1. Rodolphe Devillers (Almost) everything you always wanted to know (or maybe not…) about Geographically Weighted Regressions JCU Stats Group, March 2012

2. Outline Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (GWR)

3. Outline Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (GWR)

4. Background

5. Decrease in cod populations 1984

6. 1985 Decrease in cod populations

7. 1986 Decrease in cod populations

8. 1987 Decrease in cod populations

9. 1988 Decrease in cod populations

10. 1989 Decrease in cod populations

11. 1990 Decrease in cod populations

12. 1991 Decrease in cod populations

13. 1992 Decrease in cod populations

14. 1993 Decrease in cod populations

15. 1994 Decrease in cod populations

16. Scientific surveys Fisheries observers 4 species > 800 000 records GeoCod Project (2006-…) Biological Data 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 Temperature Salinity Remote Sensing > 300 GB

17. Fisheries data Collection Environmental data Other data(Bathy, etc.) IntegrationAnalysis Normalized database Visualization 1234 GeoCod project

18. Context A number of statistical methods can be used Testing spatial statistics SpeciesEnvironnement ?

19. Outline Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (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) Ecography 30) If spatial autocorrelation is not stationary: GWR

27. Outline Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (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-Ord Gi* (hotspot analysis) Look at GeoDa (free software from Luc Anselin - http://geodacenter.asu.edu/) http://geodacenter.asu.edu/ Local regressions: GWR

31. Outline Background Spatial autocorrelation Spatial non-stationarity Geographically Weighted Regressions (GWR)

32. Brunsdon, Fortheringham and Charlton GWR

33. 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. 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 GWR

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: dependent variable, x 1 to x p : independent variables, β 0: origin, β 1 to β p : coefficients, ε: error.

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

38. Method – Data interpolation

39. Method

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

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

42. Fixed Variable

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

44. Results spatial stationarity Windle et 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

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

51. Results Min and max GWR coefficients (R 2 ) Model power decreases with years

52. GWR coefficients– Capelan 1985 1986 1987 1988 1989 1990 1991 1992

53. GWR coefficients – Catch per Unit Effort 1985 1986 1987 1988 1989 1990 1991 1992

54. Conclusions The spatial structure of data matters Ecology (and mostly marine ecology) is still in the process of adopting such methods GWR is an interesting method but can be hard to interpret and should be used together with other methods

55. Questions? http://www.ucs.mun.ca/~rdeville/geocod Technical questions beyond my knowledge: Matt Windle (Matthew.Windle@mi.mun.ca) Technical questions beyond Matt’s knowledge: abc123@gmail.com (allow for several months for an answer)