Random Forest and generalizations (in particular, Generalized Random Forests (GRF) and Distributional Random Forests (DRF) ) are powerful and easy-to-use machine learning methods that should not be absent in the toolbox of any data scientist. They not only show robust performance over a large range of datasets without the need for tuning, but can also easily handle missing values, and even provide confidence intervals. In this article, we focus on another feature they are able to provide: notions of feature importance. In particular, we focus on:

- Traditional Random Forest (RF), which is used to predict the conditional expectation of a variable
*Y*given p predictors.*X* - The Distributional Random Forest, which is used to predict the whole conditional distribution of a d-variate
given*Y**p*predictors.*X*

Unfortunately, like many modern machine learning methods, both forests lack interpretability. That is, there are so many operations involved, it seems impossible to determine what the functional relationship between the predictors and *Y* actually is. A common way to tackle this problem is to define Variable Importance measures (VIMP), that at least help decide which predictors are important. Generally, this has two different objectives:

(1) finding a small number of variables with maximal accuracy,

(2) detecting and ranking all influential variables to focus on for further exploration.

The difference between (1) and (2) matters as soon as there is dependence between the elements in ** X** (so pretty much always). For example, if two variables are highly correlated together and with

*Y*, one of the two inputs can be removed without hurting accuracy for objective (1), since both variables convey the same information. However, both should be included for objective (2), since these two variables may have different meanings in practice for domain experts.

Today we focus on (1) and try to find a smaller number of predictors that display more or less the same predictive accuracy. For instance, in the wage example below, we are able to reduce the number of predictors from 79 to about 20, with only a small reduction in accuracy. These most important predictors contain variables such as age and education which are well-known to influence wages. There are also many great articles on medium about (2), using Shapley values such as this one or this one. There is also very recent and exciting academic literature on how to efficiently calculate Shapley values with Random Forest. But this is material for a second article.

The two measures we look at today are actually more general variable importance measures that can be used for any method, based on the drop-and-relearn principle which we will look at below. We focus exclusively on tree-based methods here, however. Moreover, we don’t go into great detail explaining the methods, but rather try to focus on their applications and why newer versions are preferable to the more traditional ones.

## The Beginnings

Variable importance measures for RFs are in fact as old as RF itself. The first accuracy the **Mean Decrease Accuracy (MDA)** was proposed by Breiman in his seminal Random Forest paper [1]. The idea is simple: For every dimension *j=1,…,p*, one compares the accuracy of the full prediction with the accuracy of the prediction when *X_j* is randomly permuted. The idea of this is to break the relationship between *X_j* and *Y* and compare the accuracy when *X_j* is not helping to predict *Y* by design, to the case when it is potentially of use.

There are various different versions of MDA implemented in R and Python:

Unfortunately, permuting variable *X_j* in this way not only breaks its relationship to *Y*, but also to the other variables in ** X**. This is not a problem if

*X_j*is independent from all other variables, but it becomes a problem once there is dependence. Consequently, [3] is able to show that as soon as there is dependence in

**, the MDA converges to something nonsensical. In particular, MDA can give high importance to a variable**

*X**X_j*that is not important to predict

*Y*, but is highly correlated with another variable, say

*X_l*, that is actually important for predicting

*Y*(as demonstrated in the example below). At the same time, it can fail to detect variables that are actually relevant, as demonstrated by a long list of papers in [3, Section 2.1]. Intuitively, what we would want to measure is the performance of the model if

*X_j*is not included, and instead, we measure the performance of a model with a permuted

*X_j*variable.

The second traditional accuracy measure is **Mean Decrease Impurity (MDI)**, which sums the weighted decreases of impurity over all nodes that split on a given covariate, averaged over all trees in the forest. Unfortunately, MDI is ill-defined from the start (it’s not clear what it should measure) and several papers highlight the practical problem of this approach (e.g. [5]) As such, we will not go into detail about MDI, as MDA is often the preferred choice.

