Introduction
Too often machine learning (ML) models are summarized using a single metric (e.g., cross-validated accuracy) and then put into production. Although we often care about the predictions from these models, it is becoming routine (and good practice) to also better understand the predictions! Understanding how an ML model makes its predictions helps build trust in the model and is the fundamental idea of the emerging field of interpretable machine learning (IML).1 For an in-depth discussion on IML, see Molnar (2019b). In this paper, we focus on global methods for quantifying the importance2 of features in an ML model; that is, methods that help us understand the global contribution each feature has to a model’s predictions. Computing variable importance (VI) and communicating them through variable importance plots (VIPs) is a fundamental component of IML and is the main topic of this paper.
While many of the procedures discussed in this paper apply to any model that makes predictions, it should be noted that these methods heavily depend on the accuracy and importance of the fitted model; hence, unimportant features may appear relatively important (albeit not predictive) in comparison to the other included features. For this reason, we stress the usefulness of understanding the scale on which VI scores are calculated and take that into account when assessing the importance of each feature and communicating the results to others. Also, we should point out that this work focuses mostly on post-hoc interpretability where a trained model is given and the goal is to understand what features are driving the model’s predictions. Consequently, our work focuses on functional understanding of the model in contrast to the lower-level mechanistic understanding (Montavon et al. 2018). That is, we seek to explain the relationship between the model’s prediction behavior and features without explaining the full internal representation of the model.3
VI scores and VIPs can be constructed for general ML models using a
number of available packages. The
iml package (Molnar 2019a)
provides the FeatureImp() function which computes feature importance
for general prediction models using the permutation approach (discussed
later). It is written in R6
(Chang 2019) and allows the user to specify a generic loss function or select
one from a pre-defined list (e.g., loss = "mse" for mean squared
error). It also allows the user to specify whether importance is
measured as the difference or as the ratio of the original model error
and the model error after permutation. The user can also specify the
number of repetitions used when permuting each feature to help stabilize
the variability in the procedure. The iml::FeatureImp() function can
also be run in parallel using any parallel backend supported by the
foreach package
(Revolution Analytics and Weston).
The ingredients
package (Biecek et al. 2019a) also provides permutation-based VI scores
through the feature_importance() function. (Note that this function
recently replaced the now deprecated
DALEX function
variable_importance() (Biecek 2019).) Similar to iml::FeatureImp(),
this function allows the user to specify a loss function and how the
importance scores are computed (e.g., using the difference or ratio). It
also provides an option to sample the training data before shuffling the
data to compute importance (the default is to use n_sample = 1000),
which can help speed up computation.
The mmpf package (Jones 2018)
also provides permutation-based VI scores via the
mmpf::permutationImportance() function. Similar to the iml and
ingredients implementation, this function is flexible enough to be
applied to any class of ML models in R.
The varImp package (Probst 2019) extends the permutation-based method for RFs in package party (Hothorn et al. 2019) to arbitrary measures from the measures package (Probst 2018). Additionally, the functions in varImp include the option of using the conditional approach described in Strobl et al. (2008) which is more reliable in the presence of correlated features. A number of other RF-specific VI packages exist on CRAN, including, but not limited to, vita (Celik 2015), rfVarImpOOB (Loecher 2019), randomForestExplainer (Paluszynska et al. 2019), and tree.interpreter (Sun 2019).4.
The caret package
(Kuhn 2020) includes a general varImp() function for computing
model-specific and filter-based VI scores. Filter-based approaches,
which are described in Kuhn and Johnson (2013), do not make use of the fitted
model to measure VI. They also do not take into account the other
predictors in the model. For regression problems, a popular filter-based
approach to measuring the VI of a numeric predictor \(x\) is to first fit
a flexible nonparametric model between \(x\) and the target \(Y\); for
example, the locally-weighted polynomial regression (LOWESS) method
developed by Cleveland (1979). From this fit, a pseudo-\(R^2\)
measure can be obtained from the resulting residuals and used as a
measure of VI. For categorical predictors, a different method based on
standard statistical tests (e.g., \(t\)-tests and ANOVAs) can be employed;
see Kuhn and Johnson (2013) for details. For classification problems, an area
under the ROC curve (AUC) statistic can be used to quantify predictor
importance. The AUC statistic is computed by using the predictor \(x\) as
input to the ROC curve. If \(x\) can reasonably separate the classes of
\(Y\), that is a clear indicator that \(x\) is an important predictor (in
terms of class separation) and this is captured in the corresponding AUC
statistic. For problems with more than two classes, extensions of the
ROC curve or a one-vs-all approach can be used.
If you use the mlr interface
for fitting ML models (Bischl et al. 2020), then you can use the
getFeatureImportance() function to extract model-specific VI scores
from various tree-based models (e.g., RFs and GBMs). Unlike caret, the
model needs to be fit via the mlr interface; for instance, you cannot
use getFeatureImportance() on a
ranger (Wright et al. 2020) model
unless it was fit using mlr.
While the iml and DALEX packages provide model-agnostic approaches to computing VI, caret, and to some extent, mlr, provide model-specific approaches (e.g., using the absolute value of the \(t\)-statistic for linear models) as well as less accurate filter-based approaches. Furthermore, each package has a completely different interface (e.g., iml is written in R6). The vip package (Greenwell et al. 2019) strives to provide a consistent interface to both model-specific and model-agnostic approaches to feature importance that is simple to use. The three most important functions exported by vip are described below:
vi()computes VI scores using model-specific or model-agnostic approaches (the results are always returned as a tibble (Müller and Wickham 2019));vip()constructs VIPs using model-specific or model-agnostic approaches with ggplot2-style graphics (Wickham et al. 2019);add_sparklines()adds a novel sparkline representation of feature effects (e.g., partial dependence plots) to any VI table produced byvi().
