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Applications of spatial statistical network models to stream data

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Abstract Streams and rivers host a significant portion of Earth's biodiversity and provide important ecosystem services for human populations. Accurate information regarding the status and trends of stream resources is vital for their effective conservation and management. Most statistical techniques applied to data measured on stream networks were developed for terrestrial applications and are not optimized for streams. A new class of spatial statistical model, based on valid covariance structures for stream networks, can be used with many common types of stream data (e.g., water quality attributes, habitat conditions, biological surveys) through application of appropriate distributions (e.g., Gaussian, binomial, Poisson). The spatial statistical network models account for spatial autocorrelation (i.e., nonindependence) among measurements, which allows their application to databases with clustered measurement locations. Large amounts of stream data exist in many areas where spatial statistical analyses could be used to develop novel insights, improve predictions at unsampled sites, and aid in the design of efficient monitoring strategies at relatively low cost. We review the topic of spatial autocorrelation and its effects on statistical inference, demonstrate the use of spatial statistics with stream datasets relevant to common research and management questions, and discuss additional applications and development potential for spatial statistics on stream networks. Free software for implementing the spatial statistical network models has been developed that enables custom applications with many stream databases. This article is categorized under: Water and Life > Nature of Freshwater Ecosystems
Example databases with spatially clustered measurements that could be modeled using spatial statistical techniques. Stream temperature measurements from the Boise River in central Idaho (a; Source: Ref ), water quality measurements across Maryland (b; Source: Ref ), fish sampling locations across the western U.S. (c; modified Source: Ref ), and (d) nitrate measurements from the Meuse River in France (Source: Ref ).
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Stream sites sampled for occurrence of bull trout in Idaho and Montana (a) that were used to develop a nonspatial logistic regression model (b) and a spatial logistic regression model (c) for predicting the distribution of this species. Box in panel a highlights the area shown in subsequent panels.
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Power curves for elevation regression coefficient in a spatial stream‐network model fit to temperature measurements using different autocovariance functions.
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Stream temperature estimates based on block kriging and simple random sampling that were derived from measurements recorded across a river network in central Idaho (a). Red shading shows stream segments where mean temperatures were estimated (b). Error bars associated with estimates are 95% confidence intervals.
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Kriged map of mean stream temperatures and prediction standard errors in the upper Bitterroot River of western Montana (a; Source: Ref ). Box in panel a highlights the area shown in panels b and c. Note that standard errors vary in size relative to sensor measurement locations in the spatial model (b) but not in the nonspatial model (c).
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Variance decomposition of stream temperature data from the Snoqualmie River in western Washington (a). Bargraphs show the proportion of total variation described by predictor variables, spatial structure modeled by the autocovariance function, and residual error for nonspatial regression models (b) and spatial regression models (c).
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Torgegrams for electrical conductivity (b) and pH (c) developed from water quality measurements across a stream network in southeast Queensland, Australia (a) Symbol sizes are proportional to the number of data pairs averaged for each value. (Reprinted with permission from Ref . Copyright 2010 American Statistical Association)
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Flow‐connected (a and b) and flow‐unconnected (c and d) spatial relationships on a stream network (Source: Ref ). Moving‐average functions for tail‐up (a and c) and tail‐down (b and d) relationships are shown in gray. Note that tail‐up models restrict autocorrelation to flow‐connected locations, whereas tail‐down models permit correlation between flow‐connected and flow‐unconnected locations.
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Counts of cutthroat trout in habitat units along two forks of Hinkle Creek in western Oregon (a; Source: Ref ). Semivariograms calculated from counts in the South Fork (black dots) show evidence of spatial autocorrelation, but there is no evidence of autocorrelation in the North Fork (b; red dots).
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