## Modern Developments I: Sobol-MDA

For the longest time, I thought these somewhat informal measures were the best we could do. One paper that changed that, came out only very recently. In this paper, the authors demonstrate theoretically that the popular measures above are actually quite flawed and do not measure what we want to measure. So the first question might be: What do we actually want to measure? One potential answer: The Sobol-index (originally proposed in the computer science literature):

Let’s unpack this. First, *tau(**X**)=E[ Y | **X**] *is the conditional expectation function we would like to estimate. This is a random variable because it is a function of the random ** X**. Now

*X**^{(-j)}*is the

*p-1*vector with covariate

*j*removed. Thus

*ST^{(j)}*is the reduction in output explained variance if the

*j*th output variable is removed.

The above is the more traditional way of writing the measure. However, for me writing:

is much more intuitive. Here *d* is a distance between two random vectors and for the *ST^{(j)}* above, this distance is simply the usual Euclidean distance. Thus the upper part of *ST^{(j)}* is simply measuring the average squared distance between what we want (*tau(**X**)*) and what we get without variable *j*. The latter is

The question becomes how to estimate this efficiently. It turns out that the intuitive drop-and-relearn principle would be enough: Simply estimating *tau(**X**) *using RF and then dropping *X_j* and refitting the RF to obtain an estimate of *tau(**X^{(-j)}**), *one obtains the consistent estimator*:*

where *tau_n(**X**_i)* is the RF estimate for a test point *X**_i* using all *p* predictors and similarly *tau_n(**X**_i^{(-j)})* is the refitted forest using only *p-1* predictors.

However, this means the forest needs to be refitted *p* times, not very efficient when *p* is large! As such the authors in [3] develop what they call the **Sobol-MDA**. Instead of refitting the forest each time, the forest is only fitted once. Then test points are dropped down the same forest and the resulting prediction is “projected” to form the measure in (1). That is, splits on *X_j* are simply ignored (remember the goal is to obtain an estimate without *X_j*). The authors are able to show that calculating (1) above with this projected approach also results in a consistent estimator! This is a beautiful idea indeed and renders the algorithm applicable even in high dimensions.

The method is implemented in R in the soboldMDA package, based on the very fast ranger package.

## Modern Developments II: MMD-based sensitivity index

Looking at the formulation using the distance *d*, a natural question is to ask whether different distances could be used to get variable importance measures for more difficult problems. One such recent example is to use the MMD distance as *d:*

The MMD distance is a wonderful tool, that allows to quite easily build a distance between distributions using a kernel *k* (such as the Gaussian kernel):

For the moment I leave the details to further articles. The most important takeaway is simply that *I^{(j)}* considers a more general target than the conditional expectation. It recognizes a variable *X_j *as important, as soon as it influences the distribution of *Y* in any way. It might be that *X_j* only changes the variance or the quantiles and leaves the conditional mean of *Y* untouched (see example below). In this case, the Sobol-MDA would not recognize *X_j* as important, but the MMD method would. This doesn’t necessarily make it better, it is simply a different tool: If one is interested in predicting the conditional expectation, *ST^{(j)}* is the right measure. However, if one is interested in predicting other aspects of the distribution, especially quantiles, *I^{(j)}* would be better suited. Again *I^{(j)}* can be consistently estimated using the drop-and-relearn principle (refitting DRF for *j=1,…,p* eacht time with variable $j$ removed), or the same projection approach as for Sobol-MDA can be used. A drop-and-relearn-based implementation is attached at the end of this article. We refer to this method here as **MMD-MDA**.

## Simulated Data

We now illustrate these two modern measures on a simple simulated example: We first download and install the Sobol-MDA package from Gitlab and then load all the packages necessary for this example:

`library(kernlab)`

library(drf)

library(Matrix)

library(DescTools)

library(mice)

library(sobolMDA)

source("compute_drf_vimp.R") ##Contents of this file can be found below

source("evaluation.R") ##Contents of this file can be found below

Then we simulate from this simple example: We take *X_1, X_2, X_4, …, X_10* independently uniform between (-1,1) and create dependence between *X_1* and *X_3* by taking *X_3=X_1 + uniform error*. Then we simulate *Y* as

`##Simulate Data that experiences both a mean as well as sd shift`# Simulate from X

x1 <- runif(n,-1,1)

x2 <- runif(n,-1,1)

