Modeling Function-Valued Processes with Nonseparable and/or Nonstationary Covariance Structure

TL;DR

This paper introduces a Bayesian framework for modeling complex multidimensional function-valued processes with nonseparable and nonstationary covariance structures, using a convolution-based approach and local empirical Bayesian estimation.

Contribution

It proposes a novel spherical parametrization for the varying anisotropy matrix, enabling flexible, interpretable, and nonparametric modeling of nonstationary covariance structures.

Findings

Effective modeling of nonseparable, nonstationary covariance structures.

Successful application to wind intensity data.

Simulation studies validate the approach.

Abstract

We discuss a general Bayesian framework on modeling multidimensional function-valued processes by using a Gaussian process or a heavy-tailed process as a prior, enabling us to handle nonseparable and/or nonstationary covariance structure. The nonstationarity is introduced by a convolution-based approach through a varying anisotropy matrix, whose parameters vary along the input space and are estimated via a local empirical Bayesian method. For the varying matrix, we propose to use a spherical parametrization, leading to unconstrained and interpretable parameters. The unconstrained nature allows the parameters to be modeled as a nonparametric function of time, spatial location or other covariates. The interpretation of the parameters is based on closed-form expressions, providing valuable insights into nonseparable covariance structures. Furthermore, to extract important information in data with complex covariance structure, the Bayesian framework can decompose the function-valued processes using the eigenvalues and eigensurfaces calculated from the estimated covariance structure. The results are demonstrated by simulation studies and by an application to wind intensity data. Supplementary materials for this article are available online.