Bayesian Smoothing with Gaussian Processes Using Fourier Basis Functions in the spectralGP Package

Area of Science:

• Statistics
• Computational Statistics
• Spatial Statistics

Background:

• Stationary Gaussian processes are fundamental in statistical modeling for spatial data.
• Efficient computation is crucial for applying these models to large datasets.
• Existing methods may face challenges with data not on regular grids.

Purpose of the Study:

• Introduce the spectralGP R package for spectral representation of Gaussian processes.
• Provide a Bayesian approach using shrinkage for efficient computation.
• Address modeling for data on irregular grids and improve Markov chain Monte Carlo (MCMC) sampling.

Main Methods:

• Utilize the Fourier basis for spectral representation of stationary Gaussian processes.
• Implement a Bayesian approach with a parameterized prior structure for coefficient shrinkage.
• Develop and demonstrate MCMC sampling techniques within the spectralGP package.

Main Results:

• The spectralGP package offers efficient computation for spatial surfaces and regression.
• The Bayesian shrinkage approach effectively handles numerous basis coefficients.
• Methods are presented for irregular data and improved MCMC mixing.

Conclusions:

• The spectralGP package provides a powerful tool for statistical modeling with Gaussian processes.
• The Bayesian framework and MCMC strategies enhance applicability and performance.
• The package facilitates efficient analysis of spatial data in various statistical models.