X0 <- matrix(runif(7*n,-1,1), nrow=n, ncol=7)

x3 <- x1+ runif(n,-1,1)

X <- cbind(x1,x2, x3, X0)

# Simulate dependent variable Y

Y <- as.matrix(rnorm(n,mean = 0.8*(x1 > 0), sd = 1 + 1*(x2 > 0)))

colnames(X)<-paste0("X", 1:10)

head(cbind(Y,X))

We then analyze the Sobol-MDA approach to estimate the conditional expectation of *Y* given ** X**:

`## Variable importance for conditional Expectation Estimation`XY <- as.data.frame(cbind(Xfull, Y))

colnames(XY) <- c(paste('X', 1:(ncol(XY)-1), sep=''), 'Y')

num.trees <- 500

forest <- sobolMDA::ranger(Y ~., data = XY, num.trees = num.trees, importance = 'sobolMDA')

sobolMDA <- forest$variable.importance

names(sobolMDA) <- colnames(X)

sort(sobolMDA, decreasing = T)

X1 X8 X7 X6 X5 X9

0.062220958 0.021946135 0.016818860 0.016777223 -0.001290326 -0.001540919

X3 X10 X4 X2

-0.001578540 -0.007400854 -0.008299478 -0.020334150

As can be seen, it correctly identifies that *X_1* is the most important variable, while the others are ranked equally (un)important. This makes sense because the conditional expectation of *Y* is only changed by *X_1*. Crucially, the measure manages to do this despite the dependence between *X_1 *and *X_3*. Thus we successfully pursued goal (1), as explained above, in this example. On the other hand, we can also have a look at the traditional MDA:

`forest <- sobolMDA::ranger(Y ~., data = XY, num.trees = num.trees, importance = 'permutation')`

MDA <- forest$variable.importance

names(MDA) <- colnames(X)sort(MDA, decreasing = T)

X1 X3 X6 X7 X8 X2

0.464516976 0.118147061 0.063969310 0.032741521 0.029004312 -0.004494380

X4 X9 X10 X5

-0.009977733 -0.011030996 -0.014281844 -0.018062544

In this case, while it correctly identifies *X_1 *as the most important variable, it also places *X_3 *in second place, with a value that seems quite a bit higher than the remaining variables. This despite the fact, that *X_3 *is just as unimportant as *X_2, X_4,…, X_10*!

But what if we are interested in predicting the distribution of *Y* more generally, say for estimating quantiles? In this case, we need a measure that is able to recognize the influence of *X_2* on the conditional variance of *Y*. Here the MMD variable importance measure comes into play:

`MMDVimp <- compute_drf_vimp(X=X,Y=Y)`

sort(MMDVimp, decreasing = T)X2 X1 X10 X6 X8 X3

0.683315006 0.318517259 0.014066410 0.009904518 0.006859128 0.005529749

X7 X9 X4 X5

0.003476256 0.003290550 0.002417677 0.002036174

Again the measure is able to correctly identify what matters: *X_1* and *X_2* are the two most important variables. And again, it does this despite the dependence between *X_1* and *X_3*. Interestingly, it also gives the variance shift from *X_2* a higher importance than the expectation shift from *X_1*.

## Real Data

Finally, I present a real data application to demonstrate the variable importance measure. Note that with DRF, we could look even at multivariate ** Y** but to keep things more simple, we focus on a univariate setting and consider the US wage data from the 2018 American Community Survey by the US Census Bureau. In the first DRF paper, we obtained data on approximately 1 million full-time employees from the 2018 American Community Survey by the US Census Bureau from which we extracted the salary information and all covariates that might be relevant for salaries. This wealth of data is ideal to experiment with a method like DRF (in fact we will only use a tiny subset for this analysis). The data we load can be found here.