There’s also a function called vint() (for variable interactions) but
it is experimental and will not be discussed here; the interested reader
is pointed to Greenwell et al. (2018). Note that vi() is actually a
wrapper around four workhorse functions, vi_model(), vi_firm(),
vi_permute(), and vi_shap(), that compute various types of VI
scores. The first computes model-specific VI scores, while the latter
three produce model-agnostic ones. The workhorse function that actually
gets called is controlled by the method argument in vi(); the
default is method = "model" which corresponds to model-specific VI
(see ?vip::vi for details and links to further documentation).
Constructing VIPs in R
We’ll illustrate major concepts using the Friedman 1 benchmark problem described in Friedman (1991) and Breiman (1996):
\[Y_i = 10 \sin\left(\pi X_{1i} X_{2i}\right) + 20 \left(X_{3i} - 0.5\right) ^ 2 + 10 X_{4i} + 5 X_{5i} + \epsilon_i, \quad i = 1, 2, \dots, n, \label{eqn:friedman1} \tag{1}\]
where \(\epsilon_i \stackrel{iid}{\sim} N\left(0, \sigma^2\right)\). Data
from this model can be generated using the vip::gen_friedman(). By
default, the features consist of 10 independent variables uniformly
distributed on the interval \(\left[0,1\right]\); however, only 5 out of
these 10 are actually used in the true model. The code chunk below
simulates 500 observations from the model in Equation
(1) with \(\sigma = 1\); see ?vip::gen_friedman for
details.
trn <- vip::gen_friedman(500, sigma = 1, seed = 101) # simulate training data
tibble::as_tibble(trn) # inspect output#> # A tibble: 500 x 11
#> y x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 14.9 0.372 0.406 0.102 0.322 0.693 0.758 0.518 0.530 0.878 0.763
#> 2 15.3 0.0438 0.602 0.602 0.999 0.776 0.533 0.509 0.487 0.118 0.176
#> 3 15.1 0.710 0.362 0.254 0.548 0.0180 0.765 0.715 0.844 0.334 0.118
#> 4 10.7 0.658 0.291 0.542 0.327 0.230 0.301 0.177 0.346 0.474 0.283
#> 5 17.6 0.250 0.794 0.383 0.947 0.462 0.00487 0.270 0.114 0.489 0.311
#> 6 18.3 0.300 0.701 0.992 0.386 0.666 0.198 0.924 0.775 0.736 0.974
#> 7 14.6 0.585 0.365 0.283 0.488 0.845 0.466 0.715 0.202 0.905 0.640
#> 8 17.0 0.333 0.552 0.858 0.509 0.697 0.388 0.260 0.355 0.517 0.165
#> 9 8.54 0.622 0.118 0.490 0.390 0.468 0.360 0.572 0.891 0.682 0.717
#> 10 15.0 0.546 0.150 0.476 0.706 0.829 0.373 0.192 0.873 0.456 0.694
#> # ... with 490 more rowsFrom Equation (1), it should be clear that features \(X_1\)–\(X_5\) are the most important! (The others don’t influence \(Y\) at all.) Also, based on the form of the model, we’d expect \(X_4\) to be the most important feature, probably followed by \(X_1\) and \(X_2\) (both comparably important), with \(X_5\) probably being less important. The influence of \(X_3\) is harder to determine due to its quadratic nature, but it seems likely that this nonlinearity will suppress the variable’s influence over its observed range (i.e., 0–1).
1 Model-specific VI
Some machine learning algorithms have their own way of quantifying the importance of each feature, which we refer to as model-specific VI. We describe some of these in the subsections that follow. One particular issue with model-specific VI scores is that they are not necessarily comparable across different types of models. For example, directly comparing the impurity-based VI scores from tree-based models to the the absolute value of the \(t\)-statistic in linear models.
Decision trees and tree ensembles
Decision trees probably offer the most natural model-specific approach to quantifying the importance of each feature. In a binary decision tree, at each node \(t\), a single predictor is used to partition the data into two homogeneous groups. The chosen predictor is the one that maximizes some measure of improvement \(i^t\). The relative importance of predictor \(X\) is the sum of the squared improvements over all internal nodes of the tree for which \(X\) was chosen as the partitioning variable; see Breiman et al. (1984) for details. This idea also extends to ensembles of decision trees, such as RFs and GBMs. In ensembles, the improvement score for each predictor is averaged across all the trees in the ensemble. Fortunately, due to the stabilizing effect of averaging, the improvement-based VI metric is often more reliable in large ensembles; see Hastie et al. (2009 368).
RFs offer an additional method for computing VI scores. The idea is to
use the leftover out-of-bag (OOB) data to construct validation-set
errors for each tree. Then, each predictor is randomly shuffled in the
OOB data and the error is computed again. The idea is that if variable
\(X\) is important, then the validation error will go up when \(X\) is
perturbed in the OOB data. The difference in the two errors is recorded
for the OOB data then averaged across all trees in the forest. Note that
both methods for constructing VI scores can be unreliable in certain
situations; for example, when the predictor variables vary in their
scale of measurement or their number of categories (Strobl et al. 2007), or
when the predictors are highly correlated (Strobl et al. 2008).
