Defining probability density for a distribution of random functions

TL;DR

This paper introduces a new way to define and estimate a surrogate probability density for functional data using principal component analysis, enabling meaningful density approximation in infinite-dimensional spaces.

Contribution

It develops a novel density concept based on PCA eigenfunctions for functional data, providing a practical and interpretable density surrogate with estimable properties.

Findings

The proposed density surrogate accurately reflects small-ball density features.

Estimates of principal component score densities reveal new shape differences.

The method is practically estimable and applicable to real data.

Abstract

The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis. This leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension. This density approximation is estimable readily from data. It accurately represents, in a monotone way, key features of small-ball approximations to density. Our results on estimators of the densities of principal component scores are also of independent interest; they reveal interesting shape differences that have not previously been considered. The statistical implications of these results and properties are identified and discussed, and practical ramifications are illustrated in numerical work.