`# Load data (https://github.com/lorismichel/drf/blob/master/applications/wage_data/data/datasets/wage_benchmark.Rdata)`

load("wage_benchmark.Rdata")##Define the training data

n<-1000

Xtrain<-X[1:n,]

Ytrain<-Y[1:n,]

Xtrain<-cbind(Xtrain,Ytrain[,"male"])

colnames(Xtrain)[ncol(Xtrain)]<-"male"

Ytrain<-Ytrain[,1, drop=F]

##Define the test data

ntest<-2000

Xtest<-X[(n+1):(n+ntest),]

Ytest<-Y[(n+1):(n+ntest),]

Xtest<-cbind(Xtest,Ytest[,"male"])

colnames(Xtest)[ncol(Xtest)]<-"male"

Ytest<-Ytest[,1, drop=F]

We now calculate both variable importance measures (this will take a while as only the drop-and-relearn method is implemented for DRF):

`# Calculate variable importance for both measures`

# 1. Sobol-MDA

XY <- as.data.frame(cbind(Xtrain, Ytrain))

colnames(XY) <- c(paste('X', 1:(ncol(XY)-1), sep=''), 'Y')

num.trees <- 500

forest <- sobolMDA::ranger(Y ~., data = XY, num.trees = num.trees, importance = 'sobolMDA')

SobolMDA <- forest$variable.importance

names(SobolMDA) <- colnames(Xtrain)# 2. MMD-MDA

MMDVimp <- compute_drf_vimp(X=Xtrain,Y=Ytrain,silent=T)

print("Top 10 most important variables for conditional Expectation estimation")

sort(SobolMDA, decreasing = T)[1:10]

print("Top 5 most important variables for conditional Distribution estimation")

sort(MMDVimp, decreasing = T)[1:10]

`Sobol-MDA:`education_level age male

0.073506769 0.027079349 0.013722756

occupation_11 occupation_43 industry_54

0.013550320 0.010025332 0.007744589

industry_44 occupation_23 occupation_15

0.006657918 0.005772662 0.004610835

marital_never married

0.004545964

`MMD-MDA:`education_level age male

0.420316085 0.109212519 0.027356393

occupation_43 occupation_11 marital_never married

0.016861954 0.014122583 0.003449910

occupation_29 marital_married industry_81

0.002272629 0.002085207 0.001152210

industry_72

0.000984725

In this case, the two variable importance measures agree quite a bit on which variables are important. While this is not a causal analysis, it is also nice that variables that are known to be important to predict wages, specifically “age”, “education_level” and “gender”, are indeed seen as very important by the two measures.

To obtain a small set of predictive variables, one could now for *j=1,…p-1,*

(I) Remove the least important variable

(II) Calculate the loss (e.g. mean squared error) on a test set

(III) Recalculate the variable importance for the remaining variable

(IV) Repeat until a certain stopping criterion is met

One could stop, for instance, if the loss increased by more than 5%. To make my life easier in this article, I just use the same variable importance values saved in “SobolMDA” and “MMDVimp” above. That is, I ignore step (III) and only consider (I), (II) and (IV). When the goal of estimation is the full conditional distribution, step (II) is also not entirely clear. We use what we refer to as MMD loss, described in more detail in our paper ([4]). This loss considers the error we are making in the prediction of the distribution. For the conditional mean, we simply use the mean-squared error. This is done in the function “evalall” found below:

`# Remove variables one-by-one accoring to the importance values saved in SobolMDA`

# and MMDVimp.

evallistSobol<-evalall(SobolMDA, X=Xtrain ,Y=Ytrain ,Xtest, Ytest, metrics=c("MSE"), num.trees )

evallistMMD<-evalall(MMDVimp, X=Xtrain ,Y=Ytrain ,Xtest, Ytest, metrics=c("MMD"), num.trees )plot(evallistSobol$evalMSE, type="l", lwd=2, cex=0.8, col="darkgreen", main="MSE loss" , xlab="Number of Variables removed", ylab="Values")

plot(evallistMMD$evalMMD, type="l", lwd=2, cex=0.8, col="darkgreen", main="MMD loss" , xlab="Number of Variables removed", ylab="Values")

This results in the following two pictures:

Notice that both have somewhat wiggly lines, which is first due to the fact that I did not recalculate the importance measure, e.g., left out step (III), and second due to the randomness of the forests. Aside from this, the graphs nicely show how the errors successively increase with each variable that is removed. This increase is first slow for the least important variables and then gets quicker for the most important ones, exactly as one would expect. In particular, the loss in both cases remains virtually unchanged if one removes the 50 least important variables! In fact, one could remove about 70 variables in both cases without increasing the loss by more than 6%. One has to note though that many predictors are part of one-hot encoded categorical variables and thus one needs to be somewhat careful when removing predictors, as they correspond to levels of one categorical variable. However, in an actual application, this might still be desirable.

