R. La X1 Xish Ra X1 R. FIGURE 8.3. Top Left: A partition of two-dimensional feature space that could not result from recursive binary splitting. Top Right: The output of recursive binary splitting on a two-dimensional example. Bottom Left: A tree corresponding to the partition in the top right panel. Bottom Right: A perspective plot of the prediction surface corresponding to that tree.1. (10 points) (i) We try to understand the relationship between regression trees and regression on step functions in Lecture 7. Take as an example- Write out the step functions of X1 and X2, say, 0,1 {X1} and 0: (X2), such that f{X) = .80 + Esq; (X1) +25%: (X2) 5' k may generate the tree structure in Figure 8.3. l(Hint: avoid the dun'nny variable trap). What is the relationship between 030, ,6}, 32,.) and my, the mean of Y on Rg, f3 = 1, 2, - - - , 5? Does this regression tree impose any restriction on (g, ,8}, 18:)? Can this regression on step functions model the regression tree structure in Figure 8.3? Explain why. (ii) Suppose a r_v_ X takes K values, 1,2, - -- , K, with probability p (k) for k E {1, 2, - -- , K}. Show that the entropy of X is E [logp(X)]_ 2. (6 points) (i) In bagging, suppose we use each bootstrap dataset as our training sample, and the original sample as our validation sample to estimate the prediction error. Do you expect this estimate will underestimate or overestimate the true prediction error? Explain why. (ii) Take another way around: if we use the original sample as our training sample, and each bootstrap dataset as our validation sample to estimate the prediction error. Do you expect this new estimate better or worse than the estimate in (i)? Explain why