Kriging models for linear networks and non‐Euclidean distances: Cautions and solutions

Abstract

AbstractThere are now many examples where ecological researchers used non‐Euclidean distance metrics in geostatistical models that were designed for Euclidean distance, such as those used for kriging. This can lead to problems where predictions have negative variance estimates. Technically, this occurs because the spatial covariance matrix, which depends on the geostatistical models, is not guaranteed to be positive definite when non‐Euclidean distance metrics are used. These are not permissible models, and should be avoided.I give a quick review of kriging and illustrate the problem with several simulated examples, including locations on a circle, locations on a linear dichotomous network (such as might be used for streams), and locations on a linear trail or road network. I re‐examine the linear network distance models from Ladle, Avgar, Wheatley, and Boyce (,Methods in Ecology and Evolution, 8, 329) and show that they are not guaranteed to have a positive definite covariance matrix.I introduce the reduced‐rank method, also called a predictive‐process model, for creating valid spatial covariance matrices with non‐Euclidean distance metrics. It has an additional advantage of fast computation for large datasets.I re‐analysed the data of Ladle et al. (,Methods in Ecology and Evolution, 8, 329), showing that fitted models that used linear network distance in geostatistical models, both with and without a nugget effect, had negative variances, poor predictive performance compared with reduced‐rank methods, and had improper coverage for the prediction intervals. The reduced‐rank approach using linear network distances provided a class of permissible models that had better predictive performance and proper coverage for the prediction intervals, and could be combined with Euclidean distance models to provide the best overall predictive performance.