Chapter 5

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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

Part B: Spatial Autocorrelation and regression modelling

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Autocorrelation

Time series correlation model

• {xt,1} t=1,2,3…n‑1 and {xt,2} t=2,3,4…n

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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

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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

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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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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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Spatial Autocorrelation
• Joins counts (50% 1’s)

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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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Spatial Autocorrelation
• Joins counts

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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

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Spatial Autocorrelation
• Irregular lattice – (x,y,z) and adjacency tables

Cell data

Cell coordinates (row/col)

x,y,z view

Cell numbering

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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

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Spatial Autocorrelation
• Moran’s I
• TSA model

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Spatial Autocorrelation

Moran I =10*16.19/(26*196.68)=0.0317  0

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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)

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Spatial Autocorrelation
• Moran I Correlogram

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

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

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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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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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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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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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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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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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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

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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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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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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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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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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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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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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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Regression modelling
• SAR models
• Pure spatial lag:
• Re-arranging:
• MRSA model:

Spatial weights matrix

Autoregression parameter

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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

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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

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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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