The varImp package discussed earlier provides methods to address these
concerns for random forests in package party, with similar
functionality also built into the
partykit package
(Hothorn and Zeileis 2019). The vip package also supports the conditional
importance described in (Strobl et al. 2008) for both party- and
partykit-based RFs; see ?vip::vi_model for details. Later on, we’ll
discuss a more general permutation method that can be applied to any
supervised learning model.
To illustrate, we fit a CART-like regression tree, RF, and GBM to the simulated training data. (Note: there are a number of different packages available for fitting these types of models, we just picked popular implementations for illustration.)
# Load required packages
library(rpart) # for fitting CART-like decision trees
library(randomForest) # for fitting RFs
library(xgboost) # for fitting GBMs
# Fit a single regression tree
tree <- rpart(y ~ ., data = trn)
# Fit an RF
set.seed(101) # for reproducibility
rfo <- randomForest(y ~ ., data = trn, importance = TRUE)
# Fit a GBM
set.seed(102) # for reproducibility
bst <- xgboost(
data = data.matrix(subset(trn, select = -y)),
label = trn$y,
objective = "reg:squarederror",
nrounds = 100,
max_depth = 5,
eta = 0.3,
verbose = 0 # suppress printing
)Each of the above packages include the ability to compute VI scores for all the features in the model; however, the implementation is rather package-specific, as shown in the code chunk below. The results are displayed in Figure 1 (the code to reproduce these plots has been omitted but can be made available upon request).
# Extract VI scores from each model
vi_tree <- tree$variable.importance
vi_rfo <- rfo$variable.importance # or use `randomForest::importance(rfo)`
vi_bst <- xgb.importance(model = bst)
As we would expect, all three methods rank the variables x1–x5 as
more important than the others. While this is good news, it is
unfortunate that we have to remember the different functions and ways of
extracting and plotting VI scores from various model fitting functions.
This is one place where vip can help…one function to rule them all!
Once vip is loaded, we can use vi() to extract a tibble of VI
scores.5
# Load required packages
library(vip)
# Compute model-specific VI scores
vi(tree) # CART-like decision tree#> # A tibble: 10 x 2
#> Variable Importance
#> <chr> <dbl>
#> 1 x4 4234.
#> 2 x2 2513.
#> 3 x1 2461.
#> 4 x5 1230.
#> 5 x3 688.
#> 6 x6 533.
#> 7 x7 357.
#> 8 x9 331.
#> 9 x8 276.
#> 10 x10 275.vi(rfo) # RF#> # A tibble: 10 x 2
#> Variable Importance
#> <chr> <dbl>
#> 1 x4 74.2
#> 2 x2 59.9
#> 3 x1 53.3
#> 4 x5 37.1
#> 5 x3 22.5
#> 6 x9 1.05
#> 7 x10 0.254
#> 8 x8 -0.408
#> 9 x7 -1.56
#> 10 x6 -2.00vi(bst) # GBM#> # A tibble: 10 x 2
#> Variable Importance
#> <chr> <dbl>
#> 1 x4 0.403
#> 2 x2 0.225
#> 3 x1 0.189
#> 4 x5 0.0894
#> 5 x3 0.0682
#> 6 x9 0.00802
#> 7 x6 0.00746
#> 8 x7 0.00400
#> 9 x10 0.00377
#> 10 x8 0.00262Notice how the vi() function always returns a tibble6 with two
columns: Variable and Importance (the exceptions are
coefficient-based models which also include a Sign column giving the
sign of the corresponding coefficient, and permutation importance
involving multiple Monte Carlo simulations, but more on that later).
Also, by default, vi() always orders the VI scores from highest to
lowest; this, among other options, can be controlled by the user (see
?vip::vi for details). Plotting VI scores with vip() is just as
straightforward. For example, the following code can be used to
reproduce Figure 1.
p1 <- vip(tree) + ggtitle("Single tree")
p2 <- vip(rfo) + ggtitle("Random forest")
p3 <- vip(bst) + ggtitle("Gradient boosting")
# Display plots in a grid (Figure 1)
grid.arrange(p1, p2, p3, nrow = 1)Notice how the vip() function always returns a "ggplot" object (by
default, this will be a bar plot). For large models with many features,
a Cleveland dot plot is more effective (in fact, a number of useful
plotting options can be fiddled with). Below we call vip() and change
a few useful options (the resulting plot is displayed in Figure
2). Note that we can also call vip() directly on a
"vi" object if it’s already been constructed.
# Construct VIP (Figure 2)
library(ggplot2) # for theme_light() function
vip(bst, num_features = 5, geom = "point", horizontal = FALSE,
aesthetics = list(color = "red", shape = 17, size = 5)) +
theme_light()
Linear models
In multiple linear regression, or linear models (LMs), the absolute
value of the \(t\)-statistic (or some other scaled variant of the
estimated coefficients) is commonly used as a measure of VI.7 The
same idea also extends to generalized linear models (GLMs). In the code
chunk below, we fit an LM to the simulated Friedman data (trn)
allowing for all main effects and two-way interactions, then use the
step() function to perform backward elimination. The resulting VIP is
displayed in Figure 3.
