Intrinsic Random Functions and Universal Kriging on the Circle

TL;DR

This paper extends the theory of intrinsic random functions to the circle, introducing a Fourier series approach for stationarity and establishing the equivalence between kriging and splines on the circle.

Contribution

It develops IRF theory on the circle using Fourier series truncation and connects kriging with splines, expanding IRF applications beyond Euclidean spaces.

Findings

Fourier series truncation is key for IRF stationarity on the circle.

Kriging and splines are shown to be equivalent on the circle.

Theoretical framework based on reproducing kernel Hilbert space.

Abstract

Intrinsic random functions (IRF) provide a versatile approach when the assumption of second-order stationarity is not met. Here, we develop the IRF theory on the circle with its universal kriging application. Unlike IRF in Euclidean spaces, where differential operations are used to achieve stationarity, our result shows that low-frequency truncation of the Fourier series representation of the IRF is required for such processes on the circle. All of these features and developments are presented through the theory of reproducing kernel Hilbert space. In addition, the connection between kriging and splines is also established, demonstrating their equivalence on the circle.