Abstract
We introduce a procedure to generate an estimator of the regression function based on a data-dependent quasi-covering of the feature space. A quasi-partition is generated from the quasi-covering and the estimator predicts the conditional empirical expectation over the cells of the quasi-partition. We provide sufficient conditions to ensure the consistency of the estimator. Each element of the quasi-covering is labeled as significant or insignificant. We avoid the condition of cell shrinkage commonly found in the literature for data-dependent partitioning estimators. This reduces the number of elements in the quasi-covering. An important feature of our estimator is that it is interpretable. The proof of the consistency is based on a control of the convergence rate of the empirical estimation of conditional expectations, which is interesting in itself.
Highlights
IntroductionWe consider the following regression setting: let (X, Y ) be a pair of random variables in Rd × R of unknown distribution Q such that
We consider the following regression setting: let (X, Y ) be a pair of random variables in Rd × R of unknown distribution Q such thatY = g∗(X) + Z, where E[Z] = 0, V(Z) = σ2 < ∞ and g∗ is a measurable function from Rd to R
One reason could be that Random Forest (RF) is a random generator of rules that may not capture the importance of X6 at every run
Summary
We consider the following regression setting: let (X, Y ) be a pair of random variables in Rd × R of unknown distribution Q such that. Y = g∗(X) + Z, where E[Z] = 0, V(Z) = σ2 < ∞ and g∗ is a measurable function from Rd to R. Y is bounded: Q(S) = 1 where S = Rd × [−L, L] for some unknown L > 0. We have g∗(X) = E [Y |X] = arg min L (g) a.s, g where the arg min ranges over all the measurable functions g with E[g(X)2] < ∞. The observations (Xi, Yi) are assumed independent and identically distributed (i.i.d.) from the distribution Q
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