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Chapter 5 Part B: Spatial Autocorrelation and regression modelling

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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

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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

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

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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

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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

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

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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

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

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3 rd editionwww.spatialanalysisonline.com10 Spatial Autocorrelation Joins count – some issues Multiple z-scores Binary or k-class data Rooks 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

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3 rd editionwww.spatialanalysisonline.com11 Spatial Autocorrelation Irregular lattice – (x,y,z) and adjacency tables ,1 1,21,3 2,12,22,3 3,13,23,3 4,1 4,24,3 xyz Cell numbering Cell dataCell coordinates (row/col)x,y,z view Adjacency matrix, total 1s=26

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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

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3 rd editionwww.spatialanalysisonline.com13 Spatial Autocorrelation Morans I TSA model

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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)=

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3 rd editionwww.spatialanalysisonline.com15 Spatial Autocorrelation Morans 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)

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

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

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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

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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

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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:

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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 )

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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

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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:

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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

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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

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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

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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

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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

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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

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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:

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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

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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

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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

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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

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

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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

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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

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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

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