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Chapter 5

Chapter 5. Part B: Spatial Autocorrelation and regression modelling. Autocorrelation. Time series correlation model { x t, 1 } t =1,2,3… n ‑1 and { x t, 2 } t =2,3,4… n. Spatial Autocorrelation. Correlation coefficient { x i } i =1,2,3… n , { y i } i =1,2,3… n

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Chapter 5

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

  2. Autocorrelation Time series correlation model • {xt,1} t=1,2,3…n‑1 and {xt,2} t=2,3,4…n www.spatialanalysisonline.com

  3. Spatial Autocorrelation • Correlation coefficient • {xi} i=1,2,3…n, {yi} i=1,2,3…n • Time series correlation model • {xt,1} t=1,2,3…n‑1 and {xt,2} t=2,3,4…n • Mean values: Lag 1 autocorrelation: large n www.spatialanalysisonline.com

  4. 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 www.spatialanalysisonline.com

  5. 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 www.spatialanalysisonline.com

  6. 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 www.spatialanalysisonline.com

  7. Spatial Autocorrelation • Joins counts (50% 1’s) www.spatialanalysisonline.com

  8. 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 www.spatialanalysisonline.com

  9. Spatial Autocorrelation • Joins counts www.spatialanalysisonline.com

  10. 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 www.spatialanalysisonline.com

  11. Spatial Autocorrelation • Irregular lattice – (x,y,z) and adjacency tables Cell data Cell coordinates (row/col) x,y,z view Cell numbering Adjacency matrix, total 1’s=26 www.spatialanalysisonline.com

  12. Spatial Autocorrelation • “Spatial” (auto)correlation coefficient • Coordinate (x,y,z) data representation for cells • Spatial weights matrix (binary or other), W={wij} • From last slide: Σ wij=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 www.spatialanalysisonline.com

  13. Spatial Autocorrelation • Moran’s I • TSA model www.spatialanalysisonline.com

  14. Spatial Autocorrelation Moran I =10*16.19/(26*196.68)=0.0317  0 www.spatialanalysisonline.com

  15. 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) www.spatialanalysisonline.com

  16. Spatial Autocorrelation • Moran I Correlogram www.spatialanalysisonline.com

  17. Spatial Autocorrelation • Geary C • Co-variation model uses squared differences rather than products • Similar approach is used in geostatistics www.spatialanalysisonline.com

  18. 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 www.spatialanalysisonline.com

  19. 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 www.spatialanalysisonline.com

  20. Regression modelling • Ordinary Least Squares (OLS) model • Minimise sum of squared errors (or residuals) • Solved for coefficients by matrix expression: www.spatialanalysisonline.com

  21. 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) www.spatialanalysisonline.com

  22. 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 www.spatialanalysisonline.com

  23. 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: www.spatialanalysisonline.com

  24. 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 www.spatialanalysisonline.com

  25. 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 www.spatialanalysisonline.com

  26. Regression modelling • Some ‘regression’ terminology • Simple linear • Multiple • Multivariate • SAR • CAR • Logistic • Poisson • Ecological • Hedonic • Analysis of variance • Analysis of covariance www.spatialanalysisonline.com

  27. Regression modelling • Spatial regression – trend surfaces and residuals (a form of ESDA) • General model: • y - observations, f( , , ) - some function, (x1,x2) - plane coordinates, w - attribute vector • Linear trend surface plot • Residuals plot • 2nd and 3rd order polynomial regression • Goodness of fit measures – coefficient of determination www.spatialanalysisonline.com

  28. 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 www.spatialanalysisonline.com

  29. 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 www.spatialanalysisonline.com

  30. 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: www.spatialanalysisonline.com

  31. 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 www.spatialanalysisonline.com

  32. 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 www.spatialanalysisonline.com

  33. 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 www.spatialanalysisonline.com

  34. 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 www.spatialanalysisonline.com

  35. Regression modelling • SAR models • Pure spatial lag: • Re-arranging: • MRSA model: Spatial weights matrix Autoregression parameter Linear regression added www.spatialanalysisonline.com

  36. Regression modelling • SAR models • Spatial error model: • Substituting and re-arranging: Linear regression + spatial error iid error vector Spatial weighted error vector Linear regression (global) iid error vector SAR lag Local trend www.spatialanalysisonline.com

  37. 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 Autoregression parameter Expected value at i weighted mean for neighbourhood of i www.spatialanalysisonline.com

  38. Regression modelling • Spatial filtering • Apply a spatial filter to the data to remove SA effects • Model the filtered data • Example: Spatial filter www.spatialanalysisonline.com

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