Compatibility of Space-Time Kernels with Full, Dynamical, or Compact Support

TL;DR

This paper compares different space-time covariance kernels with various support properties, exploring their compatibility under fixed domain asymptotics through spectral density analysis and convolution techniques.

Contribution

It introduces a method to construct space-time compact support kernels from dynamical models via temporal tapering, enabling compatibility analysis.

Findings

Models with full, dynamical, and compact support can be compatible under certain conditions.

Spectral density tails determine the compatibility of covariance models.

Implications for maximum likelihood estimation and kriging under misspecification.

Abstract

We deal with the comparison of space-time covariance kernels having, either, full, spatially dynamical, or space-time compact support. Such a comparison is based on compatibility of these covariance models under fixed domain asymptotics, having a theoretical background that is substantially coming from equivalence or orthogonality of Gaussian measures. In turn, such a theory is intimately related to the tails of the spectral densities associated with the three models. Models with space-time compact support are still elusive. We taper the temporal part of a model with dynamical support, obtaining a space-time compact support. The spectrum related to such a construction is obtained through temporal convolution of the spatially dynamical spectrum with the spectrum associated with the temporal taper. The solution of such a challenge opens the door to the compatibility-based comparison. Our findings show that indeed these three models can be compatible under some suitable parametric restrictions. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics.