The partial dependence plot is a nice tool to analyse the impact of some explanatory variables when using nonlinear models, such as a random forest, or some gradient boosting.The idea (in dimension 2), given a model m(x_1,x_2) for \mathbb{E}[Y|X_1=x_1,X_2=x_2]. The partial dependence plot for variable x_1 is model m is function p_1 defined as x_1\mapsto\mathbb{E}_{\mathbb{P}_{X_2}}[m(x_1,X_2)]. This can be approximated, using some dataset using \widehat{p}_1(x_1)=\frac{1}{n}\sum_{i=1}^n m(x_1,x_{2,i})My concern here what the interpretation of that plot when there are some (strongly) correlated covariates. Let us generate some dataset to start with
n=1000
library(mnormt)
r=.7
set.seed(1234)
X = rmnorm(n,mean = c(0,0),varcov = matrix(c(1,r,r,1),2,2))
Y = 1+X[,1]-2*X[,2]+rnorm(n)/2
df = data.frame(Y=Y,X1=X[,1],X2=X[,2])
As we can see, the true model is here is y_i=\beta_0+\beta_1 x_{1,i}+\beta_2x_{2,i}+\varepsilon_i where \beta_1 =1 but the two variables are positively correlated, and the second one has a strong negative impact. Note that here
reg = lm(Y~.,data=df)
summary(reg)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.01414 0.01601 63.35 <2e-16 ***
X1 1.02268 0.02305 44.37 <2e-16 ***
X2 -2.03248 0.02342 -86.80 <2e-16 ***
If we estimate a wrongly specified model y_i=b_0+b_1 x_{1,i}+\eta_i, we would get
reg1 = lm(Y~X1,data=df)
summary(reg1)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.03522 0.04680 22.121 <2e-16 ***
X1 -0.44148 0.04591 -9.616 <2e-16 ***
Thus, on the proper model, \widehat{\beta}_1\sim+1.02 while \widehat{b}_1\sim-0.44 on the mispecified model.
Now, let us look at the parial dependence plot of the good model, using standard R dedicated packages,
library(pdp)
pdp::partial(reg, pred.var = "X1", plot = TRUE,
plot.engine = "ggplot2")
which is the linear line y=1+x, that corresponds to y=\beta_0+\beta_1x.
library(DALEX)
plot(DALEX::single_variable(DALEX::explain(reg,
data=df),variable = "X1",type = "pdp"))
which corresponds to the previous graph. Here, it is also possible to creaste our own function to compute that partial dependence plot,
pdp1 = function(x1){
nd = data.frame(X1=x1,X2=df$X2)
mean(predict(reg,newdata=nd))
}
that will be the straight line below (the dotted line is the theoretical one y=1+x,
vx=seq(-3.5,3.5,length=101)
vpdp1 = Vectorize(pdp1)(vx)
plot(vx,vpdp1,type="l")
abline(a=1,b=1,lty=2)
which is very different from the univariate regression on x_1
abline(reg1,col="red")
Actually, the later is very consistent with a local regression, only on x_1
library(locfit)
lines(locfit(Y~X1,data=df),col="blue")
Now, to get back to the definition of the partial dependence plot, x_1\mapsto\mathbb{E}_{\mathbb{P}_{X_2}}[m(x_1,X_2)], in the context of correlated variable, I was wondering if it would not make more sense to consider some local version actually, something like x_1\mapsto\mathbb{E}_{\mathbb{P}_{X_2|X_1}}[m(x_1,X_2)]. My intuition was that, somehow, it did not make any sense to consider any X_2 while X_1 was fixed (and equal to x_1). But it would make more sense actually to look at more valid X_2‘s given the value of X_1. And a natural estimate could be some k neareast-neighbors, i.e. \tilde{p}_1(x_1)=\frac{1}{k}\sum_{i\in\mathcal{V}_k(x)}^n m(x_1,x_{2,i}) where \mathcal{V}_k(x) is the set of indices of the k x_i‘s that are the closest to x, i.e.
lpdp1 = function(x1){
nd = data.frame(X1=x1,X2=df$X2)
idx = rank(abs(df$X1-x1))
mean(predict(reg,newdata=nd[idx<50,]))
}
vlpdp1 = Vectorize(lpdp1)(vx)
lines(vx,vlpdp1,col="darkgreen",lwd=2)
Surprisingly (?), this local partial dependence plot gives a curve that corresponds to the simple regression…
