Measure Inducing Classification and Regression Trees for Functional Data

TL;DR

This paper introduces a novel tree-based method for functional data analysis that learns weighted functional spaces to improve classification and regression while maintaining interpretability.

Contribution

It develops a new algorithm that combines representation learning with multiple splitting rules for functional data, enhancing accuracy and interpretability.

Findings

Effective in classification and regression tasks

Reduces generalization error compared to traditional methods

Demonstrated on real-world datasets

Abstract

We propose a tree-based algorithm for classification and regression problems in the context of functional data analysis, which allows to leverage representation learning and multiple splitting rules at the node level, reducing generalization error while retaining the interpretability of a tree. This is achieved by learning a weighted functional $L^{2}$ space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the input functions, in order to determine the binary split for each internal node of the tree. The approach is designed to manage multiple functional inputs and/or outputs, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with scalar and categorical data, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar-valued functional inputs defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real world applications.