# Variable Importance with Correlated Features

Variable importance graphs are great tool to see, in a model, which variables are interesting. Since we usually use it with random forests, it looks like it is works well with (very) large datasets. The problem with large datasets is that a lot of features are ‘correlated’, and in that case, interpretation of the values of variable importance plots can hardly be compared. Consider for instance a very simple linear model (the ‘true’ model, used to generate data)

$Y=\beta_0+\beta_1 X_{1}+\beta_3 X_{3}+\varepsilon$

Here, we use a random forest to model the relationship between the features, but actually, we consider another feature – not used to generate the data – $\color{blue}{X_2}$, that is correlated to $\color{black}{X_1}$. And we consider a random forest on those three features, $\widehat{Y}=\text{\sffamily rf}(X_{1},\color{blue}{X_2},\color{black}{X_{3})}$.

In order to get some more robust results, I geneate 100 datasets, of size 1,000.

library(mnormt)

impact_correl=function(r=.9){
nsim=10
IMP=matrix(NA,3,nsim)
n=1000
R=matrix(c(1,r,r,1),2,2)
for(s in 1:nsim){
X1=rmnorm(n,varcov=R)
X3=rnorm(n)
Y=1+2*X1[,1]-2*X3+rnorm(n)
db=data.frame(Y=Y,X1=X1[,1],X2=X1[,2],X3=X3)
library(randomForest)
RF=randomForest(Y~.,data=db)
IMP[,s]=importance(RF)}
apply(IMP,1,mean)}

C=c(seq(0,.6,by=.1),seq(.65,.9,by=.05),.99,.999)
VI=matrix(NA,3,length(C))
for(i in 1:length(C)){VI[,i]=impact_correl(C[i])}

plot(C,VI[1,],type="l",col="red")
lines(C,VI[2,],col="blue")
lines(C,VI[3,],col="purple")

The purple line on top is the variable importance value of $X_{3}$, which is rather stable (almost constant, as a first order approximation). The red line is the variable importance function of $\color{black}{X_1}$ while the blue line is the variable importance function of $\color{blue}{X_2}$.  For instance, the importance function with two very correlated variable is

It looks like $X_{3}$ is much more important than the other two, which is – somehow – not the case. It is just that the model cannot choose between $\color{black}{X_1}$ and $\color{blue}{X_2}$: sometimes, $\color{black}{X_1}$ is slected, and sometimes it is$\color{blue}{X_2}$. I think I find that graph confusing because I would probably expect the importance of $\color{black}{X_1}$ to be constant. It looks like we have a plot of the importance of each variable, given the existence of all the other variables.

Actually, what I have in mind is what we get when we consider the stepwise procedure, and when we remove each variable from the set of features,

library(mnormt)
impact_correl=function(r=.9){
nsim=100
IMP=matrix(NA,4,nsim)
n=1000
R=matrix(c(1,r,r,1),2,2)
for(s in 1:nsim){
X1=rmnorm(n,varcov=R)
X3=rnorm(n)
Y=1+2*X1[,1]-2*X3+rnorm(n)
db=data.frame(Y=Y,X1=X1[,1],X2=X1[,2],X3=X3)
IMP[1,s]=AIC(lm(Y~X1+X2+X3,data=db))
IMP[2,s]=AIC(lm(Y~X2+X3,data=db))
IMP[3,s]=AIC(lm(Y~X1+X3,data=db))
IMP[4,s]=AIC(lm(Y~X1+X2,data=db))
}
apply(IMP,1,mean)}

Here, if we uses the same code as previously,

C=c(seq(0,.6,by=.1),seq(.65,.9,by=.05),.99,.999)
VI=matrix(NA,3,length(C))
for(i in 1:length(C)){VI[,i]=impact_correl(C[i])}


we get the following graph

plot(C,VI[2,],type="l",col="red")
lines(C,VI2[3,],col="blue")
lines(C,VI2[4,],col="purple")

The purple line is obtained when we remove $X_{3}$ : it is the worst model. When we keep$\color{black}{X_1}$ and $X_{3}$, we get the blue line. And this line is constant: the quality of the does not depend on $\color{blue}{X_2}$ (this is what puzzled me in the previous graph, that having $\color{blue}{X_2}$ does have an impact on the importance of$\color{black}{X_1}$). The red line is what we get when we remove $\color{black}{X_1}$. With 0 correlation, it is the same as the purple line, we get a poor model. With a correlation close to 1, it is same as having $\color{black}{X_1}$,  and we get the same as the blue line.

Nevertheless, discussing the importance of features, when we have a lot of correlation features is not that intuitive…

## One thought on “Variable Importance with Correlated Features”

1. Blazko says:

Hi Arthur, nice discussion this one about feature importance. And what you did is quite similar the way I used to do in my studies and projects. I call this measure as “RLPP – Relative Loss of Prediction Power”, and I also use this metric as a variable selection method alternative. The results are very good in the end.
The RLPP calculation can be done using the AUC (my favorite), KS, Gini or any other goodness of fit metric. It consist in the following steps:
1. Train you model as usual, using your favorite technique
2. After selecting your best model, you have the AUC of reference
3. Then you calculate news AUCs (reestimating your model) for each new model discarding one feature at the time and keeping the others n-1 features in the model. For each feature discarded, you have his respective AUCi
4. Now all you have to do is to compare these AUCi vs AUC of reference, and check how much loss in AUCref each feature have done by using a simple formula as RLPP = (AUCref – AUCi) / AUCref.

As larger the RLPP, stronger is the impact of that feature in the prediction power of your model.

As curiosity, I have found that sometimes, some features can even make your model perform worst if you keep it as a predictor. Even that this “bad” feature had been selected as a “good” predictor in stepwise procedure!
I have already tested this approach in credit card fraud detection, churn and customer acquisition scenarios. And using RLPP let me simplify (sometimes, a lot) the final model and its deploy efforts, without lossing performance at all.
By the way, congrats for your posts. Well written and very useful. 😉

Regards, Blazko