Functional Gaussian Process Model for Bayesian Nonparametric Analysis

TL;DR

This paper introduces a novel spectral projection-based Gaussian process model that enables fast, accurate nonparametric spatial analysis, addressing computational challenges and supporting non-stationary data clustering.

Contribution

It proposes a spectral projection construction for Gaussian processes that improves computational efficiency and supports non-stationary, clustered spatial data analysis.

Findings

Provides a high-speed algorithm for Gaussian process computation.

Achieves accurate covariance parameter estimation.

Demonstrates effectiveness on large-scale spatial datasets.

Abstract

Gaussian process is a theoretically appealing model for nonparametric analysis, but its computational cumbersomeness hinders its use in large scale and the existing reduced-rank solutions are usually heuristic. In this work, we propose a novel construction of Gaussian process as a projection from fixed discrete frequencies to any continuous location. This leads to a valid stochastic process that has a theoretic support with the reduced rank in the spectral density, as well as a high-speed computing algorithm. Our method provides accurate estimates for the covariance parameters and concise form of predictive distribution for spatial prediction. For non-stationary data, we adopt the mixture framework with a customized spectral dependency structure. This enables clustering based on local stationarity, while maintains the joint Gaussianness. Our work is directly applicable in solving some of the challenges in the spatial data, such as large scale computation, anisotropic covariance, spatio-temporal modeling, etc. We illustrate the uses of the model via simulations and an application on a massive dataset.