The Fast and the Fourier-ous: Non-Parametric Autocovariance Modeling with Spline Kernels

Modeling autocovariance functions (ACFs) is fundamental in time-series, spatial, and spatio-temporal statistics. Standard parametric models can struggle with complex, non-separable dependence, and non-parametric approaches must ensure the ACF remains positive semi-definite. In this talk, I present a new family of non-parametric, closed-form ACFs that are provably dense in a broad class of continuous processes, provide optimally efficient functional representations, and extend naturally to multivariate and multidimensional settings. By avoiding rigid assumptions such as separability, the method captures realistic space-time interactions and can handle irregularly observed data. Illustrations are given from oceanographic spatio-temporal applications.