Mixture Models and Densities for Functional Data
Abstract: We propose a novel perspective to represent infinite-dimensional functional data as mixtures, where each realization of the underlying stochastic process is represented by a finite number of basis functions. In the proposed mixture representation, the number of basis
functions that constitutes the mixture may be arbitrarily large, where the number of included mixture components is specifically adapted for each random trajectory. We show that within this framework, a probability density can be well defined under mild regularity conditions, without the need for finite truncation or approximation, while genuine probability densities do not exist in the usually considered infinite-dimensional function space due to the low small ball probabilities. Unlike traditional functional principal component analysis, in the proposed approach individual trajectories possess a trajectory-specific dimension that may be treated as a latent random variable. We establish a notion of consistency for estimating the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation-maximization algorithm. Simulations confirm that in comparison to traditional functional principal component analysis the proposed method achieves similar or better data recovery while using fewer components on average. The practical merits are demonstrated inan analysis of egg-laying trajectories for medflies.