# Fit a LM
linmod <- lm(y ~ .^2, data = trn)
backward <- step(linmod, direction = "backward", trace = 0)
# Extract VI scores
(vi_backward <- vi(backward))#> # A tibble: 21 x 3
#> Variable Importance Sign
#> <chr> <dbl> <chr>
#> 1 x4 14.2 POS
#> 2 x2 7.31 POS
#> 3 x1 5.63 POS
#> 4 x5 5.21 POS
#> 5 x3:x5 2.46 POS
#> 6 x1:x10 2.41 NEG
#> 7 x2:x6 2.41 NEG
#> 8 x1:x5 2.37 NEG
#> 9 x10 2.21 POS
#> 10 x3:x4 2.01 NEG
#> # ... with 11 more rows# Plot VI scores; by default, `vip()` displays the top ten features
pal <- palette.colors(2, palette = "Okabe-Ito") # colorblind friendly palette
vip(vi_backward, num_features = length(coef(backward)), # Figure 3
geom = "point", horizontal = FALSE, mapping = aes(color = Sign)) +
scale_color_manual(values = unname(pal)) +
theme_light() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
A major limitation of this approach is that a VI score is assigned to each term in the model, rather than to each individual feature! We can solve this problem using one of the model-agnostic approaches discussed later.
Multivariate adaptive regression splines (MARS), which were introduced
in Friedman (1991), is an automatic regression technique and
can be seen as a generalization of LMs and GLMs. In the MARS algorithm,
the contribution (or VI score) for each predictor is determined using a
generalized cross-validation (GCV) statistic (though, other statistics
can also be used; see ?vip::vi_model for details). An example using
the earth package
(Milborrow 2019) is given below (the results are plotted in Figure
4):
# Load required packages
library(earth)
# Fit a MARS model
mars <- earth(y ~ ., data = trn, degree = 2, pmethod = "exhaustive")
# Extract VI scores
vi(mars, type = "gcv")#> # A tibble: 10 x 2
#> Variable Importance
#> <chr> <dbl>
#> 1 x4 100
#> 2 x1 83.2
#> 3 x2 83.2
#> 4 x5 59.3
#> 5 x3 43.5
#> 6 x6 0
#> 7 x7 0
#> 8 x8 0
#> 9 x9 0
#> 10 x10 0# Plot VI scores (Figure 4)
vip(mars)
To access VI scores directly in earth, you can use the
earth::evimp() function.
Neural networks
For neural networks (NNs), two popular methods for constructing VI
scores are the Garson algorithm (Garson 1991), later
modified by Goh (1995), and the Olden algorithm
(Olden et al. 2004). For both algorithms, the basis of these VI
scores is the network’s connection weights. The Garson algorithm
determines VI by identifying all weighted connections between the nodes
of interest. Olden’s algorithm, on the other hand, uses the products of
the raw connection weights between each input and output neuron and sums
these products across all hidden neurons. This has been shown to
outperform the Garson method in various simulations. For DNNs, a similar
method due to Gedeon (1997) considers the weights connecting the
input features to the first two hidden layers (for simplicity and
speed); but this method can be slow for large networks. We illustrate
these two methods below using vip() with the
nnet package (Ripley 2016) (see
the results in Figure 5).
# Load required packages
library(nnet)
# Fit a neural network
set.seed(0803) # for reproducibility
nn <- nnet(y ~ ., data = trn, size = 7, decay = 0.1,
linout = TRUE, trace = FALSE)
# Construct VIPs
p1 <- vip(nn, type = "garson")
p2 <- vip(nn, type = "olden")
# Display plots in a grid (Figure 5)
grid.arrange(p1, p2, nrow = 1)
2 Model-agnostic VI
Model-agnostic interpretability separates interpretation from the model. Compared to model-specific approaches, model-agnostic VI methods are more flexible and can be applied to any supervised learning algorithm. In this section, we discuss model-agnostic methods for quantifying global feature importance using three different approaches: 1) a simple variance-based approach, 2) permutation-based feature importance, and 3) Shapley-based feature importance.
Variance-based methods
Our first model-agnostic method is based on a simple feature importance
ranking measure (FIRM); for details, see Greenwell et al. (2018),
Zien et al. (2009), and Scholbeck et al. (2019). The specific approach
used here is based on quantifying the “flatness” of the effects of each
feature.8 Feature effects can be assessed using partial dependence
plots (PDPs) (Friedman 2001) or individual conditional
expectation (ICE) curves (Goldstein et al. 2015). PDPs and ICE curves
help visualize the effect of low cardinality subsets of the feature
space on the estimated prediction surface (e.g., main effects and
two/three-way interaction effects.). They are also model-agnostic and
can be constructed in the same way for any supervised learning
algorithm. Below, we fit a projection pursuit regression (PPR) model
(see ?stats::ppr for details and references) and construct PDPs for
each feature using the pdp
package Greenwell (2017). The results are displayed in Figure 6.
Notice how the PDPs for the uninformative features are relatively flat
compared to the PDPs for features x1–x5!
Next, we compute PDP-based VI scores for the fitted PPR and NN models.
The PDP method constructs VI scores that quantify the relative
“flatness” of each PDP (by default, this is defined by computing the
standard deviation of the \(y\)-axis values for each PDP). To use the PDP
method, specify method = "firm" in the call to vi() or vip() (or
just use vi_firm() directly):
# Fit a PPR model (nterms was chosen using the caret package with 5 repeats of
# 5-fold cross-validation)
pp <- ppr(y ~ ., data = trn, nterms = 11)
# Construct VIPs
p1 <- vip(pp, method = "firm") + ggtitle("PPR")
p2 <- vip(nn, method = "firm") + ggtitle("NN")
# Display plots in a grid (Figure 7)
grid.arrange(p1, p2, ncol = 2)
In Figure 7 we display the PDP-based feature importance for the previously obtained PPR and NN models. These VI scores essentially capture the variability in the partial dependence values for each main effect.
