Modeling spatial correlation that grows on trees, with a stream network application

Abstract Spatial data on a network, like spatial data on a Euclidean domain, may exhibit nonstationarity. This article develops two classes of nonstationary models for continuously indexed data on directed tree networks, such as stream networks, that are adaptations of models used previously for nonstationary temporal or spatial data on Euclidean domains. These classes, called elastic models and spatially varying moving average models, allow the spatial dependence between observations at sites any fixed distance apart to grow monotonically as one moves either up or down the network. The process variance, or components thereof, may also be allowed to grow monotonically. An example of trout density data from a stream network in Wyoming, USA indicates that the proposed nonstationary models fit those data much better than their existing stationary or quasi-stationary counterparts.