AI 中文总结
本研究提出结合时间离散傅里叶变换与希尔伯特空间高斯过程近似的谱似然方法，用于高效重建时空高斯过程，在保持重建精度的同时将拟合时间减少69%。
AI 中文摘要
在多个站点共享规律时间记录的情况下，重建未观测位置的平稳时空高斯过程代价高昂。我们开发了一种谱似然方法，将时间离散傅里叶变换（DFT）与频率特定空间协方差的希尔伯特空间高斯过程（HSGP）表示相结合。我们推导了有限记录DFT系数的精确协方差，并使用Whittle似然将不同频率近似为独立的空间问题。在每个时间频率上，HSGP通过在保留的拉普拉斯特征频率处评估采样谱乘子来近似空间协方差。对于由联合谱密度指定且半谱缺乏便捷闭式形式的模型，这种构造避免了重复的傅里叶反演。固定的空间基还允许缓存特征投影，在似然拟合和保留站点重建中重用，近似精度取决于域扩展和基大小。在对此类模型的模拟研究中，HSGP实现了与高精度求积参考相当的重建精度，同时将平均拟合时间减少了69%。对风场重建任务的额外应用考察了基秩、时间记录长度和可分性的影响，并证明了在使用足够丰富的基时，对保留测试站点的准确推断。综合来看，结果表明当谱乘子可以直接评估、数值空间反演代价高昂且适当基的秩低于拟合站点数量时，可以实现计算节省。
英文摘要
Reconstructing stationary space-time Gaussian processes at unobserved locations is costly when many sites share regular temporal records. We develop a spectral likelihood combining a temporal discrete Fourier transform (DFT) with a Hilbert-space Gaussian process (HSGP) representation of frequency-specific spatial covariance. We derive the exact covariance of the finite-record DFT coefficients and use a Whittle likelihood that approximates distinct frequencies as independent spatial problems. At each temporal frequency, HSGP approximates spatial covariance by evaluating the sampled spectral multiplier at retained Laplacian eigenfrequencies. For models specified by a joint spectral density whose half-spectrum lacks a convenient closed form, this construction avoids repeated Fourier inversion. The fixed spatial basis also permits cached feature projections to be reused in likelihood fitting and held-site reconstruction, with approximation accuracy depending on domain extension and basis size. In a simulation study of such a model, HSGP achieved reconstruction accuracy comparable to a high-accuracy quadrature reference while reducing mean fitting time by 69%. Additional applications to wind-field reconstruction tasks examine the effects of basis rank, temporal record length, and separability, and demonstrate accurate inference on held-out test sites when sufficiently rich bases are used. Taken together, the results indicate that computational savings are attainable when the spectral multiplier can be evaluated directly, numerical spatial inversion is costly, and an adequate basis has rank lower than the number of fitting sites.
Comments29 pages, 7 figures