## Conclusion

In this article, we looked at modern approaches to variable importance in Random Forests, with the goal of obtaining a small set of predictors or covariates, both with respect to the conditional expectation and for the conditional distribution more generally. We have seen in the wage data example, that this can lead to a substantial reduction in predictors with virtually the same accuracy.

As noted above the measures presented are not strictly constrained to Random Forest, but can be used more generally in principle. However, forests allow for the elegant projection approach that allows for the calculation of the importance measure for all variables *j*, without having to refit the forest each time (!) This is described in both [3] and [4].

## Literature

[1] Breiman, L. (2001). Random forests. Machine learning, 45(1):5–32.

[2] Breiman, L. (2003a). Setting up, using, and understanding random forests v3.1. Technical report, UC Berkeley, Department of Statistics

[3] Bénard, C., Da Veiga, S., and Scornet, E. (2022). Mean decrease accuracy for random forests: inconsistency, and a practical solution via the Sobol-MDA. Biometrika, 109(4):881–900.

[4] Clément Bénard, Jeffrey Näf, and Julie Josse. MMD-based variable importance for distributional random forest, 2023.

[5] Strobl, C., Boulesteix, A.-L., Zeileis, A., and Hothorn, T. (2007). Bias in random forest variable importance measures: illustrations, sources and a solution. BMC Bioinformatics, 8:25.

## Appendix : Code

#### Contents of compute_drf_vimp.R #######' Variable importance for Distributional Random Forests

#'

#' @param X Matrix with input training data.

#' @param Y Matrix with output training data.

#' @param X_test Matrix with input testing data. If NULL, out-of-bag estimates are used.

#' @param num.trees Number of trees to fit DRF. Default value is 500 trees.

#' @param silent If FALSE, print variable iteration number, otherwise nothing is print. Default is FALSE.

#'

#' @return The list of importance values for all input variables.

#' @export

#'

#' @examples

compute_drf_vimp <- function(X, Y, X_test = NULL, num.trees = 500, silent = FALSE){

# fit initial DRF

bandwidth_Y <- drf:::medianHeuristic(Y)

k_Y <- rbfdot(sigma = bandwidth_Y)

K <- kernelMatrix(k_Y, Y, Y)

DRF <- drf(X, Y, num.trees = num.trees)

wall <- predict(DRF, X_test)$weights

# compute normalization constant

wbar <- colMeans(wall)

wall_wbar <- sweep(wall, 2, wbar, "-")

I0 <- as.numeric(sum(diag(wall_wbar %*% K %*% t(wall_wbar))))

# compute drf importance dropping variables one by one

I <- sapply(1:ncol(X), function(j) {

if (!silent){print(paste0('Running importance for variable X', j, '...'))}

DRFj <- drf(X = X[, -j, drop=F], Y = Y, num.trees = num.trees)

DRFpredj <- predict(DRFj, X_test[, -j])

wj <- DRFpredj$weights

Ij <- sum(diag((wj - wall) %*% K %*% t(wj - wall)))/I0

return(Ij)

})

# compute retraining bias

DRF0 <- drf(X = X, Y = Y, num.trees = num.trees)

DRFpred0 = predict(DRF0, X_test)

w0 <- DRFpred0$weights

vimp0 <- sum(diag((w0 - wall) %*% K %*% t(w0 - wall)))/I0

# compute final importance (remove bias & truncate negative values)

vimp <- sapply(I - vimp0, function(x){max(0,x)})

names(vimp)<-colnames(X)

return(vimp)

}

`#### Contents of evaluation.R ######`compute_mmd_loss <- function(Y_train, Y_test, weights){

