Efficient Evaluation of Spectral Densities for Gaussian Process Modeling

Core Concepts

An adaptive Fourier integration framework is introduced that enables efficient and accurate evaluation of Gaussian process covariance functions and their derivatives directly from any continuous, integrable spectral density.

Abstract

The content presents an adaptive integration framework for efficiently and accurately evaluating Gaussian process covariance functions and their derivatives from any continuous, integrable spectral density. Key highlights: The framework employs high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic to achieve orders of magnitude speedup compared to naive uniform quadrature approaches. This allows evaluating covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. The authors derive novel algebraic truncation error bounds to monitor convergence of the adaptive integration. The framework facilitates gradient-based maximum likelihood estimation using previously numerically infeasible long-memory spectral models. The authors demonstrate the effectiveness of the approach on a singular Matérn spectral density model and apply it to fit a long-memory model for high-frequency wind velocity profiles.

Stats

The content does not provide any specific numerical data or statistics. It focuses on the methodological aspects of the adaptive Fourier integration framework.

Quotes

"For stationary processes, however, there is a much more flexible way to construct valid covariance models by instead specifying their Fourier transform." "Bochner's theorem states that K(r) := F{S}(r) is a positive definite function for any integrable positive function S, which we refer to as a spectral density."

Key Insights Distilled From

Fast Adaptive Fourier Integration for Spectral Densities of Gaussian Processes

by Paul G. Beck... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19053.pdf

Deeper Inquiries

How can the proposed framework be extended to handle non-stationary Gaussian processes?

The proposed framework for adaptive Fourier integration can be extended to handle non-stationary Gaussian processes by incorporating time-varying spectral densities. In the context of Gaussian processes, non-stationarity refers to the covariance function depending on both the spatial distance between points and the specific points themselves. This can be achieved by introducing additional parameters or functions in the spectral density that capture the non-stationarity. For example, one could introduce time-dependent parameters in the spectral density function to model temporal variations in the process. By allowing the spectral density to vary with time or other relevant variables, the framework can adapt to non-stationary processes and provide accurate evaluations of the covariance function at irregular locations.

What are the theoretical limits of the accuracy and efficiency of the adaptive Fourier integration approach, and how do they depend on the properties of the spectral density?

The accuracy and efficiency of the adaptive Fourier integration approach depend on several factors related to the properties of the spectral density. The theoretical limits of accuracy are primarily determined by the smoothness and decay properties of the spectral density. For slowly decaying spectral densities or those with singularities, the accuracy may be limited by numerical precision and the ability to resolve oscillations in the integrand. The efficiency of the approach is influenced by the choice of quadrature rule, the panel selection heuristic, and the computational cost of the NUFFT. The efficiency of the method is also affected by the Nyquist-informed panel selection heuristic, which determines the size of the panels in the spectral domain based on the frequency content of the spectral density. By adaptively choosing panel sizes, the method can focus computational resources where they are most needed, improving efficiency. Additionally, the efficiency of the approach is influenced by the computational cost of the NUFFT, which accelerates the computation of the Fourier integrals.

Can the ideas behind the Nyquist-informed panel selection heuristic be applied to other numerical integration problems involving oscillatory integrands?

Yes, the ideas behind the Nyquist-informed panel selection heuristic can be applied to other numerical integration problems involving oscillatory integrands. The key concept of adapting the panel sizes based on the frequency content of the integrand can be generalized to various numerical integration problems where oscillations play a significant role. By dynamically adjusting the panel sizes to capture the oscillations accurately, the method can improve the efficiency and accuracy of the integration process. The Nyquist-informed panel selection heuristic can be particularly useful in problems where the integrand exhibits rapid oscillations or singularities, as it allows for a more targeted allocation of computational resources. By focusing on regions of the integrand that require higher resolution, the method can optimize the integration process and provide more accurate results. This adaptive approach can be beneficial in a wide range of numerical integration problems, especially those involving oscillatory functions or singularities.