1. www.spatialanalysisonline.com Chapter 5 Part B: Spatial Autocorrelation and regression modelling

2. 3 rd editionwww.spatialanalysisonline.com2 Autocorrelation Time series correlation model {x t,1 } t=1,2,3…n ‑ 1 and {x t,2 } t=2,3,4…n

3. 3 rd editionwww.spatialanalysisonline.com3 Spatial Autocorrelation Correlation coefficient {x i } i=1,2,3…n, {y i } i=1,2,3…n  Time series correlation model {x t,1 } t=1,2,3…n ‑ 1 and {x t,2 } t=2,3,4…n Mean values: Lag 1 autocorrelation: large n

4. 3 rd editionwww.spatialanalysisonline.com4 Spatial Autocorrelation Classical statistical model assumptions Independence vs dependence in time and space Tobler’s first law: “All things are related, but nearby things are more related than distant things” Spatial dependence and autocorrelation Correlation and Correlograms

5. 3 rd editionwww.spatialanalysisonline.com5 Spatial Autocorrelation Covariance and autocovariance Lags – fixed or variable interval Correlograms and range Stationary and non-stationary patterns Outliers Extending concept to spatial domain  Transects  Neighbourhoods and distance-based models

6. 3 rd editionwww.spatialanalysisonline.com6 Spatial Autocorrelation Global spatial autocorrelation  Dataset issues: regular grids; irregular lattice (zonal) datasets; point samples Simple binary coded regular grids – use of Joins counts Irregular grids and lattices – extension to x,y,z data representation Use of x,y,z model for point datasets Local spatial autocorrelation  Disaggregating global models

7. 3 rd editionwww.spatialanalysisonline.com7 Spatial Autocorrelation Joins counts (50% 1’s) A. Completely separated pattern (+ve)B. Evenly spaced pattern (-ve) C. Random pattern

8. 3 rd editionwww.spatialanalysisonline.com8 Spatial Autocorrelation Joins count  Binary coding  Edge effects  Double counting  Free vs non-free sampling  Expected values (free sampling) 1-1 = 15/60, 0-0 = 15/60, 0-1 or 1-0 = 30/60

9. 3 rd editionwww.spatialanalysisonline.com9 Spatial Autocorrelation Joins counts A. Completely separated (+ve)B. Evenly spaced (-ve) C. Random

10. 3 rd editionwww.spatialanalysisonline.com10 Spatial Autocorrelation Joins count – some issues  Multiple z-scores  Binary or k-class data  Rook’s move vs other moves  First order lag vs higher orders  Equal vs unequal weights  Regular grids vs other datasets  Global vs local statistics  Sensitivity to model components

11. 3 rd editionwww.spatialanalysisonline.com11 Spatial Autocorrelation Irregular lattice – (x,y,z) and adjacency tables +4.55+5.54 +2.24-5.15+9.02 +3.10-4.39-2.09 +0.46-3.06 1,1 1,21,3 2,12,22,3 3,13,23,3 4,1 4,24,3 xyz 124.55 135.54 212.24 22 ‑ 5.15 239.02 313.1 32 ‑ 4.39 33 ‑ 2.09 420.46 43 ‑ 3.06 37 148 259 610 Cell numbering Cell dataCell coordinates (row/col)x,y,z view Adjacency matrix, total 1’s=26

12. 3 rd editionwww.spatialanalysisonline.com12 Spatial Autocorrelation “Spatial” (auto)correlation coefficient  Coordinate (x,y,z) data representation for cells  Spatial weights matrix (binary or other), W={w ij } From last slide: Σ w ij =26  Coefficient formulation – desirable properties Reflects co-variation patterns Reflects adjacency patterns via weights matrix Normalised for absolute cell values Normalised for data variation Adjusts for number of included cells in totals

13. 3 rd editionwww.spatialanalysisonline.com13 Spatial Autocorrelation Moran’s I TSA model

14. 3 rd editionwww.spatialanalysisonline.com14 Spatial Autocorrelation A. Computation of variance/covariance-like quantities, matrix C B. C*W: Adjustment by multiplication of the weighting matrix, W Moran I =10*16.19/(26*196.68)=0.0317  0

15. 3 rd editionwww.spatialanalysisonline.com15 Spatial Autocorrelation Moran’s I Modification for point data Replace weights matrix with distance bands, width h Pre-normalise z values by subtracting means Count number of other points in each band, N(h)

16. 3 rd editionwww.spatialanalysisonline.com16 Spatial Autocorrelation Moran I Correlogram Source data pointsLag distance bands, hCorrelogram

17. 3 rd editionwww.spatialanalysisonline.com17 Spatial Autocorrelation Geary C  Co-variation model uses squared differences rather than products  Similar approach is used in geostatistics

18. 3 rd editionwww.spatialanalysisonline.com18 Spatial Autocorrelation Extending SA concepts  Distance formula weights vs bands  Lattice models with more complex neighbourhoods and lag models (see GeoDa)  Disaggregation of SA index computations (row- wise) with/without row standardisation (LISA)  Significance testing Normal model Randomisation models Bonferroni/other corrections

19. 3 rd editionwww.spatialanalysisonline.com19 Regression modelling Simple regression – a statistical perspective  One (or more) dependent (response) variables  One or more independent (predictor) variables  Linear regression is linear in coefficients:  Vector/matrix form often used  Over-determined equations & least squares

20. 3 rd editionwww.spatialanalysisonline.com20 Regression modelling Ordinary Least Squares (OLS) model  Minimise sum of squared errors (or residuals)  Solved for coefficients by matrix expression:

