Flexible regression models over river networks

Abstract

Many statistical models are available for spatial data but the vast majority of these assume that spatial separation can be measured by Euclidean distance. Data which are collected over river networks constitute a notable and commonly occurring exception, where distance must be measured along complex paths and, in addition, account must be taken of the relative flows of water into and out of confluences. Suitable models for this type of data have been constructed based on covariance functions. The aim of the paper is to place the focus on underlying spatial trends by adopting a regression formulation and using methods which allow smooth but flexible patterns. Specifically, kernel methods and penalized splines are investigated, with the latter proving more suitable from both computational and modelling perspectives. In addition to their use in a purely spatial setting, penalized splines also offer a convenient route to the construction of spatiotemporal models, where data are available over time as well as over space. Models which include main effects and spatiotemporal interactions, as well as seasonal terms and interactions, are constructed for data on nitrate pollution in the River Tweed. The results give valuable insight into the changes in water quality in both space and time.

Keywords: Flexible regression; Kernels; Network; Penalized splines; Smoothing; Spatial separation; Spatiotemporal models; Water quality.

Figure 1

River Tweed catchment, with sampling stations colour coded by nitrate level recorded in February 2004 (○, stations where no measurements were available at this time point): the scale of the map is approximately 110 km in each direction; the plot was produced by using the R package OpenStreetMap (Fellows, 2012) and the underlying map image was produced by Esri (www.esri.com) and its data providers

Figure 2

(a) Decomposition of the river network into a large number of small stream units by using different colours and (b) a schematic representation of a confluence, with model parameters (, ), flows (, ) and the corresponding outgoing versions (, )

Figure 3

Smoothing based on (a) Euclidean distance and (b) flow-weighted distance by using data from February 2004 and with 12 degrees of freedom

Figure 4

Network smoothing with (a) 6, (b) 19 (optimal) and (c) 48 degrees of freedom, using the data from February 2004

Figure 5

(a)–(d) Main effects of space, year and day of the year, plus the interaction of these last two terms, and (e), (f) fitted values at two specific spatial locations, namely Gala Water Foot and Norham Gauging Station respectively, including a comparison of the simple main effects model (– – –) and the interaction model (_____) in each case: ……, 2 standard errors under the fitted covariance model; ¢, 2 standard errors under the independence model

Figure 6

Estimated spatial effects at (a) January, (b) May and (c) October 2005, indicated by colour and scaling of ‘nodes’ located at the stream units