The specification of a covariance function is of paramount importance when employing Gaussian process models, but the requirement of positive definiteness severely limits those used in practice. Designing flexible stationary covariance functions is, however, straightforward in the spectral domain, where one needs only to supply a positive and symmetric spectral density. In this work, we introduce an adaptive integration framework for efficiently and accurately evaluating covariance functions and their derivatives at irregular locations directly from \textit{any} continuous, integrable spectral density. In order to make this approach computationally tractable, we employ high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic, and derive novel algebraic truncation error bounds which are used to monitor convergence. As a result, we demonstrate several orders of magnitude speedup compared to naive uniform quadrature approaches, allowing us to evaluate covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. We then apply our methodology to perform gradient-based maximum likelihood estimation using a previously numerically infeasible long-memory spectral model for wind velocities below the atmospheric boundary layer.

翻译：协方差函数的设定在使用高斯过程模型时至关重要，但正定性的要求严重限制了实际应用中的协方差函数选择。然而，在谱域中设计灵活的平稳协方差函数却十分直接：只需提供一个正且对称的谱密度即可。本文提出了一种自适应积分框架，能够高效且精确地从任意连续、可积的谱密度中直接计算不规则位置上的协方差函数及其导数。为使该方法在计算上可行，我们采用了高阶面板求积法、非均匀快速傅里叶变换以及基于奈奎斯特准则的面板选择启发式策略，并推导了用于监控收敛的新型代数截断误差界。结果表明，与朴素均匀求积方法相比，我们实现了数个数量级的加速，从而能够在笔记本电脑上以数秒时间，针对数百万个位置，从缓慢衰减的奇异谱密度中计算协方差函数，并达到用户指定的容差。我们随后将该方法应用于基于梯度的最大似然估计中，该估计使用了先前因数值不可行而无法应用的风速大气边界层以下长记忆谱模型。