21. 3 rd editionwww.spatialanalysisonline.com21 Regression modelling OLS – models and assumptions  Model – simplicity and parsimony  Model – over-determination, multi-collinearity and variance inflation  Typical assumptions Data are independent random samples from an underlying population Model is valid and meaningful (in form and statistical) Errors are iid Independent; No heteroskedasticity; common distribution Errors are distributed N(0,  2 )

22. 3 rd editionwww.spatialanalysisonline.com22 Regression modelling Spatial modelling and OLS  Positive spatial autocorrelation is the norm, hence dependence between samples exists  Datasets often non-Normal >> transformations may be required (Log, Box-Cox, Logistic)  Samples are often clustered >> spatial declustering may be required  Heteroskedasticity is common  Spatial coordinates (x,y) may form part of the modelling process

23. 3 rd editionwww.spatialanalysisonline.com23 Regression modelling OLS vs GLS  OLS assumes no co-variation Solution:  GLS models co-variation: y~ N( ,C) where C is a positive definite covariance matrix y=X  +u where u is a vector of random variables (errors) with mean 0 and variance-covariance matrix C Solution:

24. 3 rd editionwww.spatialanalysisonline.com24 Regression modelling GLS and spatial modelling  y~ N( ,C) where C is a positive definite covariance matrix (C must be invertible)  C may be modelled by inverse distance weighting, contiguity (zone) based weighting, explicit covariance modelling… Other models  Binary data – Logistic models  Count data – Poisson models

25. 3 rd editionwww.spatialanalysisonline.com25 Regression modelling Choosing between models  Information content perspective and AIC where n is the sample size, k is the number of parameters used in the model, and L is the likelihood function

26. 3 rd editionwww.spatialanalysisonline.com26 Regression modelling Some ‘regression’ terminology Simple linear Multiple Multivariate SAR CAR Logistic Poisson Ecological Hedonic Analysis of variance Analysis of covariance

27. 3 rd editionwww.spatialanalysisonline.com27 Regression modelling Spatial regression – trend surfaces and residuals (a form of ESDA)  General model: y - observations, f(,, ) - some function, (x 1,x 2 ) - plane coordinates, w - attribute vector Linear trend surface plot Residuals plot 2 nd and 3 rd order polynomial regression Goodness of fit measures – coefficient of determination

28. 3 rd editionwww.spatialanalysisonline.com28 Regression modelling Regression & spatial autocorrelation (SA)  Analyse the data for SA  If SA ‘significant’ then Proceed and ignore SA, or Permit the coefficient, , to vary spatially (GWR), or Modify the regression model to incorporate the SA

29. 3 rd editionwww.spatialanalysisonline.com29 Regression modelling Regression & spatial autocorrelation (SA)  Analyse the data for SA  If SA ‘significant’ then Proceed and ignore SA, or Permit the coefficient, , to vary spatially (GWR) or Modify the regression model to incorporate the SA

30. 3 rd editionwww.spatialanalysisonline.com30 Regression modelling Geographically Weighted Regression (GWR)  Coefficients, , allowed to vary spatially,  (t)  Model:  Coefficients determined by examining neighbourhoods of points, t, using distance decay functions (fixed or adaptive bandwidths)  Weighting matrix, W(t), defined for each point  Solution: GLS:

31. 3 rd editionwww.spatialanalysisonline.com31 Regression modelling Geographically Weighted Regression  Sensitivity – model, decay function, bandwidth, point/centroid selection  ESDA – mapping of surface, residuals, parameters and SEs  Significance testing Increased apparent explanation of variance Effective number of parameters AICc computations

32. 3 rd editionwww.spatialanalysisonline.com32 Regression modelling Geographically Weighted Regression  Count data – GWPR use of offsets Fitting by ILSR methods  Presence/Absence data – GWLR True binary data Computed binary data - use of re-coding, e.g. thresholding Fitting by ILSR methods

33. 3 rd editionwww.spatialanalysisonline.com33 Regression modelling Regression & spatial autocorrelation (SA)  Analyse the data for SA  If SA ‘significant’ then Proceed and ignore SA, or Permit the coefficient, , to vary spatially (GWR) or Modify the regression model to incorporate the SA

34. 3 rd editionwww.spatialanalysisonline.com34 Regression modelling Regression & spatial autocorrelation (SA)  Modify the regression model to incorporate the SA, i.e. produce a Spatial Autoregressive model (SAR)  Many approaches – including: SAR – e.g. pure spatial lag model, mixed model, spatial error model etc. CAR – a range of models that assume the expected value of the dependent variable is conditional on the (distance weighted) values of neighbouring points Spatial filtering – e.g. OLS on spatially filtered data

35. 3 rd editionwww.spatialanalysisonline.com35 Regression modelling SAR models  Pure spatial lag:  Re-arranging:  MRSA model: Autoregression parameter Spatial weights matrix Linear regression added

36. 3 rd editionwww.spatialanalysisonline.com36 Regression modelling SAR models  Spatial error model: Substituting and re-arranging: Spatial weighted error vector Linear regression + spatial error iid error vector Linear regression (global) SAR lag Local trend

37. 3 rd editionwww.spatialanalysisonline.com37 Regression modelling CAR models  Standard CAR model:  Local weights matrix – distance or contiguity  Variance : Different models for W and M provide a range of CAR models weighted mean for neighbourhood of i Autoregression parameter Expected value at i

38. 3 rd editionwww.spatialanalysisonline.com38 Regression modelling Spatial filtering  Apply a spatial filter to the data to remove SA effects  Model the filtered data  Example: Spatial filter