Kernel Mean Embedding of Probability Measures and its Applications to Functional Data Analysis

TL;DR

This paper introduces kernel mean embedding for probability measures in infinite-dimensional spaces, enabling new statistical tests for functional data analysis with improved performance in regression, ANOVA, and covariance equality.

Contribution

It presents a novel framework using kernel mean embedding and MMD for statistical inference in functional data analysis, including new tests for key problems.

Findings

New tests outperform competitors in functional data analysis tasks.

Kernel mean embedding effectively captures probability measure concentration.

Framework applicable to various functional data analysis problems.

Abstract

This study intends to introduce kernel mean embedding of probability measures over infinite-dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of probability measures in small open neighborhoods, which identifies a pseudo-likelihood and fosters a rich framework for statistical inference. Utilizing Maximum Mean Discrepancy, we devise new tests in functional response models. The performance of new derived tests is evaluated against competitors in three major problems in functional data analysis including function-on-scalar regression, functional one-way ANOVA, and equality of covariance operators.