Regularisation of CART trees by summation of $p$-values

TL;DR

This paper introduces a deterministic, p-value-based stopping rule for CART regression trees, improving efficiency and interpretability by avoiding cross-validation and enabling in-sample complexity control.

Contribution

It proposes a novel in-sample, p-value-based method for stopping CART tree growth, grounded in change point detection, applicable to high-dimensional data.

Findings

The method effectively detects signals with high probability given sufficient sample size.

It bounds the p-value of the entire tree, ensuring statistical validity.

Demonstrated on simulated and real data, showing practical utility.

Abstract

The standard procedure to decide on the complexity of a CART regression tree is to use cross-validation with the aim of obtaining a predictor that generalises well to unseen data. The randomness in the selection of folds implies that the selected CART regression tree is not a deterministic function of the data. Moreover, the cross-validation procedure may become time consuming and result in inefficient use of training data. We propose a simple deterministic in-sample method that can be used for stopping the growing of a CART regression tree based on node-wise statistical tests. This testing procedure is derived using a connection to change point detection, where the null hypothesis corresponds to no signal. The suggested $p$-value based procedure allows us to consider covariate vectors of arbitrary dimension and allows us to bound the $p$-value of an entire tree from above. Further, we show that the test detects a not too weak signal with a high probability, given a not too small sample size. We illustrate our methodology and the asymptotic results on both simulated and real world data. Additionally, we illustrate how the $p$-value based method can be used to construct a deterministic piece-wise constant auto-calibrated predictor based on a given black-box predictor.