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Title: Mixture inner product spaces and their application to functional data analysis Authors:  Zhenhua Lin - University of California, Davis (United States) [presenting]
Hans-Georg Mueller - University of California Davis (United States)
Fang Yao - Peking University, University of Toronto (Canada)
Abstract: The aim is to introduce the concept of mixture inner product spaces associated with a given separable Hilbert space, which feature an infinite-dimensional mixture of finite-dimensional vector spaces and are dense in the underlying Hilbert space. For functional data, mixture inner product spaces provide a new perspective, where each realization of the underlying stochastic process falls into one of the component spaces and is represented by a finite number of basis functions, the number of which corresponds to the dimension of the component space. In the mixture representation of functional data, the number of included mixture components used to represent a given random element is specifically adapted to each random trajectory and may be arbitrarily large. Key benefits of this novel approach are, first, that it provides a new perspective on the construction of a probability density in function space under mild regularity conditions, and second, that individual trajectories possess a trajectory-specific dimension that corresponds to a latent random variable, making it possible to use a larger number of components for less smooth and a smaller number for smoother trajectories. This enables flexible and parsimonious modeling of heterogeneous trajectory shapes. We establish estimation consistency of the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation-maximization algorithm.