Constructing VIPs in R

We’ll illustrate major concepts using the Friedman 1 benchmark problem described in Friedman (1991) and Breiman (1996):

\[\begin{equation} Y_i = 10 \sin\left(\pi X_{1i} X_{2i}\right) + 20 \left(X_{3i} - 0.5\right) ^ 2 + 10 X_{4i} + 5 X_{5i} + \epsilon_i, \quad i = 1, 2, \dots, n, \tag{1} \end{equation}\]

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 × 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
## # ℹ 490 more rows

From 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).

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, Friedman, and Charles J. Stone (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, Tibshirani, and Friedman (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 ?? (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 × 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 × 2
##    Variable Importance
##    <chr>         <dbl>
##  1 x4           72.9  
##  2 x2           61.4  
##  3 x1           55.6  
##  4 x5           37.0  
##  5 x3           22.0  
##  6 x8            1.84 
##  7 x6            1.12 
##  8 x9            0.720
##  9 x7           -1.39 
## 10 x10          -2.61

vi(bst)   # GBM

## # A tibble: 10 × 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          0.00262

Notice 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 ??.

library(patchwork)  # for easily arranging multiple ggplot2 plots

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)
p1 + p2 + p3

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 ??. 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. Motivation for the use of the assoicated \(t\)-statistic is given in Bring (1994). 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 ??.

# 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 × 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  
## # ℹ 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 for details). An example using the earth package (Milborrow 2019) is given below (the results are plotted in Figure ??):

# 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 × 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, Joy, and Death 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 ??).

# 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)
p1 + p2

Variance-based methods

Our first model-agnostic method is based on a simple feature importance ranking measure (FIRM); for details, see B. M. Greenwell, Boehmke, and McCarthy (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) 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 B. M. Greenwell (2017). The results are displayed in Figure ??. 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", train = trn) + ggtitle("PPR")
p2 <- vip(nn, method = "firm", train = trn) + ggtitle("NN")

# Display plots in a grid (Figure 7)
p1 + p2

In Figure ?? 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 ??. 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 ??, are similar to those obtained using the PDP method.

# Construct VIPs
p1 <- vip(pp, method = "firm", ice = TRUE, train = trn) + ggtitle("PPR")
p2 <- vip(nn, method = "firm", ice = TRUE, train = trn) + ggtitle("NN")

# Display plots in a grid (Figure 9)
p1 + p2

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 ??):

# Construct PDP-based VI scores
(vis <- vi(pp, method = "firm", train = trn))

## # A tibble: 10 × 2
##    Variable Importance
##    <chr>         <dbl>
##  1 x4           2.96  
##  2 x2           2.21  
##  3 x1           2.14  
##  4 x5           1.53  
##  5 x3           1.46  
##  6 x6           0.128 
##  7 x9           0.114 
##  8 x8           0.0621
##  9 x10          0.0374
## 10 x7           0.0170

# 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)
}

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, Rudin, and Dominici (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 \(M_{orig}\) 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:

1. For \(i = 1, 2, \dots, j\):

1. Permute the values of feature \(x_i\) in the training data.
2. Recompute the performance metric on the permuted data \(M_{perm}\).
3. Record the difference from baseline using \(VI\left(x_i\right) = M_{perm} - M_{orig}\).

2. Return the VI scores \(VI\left(x_1\right), VI\left(x_2\right), \dots, VI\left(x_j\right)\).

Algorithm 1: A simple algorithm for constructing permutation-based VI scores.

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, sec. 5.5.4) argues that using the ratio \(M_{perm} / M_{orig}\) 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.

To use vi_permute() you should first define a prediction wrapper that tells the function how to generate the write predictions for your chosen metric. An example is given below for the previously fitted PPR and NN models. Here we use \(R^2\) (metric = "rsq") as the evaluation metric. The results, which are displayed in Figure ??, agree with those obtained using the PDP- and ICE-based methods.

# Prediction wrapper
pfun_ppr <- function(object, newdata) {  # needs to return a numeric vector
  stats::predict(object, newdata = newdata)
}
pfun_nnet <- function(object, newdata) {  # needs to return a numeric vector
  stats::predict(object, newdata = newdata)[, 1L, drop = TRUE]
}

# Plot VI scores
set.seed(2021)  # for reproducibility
p1 <- vip(pp, method = "permute", train = trn, target = "y", metric = "rsq",
          pred_wrapper = pfun_ppr) + ggtitle("PPR")
p2 <- vip(nn, method = "permute", train = trn, target = "y", metric = "rsq",
          pred_wrapper = pfun_nnet) + ggtitle("NN")

# Display plots in a grid (Figure 11)
p1 + p2

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 = 30 to request that each feature be permuted 30 times and the results averaged together. (Additionally, if nsim > 1, you can set 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 ?? for an example.)