The ICE curve method is similar to the PDP method, except that we measure the “flatness” of each individual ICE curve and then aggregate the results (e.g., by averaging). If there are no (substantial) interaction effects, using ICE curves will produce results similar to using PDPs (which are just averaged ICE curves). However, if strong interaction effects are present, they can obfuscate the main effects and render the PDP-based approach less useful (since the PDPs for important features can be relatively flat when certain interactions are present; see Goldstein et al. (2015) for details). In fact, it is probably safest to always use ICE curves when employing the FIRM method.
Below, we display the ICE curves for each feature in the fitted PPR
model using the same \(y\)-axis scale; see Figure 8.
Again, there is a clear difference between the ICE curves for features
x1–x5 and x6–x10; the later being relatively flat by
comparison. Also, notice how the ICE curves within each feature are
relatively parallel (if the ICE curves within each feature were
perfectly parallel, the standard deviation for each curve would be the
same and the results will be identical to the PDP method). In this
example, the interaction term between x1 and x2 does not obfuscate
the PDPs for the main effects and the results are not much different.
Obtaining the ICE-based feature importance scores is also
straightforward, just specify ice = TRUE when using the FIRM approach.
This is illustrated in the code chunk below and the results, which are
displayed in Figure 9, are similar to those
obtained using the PDP method.
# Construct VIPs
p1 <- vip(pp, method = "firm", ice = TRUE) + ggtitle("PPR")
p2 <- vip(nn, method = "firm", ice = TRUE) + ggtitle("NN")
# Display plots in a grid (Figure 9)
grid.arrange(p1, p2, ncol = 2)
When using method = "firm", the feature effect values are stored in an
attribute called "effects". This is a convenience so that the feature
effect plots (e.g., PDPs and ICE curves) can easily be reconstructed and
compared with the VI scores, as demonstrated in the example below (see
Figure 10):
# Construct PDP-based VI scores
(vis <- vi(pp, method = "firm"))#> # A tibble: 10 x 2
#> Variable Importance
#> <chr> <dbl>
#> 1 x4 2.93
#> 2 x2 2.05
#> 3 x1 2.04
#> 4 x5 1.53
#> 5 x3 1.38
#> 6 x6 0.183
#> 7 x10 0.139
#> 8 x9 0.113
#> 9 x8 0.0899
#> 10 x7 0.0558# Reconstruct PDPs for all 10 features (Figure 10)
par(mfrow = c(2, 5))
for (name in paste0("x", 1:10)) {
plot(attr(vis, which = "effects")[[name]], type = "l", ylim = c(9, 19), las = 1)
}
pdp attribute of the vis object.
Permutation method
The permutation method exists in various forms and was made popular in Breiman (2001) for RFs, before being generalized and extended in Fisher et al. (2018). The permutation approach used in vip is quite simple and is outlined in Algorithm 1 below. The idea is that if we randomly permute the values of an important feature in the training data, the training performance would degrade (since permuting the values of a feature effectively destroys any relationship between that feature and the target variable). This of course assumes that the model has been properly tuned (e.g., using cross-validation) and is not over fitting. The permutation approach uses the difference between some baseline performance measure (e.g., training \(R^2\), AUC, or RMSE) and the same performance measure obtained after permuting the values of a particular feature in the training data (Note: the model is NOT refit to the training data after randomly permuting the values of a feature). It is also important to note that this method may not be appropriate when you have, for example, highly correlated features (since permuting one feature at a time may lead to unlikely data instances).
Let \(X_1, X_2, \dots, X_j\) be the features of interest and let \(\mathcal{M}_{orig}\) be the baseline performance metric for the trained model; for brevity, we’ll assume smaller is better (e.g., classification error or RMSE). The permutation-based importance scores can be computed as follows:
For \(i = 1, 2, \dots, j\):
Permute the values of feature \(X_i\) in the training data.
Recompute the performance metric on the permuted data \(\mathcal{M}_{perm}\).
Record the difference from baseline using \(imp\left(X_i\right) = \mathcal{M}_{perm} - \mathcal{M}_{orig}\).
Return the VI scores \(imp\left(X_1\right), imp\left(X_2\right), \dots, imp\left(X_j\right)\).
Algorithm 1 can be improved or modified in a number of ways. For instance, the process can be repeated several times and the results averaged together. This helps to provide more stable VI scores, and also the opportunity to measure their variability. Rather than taking the difference in step (c), Molnar (2019b 5.5.4) argues that using the ratio \(\mathcal{M}_{perm} / \mathcal{M}_{orig}\) makes the importance scores more comparable across different problems. It’s also possible to assign importance scores to groups of features (e.g., by permuting more than one feature at a time); this would be useful if features can be categorized into mutually exclusive groups, for instance, categorical features that have been one-hot-encoded.
To use the permutation approach in vip, specify method = "permute"
in the call to vi() or vip() (or you can use vi_permute()
directly). Note that using method = "permute" requires specifying a
few additional arguments (e.g., the training data, target name or vector
of target values, a prediction function, etc.); see ?vi_permute for
details.
An example is given below for the previously fitted PPR and NN models.
Here we use \(R^2\) (metric = "rsquared") as the evaluation metric. The
results, which are displayed in Figure 11,
agree with those obtained using the PDP- and ICE-based methods.
