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Rodolphe Devillers

( Almost ) everything you always wanted to know (or maybe not…) about Geographically Weighted Regressions. Rodolphe Devillers. JCU Stats Group , March 2012. Outline. Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (GWR). Outline.

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Rodolphe Devillers

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  1. (Almost) everythingyoualwayswanted to know (or maybe not…) about GeographicallyWeightedRegressions Rodolphe Devillers JCU Stats Group, March 2012

  2. Outline • Background • Spatial autocorrelation • Spatial non-stationarity • GeographicallyWeightedRegressions (GWR)

  3. Outline • Background • Spatial autocorrelation • Spatial non-stationarity • GeographicallyWeightedRegressions (GWR)

  4. Background

  5. Decrease in cod populations 1984

  6. Decrease in cod populations 1985

  7. Decrease in cod populations 1986

  8. Decrease in cod populations 1987

  9. Decrease in cod populations 1988

  10. Decrease in cod populations 1989

  11. Decrease in cod populations 1990

  12. Decrease in cod populations 1991

  13. Decrease in cod populations 1992

  14. Decrease in cod populations 1993

  15. Decrease in cod populations 1994

  16. Biological Data GeoCod Project (2006-…) Goal: Get a better understanding of the spatial and temporal dynamics of some fish/shellfish species in the NW Atlantic region, and their relationship with the physical environmental Environmental Data Scientific surveys Fisheries observers 4 species > 800 000 records Temperature Salinity Remote Sensing > 300 GB

  17. GeoCodproject 1 2 3 4 Collection Integration Analysis Visualization Environmental data Normalized database Fisheries data Other data(Bathy, etc.)

  18. Context • A number of statistical methods can be used • Testing spatial statistics Species ? Environnement

  19. Outline • Background • Spatial autocorrelation • Spatial non-stationarity • GeographicallyWeightedRegressions (GWR)

  20. Spatial autocorrelation • “…the property of random variables taking values, at pairs of locations a certain distance apart, that are more similar (positive autocorrelation) or less similar (negative autocorrelation) than expected for randomly associated pairs of observations.” (Legendre, 1993)

  21. Spatial autocorrelation - Basics Positive (Neighbours more similar) Neutral (Random) Negative • (Neighbours less similar) http://www.spatialanalysisonline.com/

  22. Spatial autocorrelation – is it common? • Elevation • Air/water temperature • Air humidity • Disease distribution • Species abundance • Housing value • Etc.

  23. Spatial autocorrelation – why bother? • Spatial autocorrelation in the data leads to spatial autocorrelation in the residuals

  24. Spatial autocorrelation – why bother? • Most statistics are based on the assumption that the values of observations in each sample are independent of one another • Consequence: it will violate the assumption about the independence of residuals and call into question the validity of hypothesis testing • Main effect: • Standard errors are underestimated, • t-scores are overestimated (= increases the chance of a Type I error = Incorrect rejection of a Null Hypothesis) • Sometime inverts the slope of relationships.

  25. Spatial autocorrelation – how to measure it? • Measures of spatial autocorrelation: • Moran’s I • Geary’s C • Others (e.g. Getis’ G)

  26. Spatial autocorrelation – How can I deal with it? • Many ways to handle this: • Subsampling, adjusting type I error, adjusting the effective sample size, etc. (Dale and Fortin (2002) Ecoscience 9(2)) • Autocovariate regressions, spatial eigenvector mapping (SEVM), generalised least squares (GLS), conditional autoregressive models (CAR), simultaneous autoregressive models (SAR), generalised linear mixed models (GLMM), generalised estimation equations (GEE), etc.(More details: Dormann et al. (2007) Ecography30) • If spatial autocorrelation is not stationary: GWR

  27. Outline • Background • Spatial autocorrelation • Spatial non-stationarity • GeographicallyWeightedRegressions (GWR)

  28. Stationarity • Classical regression models are valid under the assumptions that phenomena are stationary temporally and spatially (=statistical parameters such as the mean, the variance or the spatial autocorrelation do not vary depending on the geographic position) • E.g. Coral bleaching = 0.55 Temperature + 0.37 Nutrients + … - … • Studies (in various fields, including terrestrial ecology) have shown that they are rarely stationary

  29. Global vs Local Statistics Simpson Paradox

  30. Local spatial statistics • Local Indicators of Spatial Association (LISA) • Local Moran’s I (used to detect clustering) • Getis-OrdGi* (hotspot analysis) • Look at GeoDa(free software from Luc Anselin- http://geodacenter.asu.edu/) • Local regressions: GWR

  31. Outline • Background • Spatial autocorrelation • Spatial non-stationarity • GeographicallyWeightedRegressions (GWR)

  32. GWR • Brunsdon, Fortheringham and Charlton

  33. GWR • Increasingly used in various fields (mostly since 2006, and even more since integrated into ArcGIS) • Sally: yes, it is also available in R… (spgwr)

  34. GWR • Criticized by some authors (e.g. Wheeler 2005, Cho et al. 2009) when using collinear data, potentially leading to: • Occasional inflation of the variance • Rare inversion of the sign of the regression

  35. Windle, M., Rose, G., Devillers, R. and Fortin, M.-J. Exploring spatial non-stationarity of fisheries survey data using geographically weighted regression (GWR): an example from the Northwest Atlantic. ICES Journal of Marine Science, 67: 145-154.

  36. GWR • Geographically Weighted Regression(GRW) • (μ,ν): geographic coordinates of the samples • Multiple regression model (global) • y: dependentvariable, x1 to xp: independentvariables, β0: origin, β1 to βp: coefficients, ε: error.

  37. Method Government fisheries scientific survey data (Fisheries and Oceans Canada) Cod presence/absence (threshold at 5 kg) for the Fall 2001

  38. Method – Data interpolation

  39. Method

  40. Method Year 2001 Temperature Cod Combining data in a single point data file Exporting data points in a file (.dbf) Crab Shrimp

  41. GWR software (version 3.0) About 25 minutes per file of 5500 points 200km used for tests

  42. Fixed Variable

  43. Results Test of spatial stationarity of independent variables used in the regression Spatial non-stationarity Spatial stationarity

  44. Results spatial stationarity Windleet al. (accepted) - MEPS Stationarity of bottom temperature used to model shrimp biomass

  45. Results Comparison of regression models

  46. Results Test of the spatial auto-correlation of the residuals

  47. Results

  48. Results

  49. Results K-means clusteringof the t values of the GWR coefficients Positive relationship between crab and shrimp, weak relationship with the coast Negative relationship with crab and distance, positive with shrimp Stronger negative relationship with crab

  50. Results Weak AIC: Akaike Information Criterion GAM systematically has lower AIC values, suggesting a non-linear relationship between cod and the variables used in the analysis Strong

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