# Use 10 Monte Carlo reps
set.seed(403)  # for reproducibility
vis <- vi(pp, method = "permute", train = trn, target = "y", metric = "rsq",
          pred_wrapper = pfun_ppr, nsim = 30)
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
## 1      accuracy                   Classification accuracy
## 2  bal_accuracy          Balanced classification accuracy
## 3        youden Youden;'s index (or Youden's J statistic)
## 4       roc_auc                      Area under ROC curve
## 5        pr_auc    Area under precision-recall (PR) curve
## 6       logloss                                  Log loss
## 7         brier                               Brier score
## 8           mae                       Mean absolute error
## 9          mape            Mean absolute percentage error
## 10         rmse                   Root mean squared error
## 11          rsq                   R-squared (correlation)
## 12     rsq_trad                   R-squared (traditional)
##                                task smaller_is_better yardstick_function
## 1  Binary/multiclass classification             FALSE       accuracy_vec
## 2  Binary/multiclass classification             FALSE   bal_accuracy_vec
## 3  Binary/multiclass classification             FALSE            j_index
## 4             Binary classification             FALSE        roc_auc_vec
## 5             Binary classification             FALSE         pr_auc_vec
## 6  Binary/multiclass classification              TRUE    mn_log_loss_vec
## 7  Binary/multiclass classification              TRUE    brier_class_vec
## 8                        Regression              TRUE            mae_vec
## 9                        Regression              TRUE           mape_vec
## 10                       Regression              TRUE           rmse_vec
## 11                       Regression             FALSE            rsq_vec
## 12                       Regression             FALSE       rsq_trad_vec

The permutation method in vip supports the vector performance functions available in yardstick (Kuhn, Vaughan, and Hvitfeldt 2023). We can also use a custom metric (i.e., loss function). Suppose for example you want to measure importance using the mean absolute error (MAE):

\[\begin{equation} MAE = \frac{1}{n}\sum_{i = 1}^n\left|y_i - \hat{f}\left(\boldsymbol{x}_i\right)\right|, \end{equation}\]

where \(\hat{f}\left(\boldsymbol{x}_i\right)\) is the predicted value of \(y_i\). A simple function implementing this metric is given below (to be consistent with yardstick functions, user-supplied metric functions require two arguments: truth and estimate).

mae <- function(truth, estimate) {
  mean(abs(truth - estimate))
}

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 ??, are similar to those in Figure ??, albeit a different scale.

# Construct VIP (Figure 13)
set.seed(2321)  # for reproducibility
p1 <- vip(nn, method = "permute", train = trn, target = "y", metric = mae,
          smaller_is_better = TRUE, pred_wrapper = pfun_nnet) +
  ggtitle("Custom loss function: MAE")
set.seed(2321)  # for reproducibility
p2 <- vip(nn, method = "permute", train = trn, target = "y",
          metric = yardstick::mae_vec, smaller_is_better = TRUE,
          pred_wrapper = pfun_nnet) +
  ggtitle("Using `yardstick`'s MAE function")
p1 + p2

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, sec. 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_nnet, 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_permutations = TRUE in the call to vip(), as demonstrated in the code chunk below (see Figure ??). 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_nnet, train = trn,
    target = "y", metric = "mae", nsim = 10, geom = "point",
    all_permutations = TRUE, jitter = TRUE) +
  ggtitle("Plotting all permutation scores")

head(t1 <- vip::titanic_mice[[1L]])

##   survived pclass   age    sex sibsp parch
## 1      yes      1 29.00 female     0     0
## 2      yes      1  0.92   male     1     2
## 3       no      1  2.00 female     1     2
## 4       no      1 30.00   male     1     2
## 5       no      1 25.00 female     1     2
## 6      yes      1 48.00   male     0     0

t1$pclass <- as.ordered(t1$pclass)  # makes more sense as an ordered factor

library(randomForest)

set.seed(2053)  # for reproducibility
(rfo <- randomForest(survived ~ ., data = t1, importance = TRUE, nPerm = 30))

## 
## Call:
##  randomForest(formula = survived ~ ., data = t1, importance = TRUE,      nPerm = 30) 
##                Type of random forest: classification
##                      Number of trees: 500
## No. of variables tried at each split: 2
## 
##         OOB estimate of  error rate: 18.79%
## Confusion matrix:
##      no yes class.error
## no  727  82   0.1013597
## yes 164 336   0.3280000

vip(rfo, include_type = TRUE)

head(predict(rfo, newdata = t1, type = "response"))  # predicted class labels

##   1   2   3   4   5   6 
## yes yes yes  no yes  no 
## Levels: no yes

head(predict(rfo, newdata = t1, type = "prob"))  # predicted class probabilities

##      no   yes
## 1 0.014 0.986
## 2 0.114 0.886
## 3 0.472 0.528
## 4 0.716 0.284
## 5 0.392 0.608
## 6 0.894 0.106

pfun_class <- function(object, newdata) {  # prediction wrapper
  predict(object, newdata = newdata, type = "response")
}

