Interpretable Machine Learning

Exercise on Functional Decomposition

In the lecture, we discussed local and global explanations as well as functional decompositions. In this exercise we want to take a deeper look via simulated data.

• Load the relevant packages.

library(mlr3)
library(mlr3learners)
library(mlr3tuning)
library(DALEX)
library(DALEXtra)
library(ALEPlot)

• We can use our own implementation of PDP

## 1-dim pdp plot at a fixed point
pdp_xk <- function(data,learner,k,x_k){
  newX <- data
  newX[,k] <- x_k
  mean(predict(learner, as.data.frame(newX)))
}

## 2-dim pdp plot at a fixed point
pdp_xjk <- function(data,learner,j,k,x_j,x_k){
  newX <- data
  newX[,k] <- x_k
  newX[,j] <- x_j
  mean(predict(learner, as.data.frame(newX)))
}

#### calcualtes 1-dim pdp plot over a equi-distant grid of length I
pdp <- function(dat,learner,k, I) {
  my_grid <-  seq(min(dat[,k]) , max(dat[,k]), length.out=I)
  return(list(x_k=my_grid, pdp_k=sapply(1:I, function(j) pdp_xk(dat,learner,k,my_grid[j]))))
}

• We generate data according to the following (noiseless) mechanism \[ Y=X_1 + X_2^2+ X_1X_2 \]
• \(X_1\) , \(X_2\) are uniformly distributed on \([0,1]\) along a segment of the line \(x_2 = x_1\) with independent \(N(0, 0.05^2)\) noise added to both predictors. We generate 1000 training observations.

m <- function(x){ x[,1]+x[,2]^2+x[,1]*x[,2] }

generate_data <- function(n){
  X1 <- X2 <- runif(n)
  X1 <- X1 + rnorm(n,sd=0.05)
  X2 <- X2 + rnorm(n,sd=0.05)
  X <- cbind(X1,X2)
  Y <-  m(X) ### no noise
  dat <- data.frame(X=X, Y=Y)
  return(dat)
}

n <- 1000
dat <- generate_data(n)

• Complete the following tasks
  1. Train an auto tuned ranger model on the data
  2. Produce and visualize a SHAP explanation for the observation dat[100,-3]
  3. Produce and plot 1-dim PDPs for feature 1 and feature 2.
  4. Evaluate 1-dim and 2-dim PDPs at the fixed point dat[100,-3].
  5. Use the result in 4. together with Lemma 7.4.4 of the lecture notes to calculate the functional components \(\hat m_0\), \(\hat m_1\), \(\hat m_2\), \(\hat m_{12}\) identified via the marginal identification.
  6. Use the functional components from 5. together with Corollary 7.4.5 from the lecture notes to calculate the interventional SHAP values \(\hat \phi_1, \hat \phi_2\). Compare the values to the result from 1.