Space-Time Covariance Models on Networks with An Application on Streams

Abstract

The second-order, small-scale dependence structure of a stochastic process defined in the space-time domain is key to prediction (or kriging). While great efforts have been dedicated to developing models for cases in which the spatial domain is either a finite-dimensional Euclidean space or a unit sphere, counterpart developments on a generalized linear network are practically non-existent. To fill this gap, we develop a broad range of parametric, non-separable space-time covariance models on generalized linear networks and then an important subgroup -Euclidean trees by the space embedding technique -in concert with the generalized Gneiting class of models and 1-symmetric characteristic functions in the literature, and the scale mixture approach.We give examples from each class of models and investigate the geometric features of these covariance functions near the origin and at infinity. We also show the linkage between different classes of space-time covariance models on Euclidean trees. We illustrate the use of models constructed by different methodologies on a daily stream temperature data set and compare model predictive performance by cross validation.

Bayesian spatio-temporal models for stream networks

“…We illustrate the use of a simple kriging method to make predictions throughout the stream network in space and time. In terms of benchmark, we found these models to produce far better RMSPE to those reported in the literature ranging 1-2.7 ○ C (Gallice, 2016;Tang & Zimmerman, 2020).…”

“…A spline-based approach to spatiotemporal modelling was also developed by O'Donnell et al (2014) and then extended by Jackson et al (2018), whereby temporal dependence was incorporated in the error term using an AR(1) process. More recently Tang & Zimmerman (2020) developed several non-separable covariance structures derived from valid spatial covariance matrices and temporal covariance matrices.…”

Nonseparable Space-Time Stationary Covariance Functions on Networks cross Time

“…Simple generalizations to the setting where generalized networks are considered as topological structures that do not evolve over time are provided by Tang and Zimmerman (2020).…”

“…In particular, Tang and Zimmerman (2020) adapted a version of the Gneiting class (Gneiting, 2002b) to generalized networks cross linear time. Unfortunately, in their formulation the temporal distance is rescaled by spatial component, while Gneiting's original class proposed the opposite (rescaling spatial distance with temporal variables), which is often more realistic for geostatistical applications.…”

“…Vertices of a Euclidean tree that are connected to one edge only are called leaves. As noted in Tang and Zimmerman (2020), an arbitrary point x belongs to G when x ∈ V ∪ e∈E . As in their paper, we assume that the topological structure of G does not evolve over time.…”