# Plot VI scores
set.seed(2021) # for reproducibility
p1 <- vip(pp, method = "permute", target = "y", metric = "rsquared",
pred_wrapper = predict) + ggtitle("PPR")
p2 <- vip(nn, method = "permute", target = "y", metric = "rsquared",
pred_wrapper = predict) + ggtitle("NN")
# Display plots in a grid (Figure 11)
grid.arrange(p1, p2, ncol = 2)
The permutation approach introduces randomness into the procedure and
therefore should be run more than once if computationally feasible. The
upside to performing multiple runs of Algorithm 1 is
that it allows us to compute standard errors (among other metrics) for
the estimated VI scores, as illustrated in the example below; here we
specify nsim = 10 to request that each feature be permuted 10 times
and the results averaged together. (Additionally, if nsim > 1, you can
set geom = "boxplot" in the call to vip() to construct boxplots of
the raw permutation-based VI scores. This is useful if you want to
visualize the variability in each of the VI estimates; see Figure
12 for an example.)
# Use 10 Monte Carlo reps
set.seed(403) # for reproducibility
vis <- vi(pp, method = "permute", target = "y", metric = "rsquared",
pred_wrapper = predict, nsim = 15)
vip(vis, geom = "boxplot") # Figure 12
All available performance metrics for regression and classification can
be listed using the list_metrics() function, for example:
list_metrics()#> Metric Description Task
#> 1 accuracy Classification accuracy Binary/multiclass classification
#> 2 error Misclassification error Binary/multiclass classification
#> 3 auc Area under (ROC) curve Binary classification
#> 4 logloss Log loss Binary classification
#> 5 mauc Multiclass area under (ROC) curve Multiclass classification
#> 6 mae Mean absolute error Regression
#> 7 mse Mean squared error Regression
#> 8 r2 R squared Regression
#> 9 rsquared R squared Regression
#> 10 rmse Root mean squared error Regression
#> 11 sse Sum of squared errors RegressionWe can also use a custom metric (i.e., loss function). Suppose for example you want to measure importance using the mean absolute error (MAE):
\[MAE = \frac{1}{n}\sum_{i = 1}^n\left|Y_i - \widehat{f}\left(\boldsymbol{X}_i\right)\right|,\]
where \(\widehat{f}\left(\boldsymbol{X}_i\right)\) is the predicted value
of \(Y_i\). A simple function implementing this metric is given below
(note that, according to the documentation in ?vi_permute,
user-supplied metric functions require two arguments: actual and
predicted).
mae <- function(actual, predicted) {
mean(abs(actual - predicted))
}To use this for computing permutation-based VI scores just pass it via
the metric argument (be warned, however, that the metric used for
computing permutation importance should be the same as the metric used
to train and tune the model). Also, since this is a custom metric, we
need to specify whether a smaller value indicates better performance by
setting smaller_is_better = TRUE. The results, which are displayed in
Figure 13, are similar to those in Figure
11, albeit a different scale.
# Construct VIP (Figure 13)
set.seed(2321) # for reproducibility
pfun <- function(object, newdata) predict(object, newdata = newdata)
vip(nn, method = "permute", target = "y", metric = mae,
smaller_is_better = TRUE, pred_wrapper = pfun) +
ggtitle("Custom loss function: MAE")
Although permutation importance is most naturally computed on the
training data, it may also be useful to do the shuffling and measure
performance on new data! This is discussed in depth in Molnar (2019b 5.2). For users interested in computing permutation importance
using new data, just supply it to the train argument in the call to
vi(), vip(), or vi_permute(). For instance, suppose we wanted to
only use a fraction of the original training data to carry out the
computations. In this case, we could simply pass the sampled data to the
train argument as follows:
# Construct VIP (Figure 14)
set.seed(2327) # for reproducibility
vip(nn, method = "permute", pred_wrapper = pfun, target = "y", metric = "rmse",
train = trn[sample(nrow(trn), size = 400), ]) + # sample 400 observations
ggtitle("Using a random subset of training data")
When using the permutation method with nsim > 1, the default is to
keep all the permutation scores as an attribute called "raw_scores";
you can turn this behavior off by setting keep = FALSE in the call to
vi_permute(), vi(), or vip(). If keep = TRUE and nsim > 1, you
can request all permutation scores to be plotted by setting
all_permutation = TRUE in the call to vip(), as demonstrated in the
code chunk below (see Figure 15). This also let’s
you visually inspect the variability in the permutation scores within
each feature.
# Construct VIP (Figure 15)
set.seed(8264) # for reproducibility
vip(nn, method = "permute", pred_wrapper = pfun, target = "y", metric = "mae",
nsim = 10, geom = "point", all_permutations = TRUE, jitter = TRUE) +
ggtitle("Plotting all permutation scores")
Benchmarks
In this section, we compare the performance of four implementations of
permutation-based VI scores: iml::FeatureImp() (version 0.10.0),
ingredients::feature_importance() (version 1.3.1),
mmpf::permutationImportance (version 0.0.5), and vip::vi() (version
0.2.2.9000).
We simulated 10,000 training observations from the Friedman 1 benchmark problem and trained a random forest using the ranger package. For each implementation, we computed permutation-based VI scores 100 times using the microbenchmark package (Mersmann 2019). For this benchmark we did not use any of the parallel processing capability available in the iml and vip implementations. The results from microbenchmark are displayed in Figure 16 and summarized in the output below. In this case, the vip package (version 0.2.2.9000) was the fastest, followed closely by ingredients and mmpf. It should be noted, however, that the implementations in vip and iml can be parallelized. To the best of our knowledge, this is not the case for ingredients or mmpf (although it would not be difficult to write a simple parallel wrapper for either). The code used to generate these benchmarks can be found at http://bit.ly/2TogXrq.