# Y_train <- scale(Y_train)

# Y_test <- scale(Y_test)

bandwidth_Y <- (1/drf:::medianHeuristic(Y_train))^2

k_Y <- rbfdot(sigma = bandwidth_Y)

K_train <- matrix(kernelMatrix(k_Y, Y_train, Y_train), ncol = nrow(Y_train))

K_cross <- matrix(kernelMatrix(k_Y, Y_test, Y_train), ncol = nrow(Y_train))

weights <- matrix(weights, ncol = ncol(weights))

t1 <- diag(weights%*%K_train%*%t(weights))

t2 <- diag(K_cross%*%t(weights))

mmd_loss <- mean(t1) - 2*mean(t2)

mmd_loss

}

evalall <- function(Vimp, X ,Y ,Xtest, Ytest, metrics=c("MMD","MSE"), num.trees ){

if (ncol(Ytest) > 1 & "MSE" %in% metrics){

metrics <- metrics[!( metrics %in% "MSE") ]

}

# Sort for increasing importance, such that the least important variables are removed first

Vimp<-sort(Vimp)

if ( is.null(names(Vimp)) ){

stop("Need names for later")

}

evalMMD<-matrix(0, nrow=ncol(X))

evalMSE<-matrix(0, nrow=ncol(X))

###Idea: Create a function that takes a variable importance measure and does this loop!!

for (j in 1:ncol(X)){

if (j==1){

if ("MMD" %in% metrics){

DRFred<- drf(X=X,Y=Y)

weights<- predict(DRFred, newdata=Xtest)$weights

evalMMD[j]<-compute_mmd_loss(Y_train=Y, Y_test=Ytest, weights)

}

if ("MSE" %in% metrics){

XY <- as.data.frame(cbind(X, Y))

colnames(XY) <- c(paste('X', 1:(ncol(XY)-1), sep=''), 'Y')

RFfull <- sobolMDA::ranger(Y ~., data = XY, num.trees = num.trees)

XtestRF<-Xtest

colnames(XtestRF) <- paste('X', 1:ncol(XtestRF), sep='')

predRF<-predict(RFfull, data=XtestRF)

evalMSE[j] <- mean((Ytest - predRF$predictions)^2)

}

}else{

if ("MMD" %in% metrics){

DRFred<- drf(X=X[,!(colnames(X) %in% names(Vimp[1:(j-1)])), drop=F],Y=Y)

weights<- predict(DRFred, newdata=Xtest[,!(colnames(Xtest) %in% names(Vimp[1:(j-1)])), drop=F])$weights

evalMMD[j]<-compute_mmd_loss(Y_train=Y, Y_test=Ytest, weights)

}

if ("MSE" %in% metrics){

XY <- as.data.frame(cbind(X[,!(colnames(X) %in% names(Vimp[1:(j-1)])), drop=F], Y))

colnames(XY) <- c(paste('X', 1:(ncol(XY)-1), sep=''), 'Y')

RFfull <- sobolMDA::ranger(Y ~., data = XY, num.trees = num.trees)

XtestRF<-Xtest[,!(colnames(Xtest) %in% names(Vimp[1:(j-1)])), drop=F]

colnames(XtestRF) <- paste('X', 1:ncol(XtestRF), sep='')

predRF<-predict(RFfull, data=XtestRF)

evalMSE[j] <- mean((Ytest - predRF$predictions)^2)

# DRFall <- drf(X=X[,!(colnames(X) %in% names(Vimp[1:(j-1)])), drop=F], Y=Y, num.trees=num.trees)

# quantpredictall<-predict(DRFall, newdata=Xtest[,!(colnames(Xtest) %in% names(Vimp[1:(j-1)])), drop=F], functional="quantile",quantiles=c(0.5))

# evalMAD[j] <- mean(sapply(1:nrow(Xtest), function(j) abs(Ytest[j] - quantpredictall$quantile[,,"q=0.5"][j]) ))

}

}

}

return(list(Vimp=Vimp, evalMMD=evalMMD, evalMSE=evalMSE ))

}