived) +titanic <- na.omit(titanic) + +# Data frame containing just the features +X <- subset(titanic, select = -Survived) +\end{Sinput} +\end{Schunk} + +Next, we'll use the stats::glm() to fit a logistic regression model with +only main effects (i.e., no tw-way interactions, etc.). + +\begin{Schunk} +\begin{Sinput} +fit <- glm(Survived ~ ., data = titanic, family = binomial) +\end{Sinput} +\end{Schunk} + +Suppose we wanted to explain the predicted survival probability for a +new passenger named Jack: + +\begin{Schunk} +\begin{Sinput} +jack <- data.frame( + Pclass = 3, + Sex = factor("male", levels = c("female", "male")), + Age = 20, + SibSp = 0, + Parch = 0, + Fare = 15, # lower end of third-class ticket prices + Embarked = factor("S", levels = c("", "C", "Q", "S")) +) +\end{Sinput} +\end{Schunk} + +Our logistic regression model predicts that Jack's log-odds of survival +is + +\begin{Schunk} +\begin{Sinput} +predict(fit, newdata = jack) +\end{Sinput} +\begin{Soutput} +#> 1 +#> -1.845561 +\end{Soutput} +\end{Schunk} + +Yikes, that's equivalent to estimated 13.64\% predicted probability of +survival! With a baseline (i.e., average) survival rate of 40.62\%, can +we explain why the model predicts Jack to be much lower? +Enter\ldots Shapley values. + +There is a growing number of R packages that provide Shapley +explanations, the two most popular arguably being \pkg{iml} and +\pkg{iBreakDown}. In this example, we'll compare those with +\pkg{fastshap}. + +To start, we need to define a few things (prediction wrapper, as well as +both \pkg{iml}- and \pkg{iBreakDown}-related helpers). + +\begin{Schunk} +\begin{Sinput} +# Prediction wrapper to compute predicted probability of survive +pfun <- function(object, newdata) { + predict(object, newdata = newdata) +} + +# DALEX-based helper for iBreakDown +explainer <- DALEX::explain(fit, data = X, y = titanic$Survived, predict_function = pfun, verbose = FALSE) + +# Helper for iml +predictor <- iml::Predictor$new(fit, data = titanic, y = "Survived", + predict.fun = pfun) +\end{Sinput} +\end{Schunk} + +Next, we call each implementation's Shapley-related function to compute +explanations for Jack's prediction using 100 Monte Carlo repetitions. + +\begin{Schunk} +\begin{Sinput} +# Compute explanations +set.seed(1039) # for reproducibility +ex1 <- iBreakDown::shap(explainer, B = 100, new_observation = jack) +ex2 <- iml::Shapley$new(predictor, x.interest = jack, sample.size = 100) +ex3 <- fastshap::explain(fit, X = X, pred_wrapper = pfun, nsim = 100, + newdata = jack) + +# Plot results +library(ggplot2) # for `autoplot()` function +p3 <- plot(ex1) + ggtitle("iBreakDown") +p2 <- plot(ex2) + ggtitle("iml") +p1 <- autoplot(ex3, type = "contribution") + ggtitle("fastshap") +fastshap::grid.arrange(p1, p2, p3, nrow = 1) +\end{Sinput} +\begin{figure}[!htb] + +{\centering \includegraphics[width=1\linewidth]{greenwell_files/figure-latex/titanic-jack-explanations-1} + +} + +\caption[TBD]{TBD.}\label{fig:titanic-jack-explanations} +\end{figure} +\end{Schunk} + +Each package comes loaded with it's own bells and whistles (e.g., +\pkg{iml} and \pkg{iBreakDown} have particularly fantastic +visualizations). The main selling point of \pkg{fastshap} is speed! For +example, all three packages (in fact, all general and practical +implementations of Shapley values) use +Algorithm\textasciitilde{}\ref{alg:SampleSHAP} which requires a large +number of Monte Carlo repetitions to achieve accurate results. Below is +a simple benchmark looking at the estimated time (in seconds) to explain +Jack's prediction as a function of the number of Monte Carlo repetitions +for each implementation. (Note that this comparison does not make use of +\pkg{fastshap}'s feature-wise parallelization.) + +\begin{Schunk} +\begin{Sinput} +nsims <- c(1, 5, 10, 25, 50, 75, seq(from = 100, to = 1000, by = 100)) +times1 <- times2 <- times3 <- numeric(length(nsims)) +set.seed(904) +for (i in seq_along(nsims)) { + message("nsim = ", nsims[i], "...") + times1[i] <- system.time({ + iBreakDown::shap(explainer, B = nsims[i], new_observation = jack) + })["elapsed"] + times2[i] <- system.time({ + iml::Shapley$new(predictor, x.interest = jack, sample.size = nsims[i]) + })["elapsed"] + times3[i] <- system.time({ + fastshap::explain(fit, X = X, newdata = jack, pred_wrapper = pfun, + nsim = nsims[i]) + })["elapsed"] +} +pal <- palette.colors(3, palette = "Okabe-Ito") # colorblind friendly palette +plot(nsims, times1, type = "b", xlab = "Number of Monte Carlo repetitions", + ylab = "Time (in seconds)", las = 1, pch = 19, col = pal[1L], + xlim = c(0, max(nsims)), ylim = c(0, max(times1, times2, times3))) +lines(nsims, times2, type = "b", pch = 19, col = pal[2L],) +lines(nsims, times3, type = "b", pch = 19, col = pal[3L],) +legend("topleft", + legend = c("iBreakDown", "iml", "fastshap"), + lty = 1, pch = 19, col = pal, inset = 0.02) +\end{Sinput} +\begin{figure}[!htb] + +{\centering \includegraphics[width=0.8\linewidth]{greenwell_files/figure-latex/titanic-benchmark-1} + +} + +\caption[Quick benchmark between three different implementations of SampleSHAP for explaining Jack's unfortunate prediction]{Quick benchmark between three different implementations of SampleSHAP for explaining Jack's unfortunate prediction.}\label{fig:titanic-benchmark} +\end{figure} +\end{Schunk} + +The message to be taken from +Figure\textasciitilde{}\ref{fig:titanic-benchmark} is that +\pkg{fastshap} scales incredibly well with \(N\) or \(R\), as long as +the corresponding \code{predict()} method does. + +Oh, and \pkg{fastshap} can produce instant (and exact) Shapley +contributions for this example. + +\begin{Schunk} +\begin{Sinput} +fastshap::explain(fit, newdata = jack, exact = TRUE) # ExactSHAP +\end{Sinput} +\begin{Soutput} +#> # A tibble: 1 x 7 +#> Pclass Sex Age SibSp Parch Fare Embarked +#> +#> 1 -0.915 -0.964 0.420 0.186 0.0260 -0.0282 -0.0919 +\end{Soutput} +\begin{Sinput} +fastshap::explain(fit, X = X, pred_wrapper = pfun, nsim = 10000, + newdata = jack) # SampleSHAP +\end{Sinput} +\begin{Soutput} +#> # A tibble: 1 x 7 +#> Pclass Sex Age SibSp Parch Fare Embarked +#> +#> 1 -0.929 -0.977 0.422 0.185 0.0257 -0.0290 -0.0865 +\end{Soutput} +\begin{Sinput} +predict(fit, newdata = jack, type = "terms") # ExactSHAP (base R) +\end{Sinput} +\begin{Soutput} +#> Pclass Sex Age SibSp Parch Fare Embarked +#> 1 -0.9153946 -0.9644851 0.4204564 0.1861824 0.02599872 -0.0281944 -0.09194646 +#> attr(,"constant") +#> [1] -0.4781785 +\end{Soutput} +\end{Schunk} + \hypertarget{example-predicing-sales-prices}{% \subsection{Example: predicing sales prices}\label{example-predicing-sales-prices}} diff --git a/rjournal/greenwell_cache/latex/__packages b/rjournal/greenwell_cache/latex/__packages new file mode 100644 index 0000000..4af209d --- /dev/null +++ b/rjournal/greenwell_cache/latex/__packages @@ -0,0 +1,2 @@ +base +ggplot2 diff --git a/rjournal/greenwell_cache/latex/titanic-benchmark_ea93b04f25ab77d3e5743f87ba471dd0.RData b/rjournal/greenwell_cache/latex/titanic-benchmark_ea93b04f25ab77d3e5743f87ba471dd0.RData new file mode 100644 index 0000000000000000000000000000000000000000..b2aa0457e4cb31d917111c150bd871ffcbb1f76b GIT binary patch literal 3590 zcmV+h4*BsPiwFP!000001HD#xJXC8RA4}9I(Sk~KOkqZ2EMv)*eFv;B=A6ximaER zP{Ldk3Y(3Xh~WH~Nxtnie@%kSN+}aJkwUxUJnnLb6P1}|x~r+T-D>sD#L69l9nL|M zzLD9Z5vSaRa9eiX@a(Fc-l14oxXY zMyFL74)?F7`()?8kjkyHb1~*@!n7K=q)-TL~|bf1zOO?%H1%qB%|UWV7pkY&2hq48mdIF(4DJ|8LisTuL#B9vQ*Ie#5n zbI_u>M)=vQ-Bug2+(Z+L&2`>}SiFKbKD&I>8~j=7r>5m=jk6mZpRO=lHN<%LcKK}| zuD@yjRp`_goPw=xO}}iMOM)mfJ!AUnh{W^cXBk_h^L=+qexM~R?F#73iZRP4)SR9r zo}h16;`Y**GVwx<5cm*;)p4*a2VEBZyFZb{=r`I6OP6S$PX9MqFfg&yHWZ3bHi zOFycC`Gv{{h<7hk^tg*&+-llq;XWSH5t;Ik})lKC*KLS-doxHV5ym+274-VO7o4@;d+YRARz0v-TnhE zs82Uc*y~~Nw>9lrv!jy?*D=o1J0fbP-17%UBnSL6*6nH0gK00>p8k^_H(it~%)jQh z1RKkdgOa28ybn~~lxnl|sBppS;~(D<)k9KB#)ST=ItmwD-juH*SM->hJc%yp_Z={qlMk$zY5&^+8TLB5Ed4S-0D!t(8WHLOJ{648Ob_%XWXoKA><7&bH33 zTmO)|B6~49A$RYCgu|0%n{}={(7RIfM%mzSg!P{yipAoSW^vlm?UxfQVm2J2qf#=$ z&KZWUR&UrUmZlP7)gWt*(JF}Ouq^#7g_1`zYa2H@^f=V3p-h$+7b*=)wl`MfmpCX^ z6gBNgcoLiM;!t+KvEi>r+N9j9fMVlXnaG~V9UeD6Uu+A)55$Muhzmq%-PKe~nu#kb zpG>Q+U7mI~&8;V)H$(K*4Eo5S$!(6_f9hN>P%qt18VGqL>@)<81C@C3Bjk>pJS|*! 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