Defining probability density for a distribution of random functions

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 s..