Abstract
We develop spatial statistical models for stream networks that can estimate relationships between a response variable and other covariates, make predictions at unsampled locations, and predict an average or total for a stream or a stream segment. There have been very few attempts to develop valid spatial covariance models that incorporate flow, stream distance, or both. The application of typical spatial autocovariance functions based on Euclidean distance, such as the spherical covariance model, are not valid when using stream distance. In this paper we develop a large class of valid models that incorporate flow and stream distance by using spatial moving averages. These methods integrate a moving average function, or kernel, against a white noise process. By running the moving average function upstream from a location, we develop models that use flow, and by construction they are valid models based on stream distance. We show that with proper weighting, many of the usual spatial models based on Euclidean distance have a counterpart for stream networks. Using sulfate concentrations from an example data set, the Maryland Biological Stream Survey (MBSS), we show that models using flow may be more appropriate than models that only use stream distance. For the MBSS data set, we use restricted maximum likelihood to fit a valid covariance matrix that uses flow and stream distance, and then we use this covariance matrix to estimate fixed effects and make kriging and block kriging predictions.
Similar content being viewed by others
References
Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) Second International Symposium on Information Theory. Akademiai Kiado, Budapest, pp 267–281
RP Barry JM Ver Hoef (1996) ArticleTitleBlackbox kriging: spatial prediction without specifying the variogram J Agricult Biol Environ Stat 1 297–322 Occurrence Handle10.2307/1400521
J-P Chiles P Delfiner (1999) Geostatistics: modeling spatial uncertainty John Wiley and Sons New York 695
N Cressie (1993) Statistics for spatial data, revised edition John Wiley and Sons New York 900
N Cressie JJ Majure (1997) ArticleTitleSpatio-temporal statistical modeling of livestock waste in streams J Agricult Biol Environ Stat 2 24–47 Occurrence Handle10.2307/1400639
N Cressie M Pavlicova (2002) ArticleTitleCalibrated spatial moving average simulations Stat Model 2 267–279 Occurrence Handle10.1191/1471082x02st035oa
Curriero F (1996) The use of non-euclidean distance in geostatistics. Ph.D. Thesis, Kansas State University
CL Dent NB Grimm (1999) ArticleTitleSpatial heterogeneity of stream water nutrient concentrations over successional time Ecology 80 2283–2298 Occurrence Handle10.2307/176910
M Fuentes (2002) ArticleTitleSpectral methods for nonstationary spatial processes Biometrika 89 197–210 Occurrence Handle10.1093/biomet/89.1.197
B Gardner PJ Sullivan AJ Lembo SuffixJr (2003) ArticleTitlePredicting stream temperatures: geostatistical model comparison using alternative distance metrics Can J Fish Aquat Sci 60 344–351 Occurrence Handle10.1139/f03-025
D Higdon (1998) ArticleTitleA process-convolution approach to modelling temperatures in the North Atlantic Ocean Environ Ecol Stat 5 173–190 Occurrence Handle10.1023/A:1009666805688
Higdon D, Swall J, Kern J (1999) Non-stationary spatial modeling. In Bayesian statistics 6, Oxford Univ Press, Oxford 761–768
RC Littell RC Milliken WW Stroup R Wolfinger (1996) SAS system for mixed models SAS publishing Cary, NC 656
LS Little D Edwards DE Porter (1997) ArticleTitleKriging in estuaries: as the crow flies, or as the fish swims J Exp Mar Biol Ecol 213 1–11 Occurrence Handle10.1016/S0022-0981(97)00006-3
SL Rathbun (1998) ArticleTitleSpatial modeling in irregularly shaped regions: kriging estuaries Environmetrics 9 109–129 Occurrence Handle1:CAS:528:DyaK1c**slaku78%3D Occurrence Handle10.1002/(SICI)1099-095X(199803/04)9:2<109::AID-ENV279>3.0.CO;2-L
G Schwarz (1978) ArticleTitleEstimating the dimension of a model Ann Stat 6 461–464
RA Smith GE Schwarz RB Alexander (1997) ArticleTitleRegional interpretation of water-quality monitoring data Water Resour Res 33 2781–2798 Occurrence Handle1:CAS:528:DyaK2sXotVensbY%3D Occurrence Handle10.1029/97WR02171
Torgersen CE, Gresswell RE, Bateman DS (2004) Pattern detection in stream networks: quantifying spatial variability in fish distribution. In: Nishida, T. (ed), Proceedings of the Second Annual International Symposium on GIS/Spatial Analyses in Fishery and Aquatic Sciences. Fishery GIS Research Group, Saitama, Japan
JM Hoef ParticleVer RP Barry (1998) ArticleTitleConstructing and fitting models for cokriging and multivariable spatial prediction J Stat Planning Inference 69 273–294
JM Hoef ParticleVer N Cressie RP Barry (2004) ArticleTitleFlexible spatial models for kriging and cokriging using moving averages and the fast Fourier transform (FFT) J Comput Graphical Stat 13 265–282 Occurrence Handle10.1198/1061860043498
LL Yuan (2004) ArticleTitleUsing spatial interpolation to estimate stressor levels in unsampled streams Environ Monit Assess 94 23–38 Occurrence Handle15141444 Occurrence Handle10.1023/B:EMAS.0000016877.52279.05
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: July 2005 / Revised: March 2006
Rights and permissions
About this article
Cite this article
Hoef, J.M.V., Peterson, E. & Theobald, D. Spatial statistical models that use flow and stream distance. Environ Ecol Stat 13, 449–464 (2006). https://doi.org/10.1007/s10651-006-0022-8
Issue Date:
DOI: https://doi.org/10.1007/s10651-006-0022-8