Nested covariance functions on graphs with Euclidean edges cross time

Abstract

Covariance functions over generalized networks have been explored to a very limited extent. We consider nested spatial or space-time covariance models, where space is a generalized network, and where time can be linear (the real line) or circular. We show sufficient conditions allowing preservation of positive semidefiniteness when at least one of the weights involved in the linear combination is negative. Several examples illustrate our findings. In particular, we show nested constructions for Euclidean trees with a finite number of leaves involving basic covariance functions with different scale parameters or different compact supports. We also provide criteria that allow one to build space-time models through half spectral modeling on graphs cross linear or circular time.