Shapley method
Although vip focuses on global VI methods, it is becoming increasing popular to asses global importance by aggregating local VI measures; in particular, Shapley explanations (Štrumbelj and Kononenko 2014). Using Shapley values (a method from coalitional game theory), the prediction for a single instance \(x^\star\) can be explained by assuming that each feature value in \(x^\star\) is a “player” in a game with a payout equal to the corresponding prediction \(\widehat{f}\left(x^\star\right)\). Shapley values tell us how to fairly distribute the “payout” (i.e., prediction) among the features. Shapley values have become popular due to the attractive fairness properties they posses (Lundberg and Lee 2017). The most popular implementation is available in the Python shap package (Lundberg and Lee 2017); although a number of implementations are now available in R; for example, iml, iBreakDown (Biecek et al. 2019b), and fastshap (Greenwell 2019).
Obtaining a global VI score from Shapley values requires aggregating the
Shapley values for each feature across the entire training set (or at
least a reasonable sample thereof). In particular, we use the mean of
the absolute value of the individual Shapley values for each feature.
Unfortunately, Shapley values can be computationally expensive, and
therefore this approach may not be feasible for large training sets
(say, >3000 observations). The fastshap package provides some relief
by exploiting a few computational tricks, including the option to
perform computations in parallel (see ?fastshap::explain for details).
Also, fast and exact algorithms (Lundberg et al. 2019) can be
exploited for certain classes of models.
Starting with vip version 0.2.2.9000 you can now use method = "shap"
in the call to vi() (or use vi_shap() directly) to compute global
Shapley-based VI scores using the method described above (provided you
have the fastshap package installed)—see ?vip::vi_shap for
details. To illustrate, we compute Shapley-based VI scores from an
xgboost model
(Chen et al. 2019) using the Friedman data from earlier; the results are
displayed in Figure 17.9 (Note: specifying
include_type = TRUE in the call to vip() causes the type of VI
computed to be displayed as part of the axis label.)
# Load required packages
library(xgboost)
# Feature matrix
X <- data.matrix(subset(trn, select = -y)) # matrix of feature values
# Fit an XGBoost model; hyperparameters were tuned using 5-fold CV
set.seed(859) # for reproducibility
bst <- xgboost(X, label = trn$y, nrounds = 338, max_depth = 3, eta = 0.1,
verbose = 0)
# Construct VIP (Figure 17)
vip(bst, method = "shap", train = X, exact = TRUE, include_type = TRUE)
3 Drawbacks of existing methods
As discussed in Hooker and Mentch (2019), permute-and-predict methods—like PDPs, ICE curves, and permutation importance—can produce results that are highly misleading.10 For example, the standard approach to computing permutation-based VI scores involves independently permuting individual features. This implicitly makes the assumption that the observed features are statistically independent. In practice, however, features are often not independent which can lead to nonsensical VI scores. One way to mitigate this issue is to use the conditional approach described in Strobl et al. (2008); Hooker and Mentch (2019) provides additional alternatives, such as permute-and-relearn importance. Unfortunately, to the best of our knowledge, this approach is not yet available for general purpose. A similar modification can be applied to PDPs [Parr and Wilson (2019)]11 which seems reasonable to use in the FIRM approach when strong dependencies among the features are present (though, we have not given this much thought or consideration).
We already mentioned that PDPs can be misleading in the presence of
strong interaction effects. This drawback, of course, equally applies to
the FIRM approach using PDPs for computing VI scores. As discussed
earlier, this can be mitigated by using ICE curves instead. Another
alternative would be to use accumulated local effect (ALE) plots
(Apley and Zhu 2016) (though we haven’t really tested this idea).
Compared to PDPs, ALE plots have the advantage of being faster to
compute and less affected by strong dependencies among the features. The
downside, however, is that ALE plots are more complicated to implement
(hence, they are not currently available when using method = "firm").
ALE plots are available in the
ALEPlot (Apley 2018) and
iml packages.
Hooker (2007) also argues that feature importance (which concern only main effects) can be misleading in high dimensional settings, especially when there are strong dependencies and interaction effects among the features, and suggests an approach based on a generalized functional ANOVA decomposition—though, to our knowledge, this approach is not widely implemented in open source.
4 Use sparklines to characterize feature effects
Starting with vip 0.1.3, we have included a new function
add_sparklines() for constructing HTML-based VI tables; however, this
feature requires the DT
package (Xie et al. 2019). The primary difference between vi() and
add_sparklines() is that the latter includes an Effect column that
displays a sparkline representation of the partial dependence function
for each feature. This is a concise way to display both feature
importance and feature effect information in a single (interactive)
table. See ?vip::add_sparklines for details. We illustrate the basic
use of add_sparklines() in the code chunk below where we fit a
ranger-based random forest using the
mlr3 package
(Lang et al. 2019).12
# Load required packages
library(mlr3)
library(mlr3learners)
# Fit a ranger-based random forest using the mlr3 package
set.seed(101)
task <- TaskRegr$new("friedman", backend = trn, target = "y")
lrnr <- lrn("regr.ranger", importance = "impurity")
lrnr$train(task)
# First, compute a tibble of VI scores using any method
var_imp <- vi(lrnr)
# Next, convert to an HTML-based data table with sparklines
add_sparklines(var_imp, fit = lrnr$model, train = trn) # Figure 18
5 Ames housing example
For illustration, we’ll use the Ames housing data (Cock 2011)
which are available in the
AmesHousing package
(Kuhn 2017). These data describe the sale of individual residential
properties in Ames, Iowa from 2006–2010. The data set contains 2930
observations, 80 features (23 nominal, 23 ordinal, 14 discrete, and 20
continuous), and a continuous target giving the sale price of the home.
The version we’ll load is a cleaned up version of the original data set
and treats all categorical variables as nominal (see
?AmesHousing::make_ames for details).
Using the R package SuperLearner (Polley et al. 2019), we trained five models using 5-fold cross-validation: a GBM using the xgboost package, an RF using the ranger package, a MARS model using the earth package, a GLMNET model using the glmnet package (Friedman et al. 2019), and a support vector regression model using the kernlab package (Karatzoglou et al. 2019). The magnitude of the coefficients from the meta learner indicate which models contribute the most (if at all) to new predictions.
# Load the Ames housing data
ames <- AmesHousing::make_ames()
X <- subset(ames, select = -Sale_Price)
y <- ames$Sale_Price
# Load required packages
library(SuperLearner)
# List of base learners
learners <- c("SL.xgboost", "SL.ranger", "SL.earth", "SL.glmnet", "SL.ksvm")
# Stack models
set.seed(840) # for reproducibility
ctrl <- SuperLearner.CV.control(V = 5L, shuffle = TRUE)
sl <- SuperLearner(Y = y, X = X, SL.library = learners, verbose = TRUE,
cvControl = ctrl)
sl#>
#> Call:
#> SuperLearner(Y = y, X = X, SL.library = learners, verbose = TRUE, cvControl = ctrl)
#>
#>
#>
#> Risk Coef
#> SL.xgboost_All 580713682 0.41384425
#> SL.ranger_All 666208088 0.08083034
#> SL.earth_All 553872844 0.50532541
#> SL.glmnet_All 908881559 0.00000000
#> SL.ksvm_All 6784289108 0.00000000In the code chunks below we request permutation-based VI scores and a sparkline representation of the PDPs for the top ten features. For this we need to define a couple of wrapper functions: one for computing predictions (for the permutation VI scores), and one for computing averaged predictions (for the PDPs).
# Prediction wrapper functions
imp_fun <- function(object, newdata) { # for permutation-based VI scores
predict(object, newdata = newdata)$pred
}
par_fun <- function(object, newdata) { # for PDPs
mean(predict(object, newdata = newdata)$pred)
}To speed up the process, we perform the computations in parallel by
setting parallel = TRUE in the calls to vi() and add_sparklines().
Note that we first need to set up a parallel backend for this to work.
Both vip and pdp use
plyr (Wickham 2019)—which
relies on foreach—so any parallel backend supported by the foreach
package should work. Below we use a socket approach with the
doParallel backend
(Corporation and Weston 2019) using a cluster of size five.
# Setup parallel backend
library(doParallel) # load the parallel backend
cl <- makeCluster(5) # use 5 workers
registerDoParallel(cl) # register the parallel backend
# Permutation-based feature importance
set.seed(278) # for reproducibility
var_imp <- vi(sl, method = "permute", train = X, target = y, metric = "rmse",
pred_wrapper = imp_fun, nsim = 5, parallel = TRUE)
# Add sparkline representation of feature effects (# Figure 19)
add_sparklines(var_imp[1L:10L, ], fit = sl, pred.fun = par_fun, train = X,
digits = 2, verbose = TRUE, trim.outliers = TRUE,
grid.resolution = 20, parallel = TRUE)
# Shut down cluster
stopCluster(cl)6 Summary
VIPs help to visualize the strength of the relationship between each feature and the predicted response, while accounting for all the other features in the model. We’ve discussed two types of VI: model-specific and model-agnostic, as well as some of their strengths and weaknesses. In this paper, we showed how to construct VIPs for various types of “black box” models in R using the vip package. We also briefly discussed related approaches available in a number of other R packages. Suggestions to avoid high execution times were discussed and demonstrated via examples. This paper is based on vip version 0.2.2.9000. In terms of future development, vip can be expanded in a number of ways. For example, we plan to incorporate the option to compute group-based and conditional permutation scores. Although not discussed in this paper, vip also includes a promising statistic (similar to the variance-based VI scores previously discussed) for measuring the relative strength of interaction between features. Although VIPs can help understand which features are driving the model’s predictions, ML practitioners should be cognizant of the fact that none of the methods discussed in this paper are uniformly best across all situations; they require an accurate model that has been properly tuned, and should be checked for consistency with human domain knowledge.
7 Acknowledgments
The authors would like to thank the anonymous reviewers and the Editor for their helpful comments and suggestions. We would also like to thank the members of the 84.51\(^{\circ}\) Interpretable Machine Learning Special Interest Group for their thoughtful discussions on the topics discussed herein.
CRAN packages used
iml, R6, foreach, ingredients, DALEX, mmpf, varImp, party, measures, vita, rfVarImpOOB, randomForestExplainer, tree.interpreter, pkgsearch, caret, mlr, ranger, vip, ggplot2, partykit, earth, nnet, vivo, pdp, microbenchmark, iBreakDown, fastshap, xgboost, ALEPlot, DT, mlr3, data.table, AmesHousing, SuperLearner, glmnet, kernlab, plyr, doParallel
CRAN Task Views implied by cited packages
Bayesian, Cluster, Databases, Econometrics, Environmetrics, Finance, HighPerformanceComputing, MachineLearning, ModelDeployment, NaturalLanguageProcessing, Optimization, Phylogenetics, ReproducibleResearch, Spatial, Survival, TeachingStatistics, TimeSeries, WebTechnologies
Note
This article is converted from a Legacy LaTeX article using the texor package. The pdf version is the official version. To report a problem with the html, refer to CONTRIBUTE on the R Journal homepage.