# Compute mean decrease in accuracy
set.seed(1359)  # for reproducibility
vi(rfo,
   method = "permute",
   train = t1,
   target = "survived",
   metric = "accuracy",  # or pass in `yardstick::accuracy_vec` directly
   # smaller_is_better = FALSE,  # no need to set for built-in metrics
   pred_wrapper = pfun_class,
   nsim = 30  # use 30 repetitions
)

## # A tibble: 5 × 3
##   Variable Importance   StDev
##   <chr>         <dbl>   <dbl>
## 1 sex          0.226  0.0111 
## 2 pclass       0.0801 0.00488
## 3 age          0.0738 0.00595
## 4 sibsp        0.0346 0.00459
## 5 parch        0.0166 0.00247

sort(rfo$importance[, "MeanDecreaseAccuracy"], decreasing = TRUE)

##        sex     pclass        age      parch      sibsp 
## 0.17102147 0.05877827 0.04408406 0.01895065 0.01583429

pfun_prob <- function(object, newdata) {  # prediction wrapper
  predict(object, newdata = newdata, type = "prob")[, "yes"]
}

# Compute mean increase in Brier score
set.seed(1411)  # for reproducibility
vi(rfo,
   method = "permute",
   train = t1,
   target = "survived",
   metric = yardstick::brier_class_vec,  # or pass in `"brier"` directly
   smaller_is_better = FALSE,  # need to set when supplying a function
   pred_wrapper = pfun_prob,
   nsim = 30  # use 30 repetitions
)

## # A tibble: 5 × 3
##   Variable Importance   StDev
##   <chr>         <dbl>   <dbl>
## 1 sex          0.209  0.00866
## 2 pclass       0.0977 0.00479
## 3 age          0.0947 0.00460
## 4 parch        0.0542 0.00271
## 5 sibsp        0.0414 0.00186

set.seed(1413)  # for reproducibility
vi(rfo,
   method = "permute",
   train = t1,
   target = "survived",
   metric = "roc_auc",
   pred_wrapper = pfun_prob,
   nsim = 30  # use 30 repetitions
)

## # A tibble: 5 × 3
##   Variable Importance   StDev
##   <chr>         <dbl>   <dbl>
## 1 parch       -0.0251 0.00351
## 2 sibsp       -0.0283 0.00211
## 3 age         -0.0850 0.00477
## 4 pclass      -0.0920 0.00533
## 5 sex         -0.229  0.0137

levels(titanic$survived)

## [1] "no"  "yes"

set.seed(1413)  # for reproducibility
vi(rfo,
   method = "permute",
   train = t1,
   target = "survived",
   metric = "roc_auc",
   event_level = "second",  # use "yes" as class label/"event" of interest
   pred_wrapper = pfun_prob,
   nsim = 30  # use 30 repetitions
)

## # A tibble: 5 × 3
##   Variable Importance   StDev
##   <chr>         <dbl>   <dbl>
## 1 sex          0.229  0.0137 
## 2 pclass       0.0920 0.00533
## 3 age          0.0850 0.00477
## 4 sibsp        0.0283 0.00211
## 5 parch        0.0251 0.00351

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 \(\hat{f}\left(x^\star\right)\). 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. 2019), and fastshap (B. 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 for details). Also, fast and exact algorithms can be exploited for certain classes of models.

Starting with vip version 0.4.1 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 [R-xgboost] using the Friedman data from earlier; the results are displayed in Figure (fig:vi-shap).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,
    geom = "point", horizontal = FALSE,
    aesthetics = list(color = "forestgreen", shape = 17, size = 5)) +
  theme_light()

Passing exact = TRUE to fastshap::explain() via the ... argument in the call to vip() (or vi() and vi_shap()) only works for lightgbm, xgboost, and additive (generalized) linear models fit using R’s internal stats package. For all other cases, a prediction wrapper must be supplied via the ... argument.

To illustrate, let’s use the previous random forest that was fit to the Titanic data set. Note that Shapley explanation do not support classification, so we’ll have to use the probability-based prediction wrapper defined before:

pfun_prob <- function(object, newdata) {  # prediction wrapper
  # For Shapley explanations, this should ALWAYS return a numeric vector
  predict(object, newdata = newdata, type = "prob")[, "yes"]
}

# Compute Shapley-based VI scores
set.seed(853)  # for reproducibility
vi_shap(rfo, train = subset(t1, select = -survived), pred_wrapper = pfun_prob,
        nsim = 30)

## # A tibble: 5 × 2
##   Variable Importance
##   <chr>         <dbl>
## 1 pclass       0.104 
## 2 age          0.0649
## 3 sex          0.272 
## 4 sibsp        0.0260
## 5 parch        0.0291

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 (D. W. 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 (D. 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.