A Kernel Two-Sample Test for Functional Data

TL;DR

This paper introduces a nonparametric two-sample test for functional data using Maximum Mean Discrepancy, with theoretical analysis and validation on synthetic and real datasets.

Contribution

It develops a kernel-based two-sample test specifically designed for functional data, including theoretical properties and practical effectiveness assessments.

Findings

The proposed test effectively distinguishes different functional data distributions.

Theoretical analysis confirms the test's consistency and efficiency.

Empirical results demonstrate superior performance over existing methods.

Abstract

We propose a nonparametric two-sample test procedure based on Maximum Mean Discrepancy (MMD) for testing the hypothesis that two samples of functions have the same underlying distribution, using kernels defined on function spaces. This construction is motivated by a scaling analysis of the efficiency of MMD-based tests for datasets of increasing dimension. Theoretical properties of kernels on function spaces and their associated MMD are established and employed to ascertain the efficacy of the newly proposed test, as well as to assess the effects of using functional reconstructions based on discretised function samples. The theoretical results are demonstrated over a range of synthetic and real world datasets.