Rodolphe Devillers

(Almost) everything you

always wanted to know (or maybe

not…) about Geographically

Weighted Regressions

JCU Stats Group, March 2012

Outline

• Background• Spatial autocorrelation• Spatial non-stationarity• Geographically Weighted Regressions (GWR)

Outline

• Background• Spatial autocorrelation• Spatial non-stationarity• Geographically Weighted Regressions (GWR)

Background

Decrease in cod populations

1984

1985

Decrease in cod populations

1986

Decrease in cod populations

1987

Decrease in cod populations

1988

Decrease in cod populations

1989

Decrease in cod populations

1990

Decrease in cod populations

1991

Decrease in cod populations

1992

Decrease in cod populations

1993

Decrease in cod populations

1994

Decrease in cod populations

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 environmentalEnvironmental Data

Temperature

Salinity

Remote Sensing

> 300 GB

Fisheries data

Collection

Environmental data

Other data(Bathy,

etc.)

Integration Analysis

Normalized database

Visualization

1 2 3 4

GeoCod project

Context

• A number of statistical methods can be used• Testing spatial statistics

SpeciesEnvironne

ment

?

Outline

• Background• Spatial autocorrelation• Spatial non-stationarity• Geographically Weighted Regressions (GWR)

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)

Spatial autocorrelation - Basics

Positive(Neighbours more similar)

Neutral(Random)

Negative(Neighbours less similar)

http://www.spatialanalysisonline.com/

Spatial autocorrelation – is it common?

• Elevation• Air/water temperature

• Air humidity• Disease distribution• Species abundance• Housing value• Etc.

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

GWR Residuals

-.76 - -.35-.34 - -.09-.08 - .09.10 - .26.27 - .56

OLS Residuals

-1.34 - -.53-.52 - -.19-.18 - .08.09 - .37.38 - .92

0 100 200 30050Kilometers

±Moran's I = 0.144 Moran's I = 0.372

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.

Spatial autocorrelation – how to measure it?• Measures of spatial autocorrelation:

• Moran’s I

• Geary’s C

• Others (e.g. Getis’ G)

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

Outline

• Background• Spatial autocorrelation• Spatial non-stationarity• Geographically Weighted Regressions (GWR)

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

Global vs Local Statistics

Simpson Paradox

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/)

• Local regressions: GWR

Outline

• Background• Spatial autocorrelation• Spatial non-stationarity• Geographically Weighted Regressions (GWR)

• Brunsdon, Fortheringham and CharltonGWR

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)

• 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

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.

GWR

• Geographically Weighted Regression (GRW)

• (μ,ν): geographic coordinates of the samples

• Multiple regression model (global)

• y: dependent variable, x1 to xp: independent variables, β0: origin, β1 to βp: coefficients, ε: error.

Cod presence/absence (threshold at 5 kg) for the Fall 2001

Method

Government fisheries scientific survey data (Fisheries and Oceans Canada)

Method – Data interpolation

Method

Combining data in a single point data file

Exporting data points in a file (.dbf)

Temperature

Cod

Crab

Shrimp

Year 2001

Method

GWR software (version 3.0)

200km used for tests

About 25 minutes per file of 5500 points

Fixed

Variable

Results

Test of spatial stationarity of independent variables used in the regression

Spatial stationarity

Spatial non-stationarity

Results spatial stationarity

Windle et al. (accepted) - MEPS

Stationarity of bottom temperatureused to model shrimp biomass

Results

Comparison of regression models

Results

Test of the spatial auto-correlation of the residuals

Results

Results

Results

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

Results

GAM systematically has lower AIC values, suggesting a non-linear relationship between cod and the variables used in the analysis

Strong

WeakAIC: Akaike Information Criterion

Results

1985 1986 1987 1988 1989 1990 1991 19920

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Logistic_R2 GAM_R2 GWR_R2_ave

Year

R2Min and

max GWR coefficients (R2)

Model power decreases with years

GWR coefficients– Capelan1985

1986

1987

1988

1989

1990

1991

1992

GWR coefficients – Catch per Unit Effort1985

1986

1987

1988

1989

1990

1991

1992

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

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)