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## Chapter 5

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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 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 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

Adjacency matrix, total 1’s=26

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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
- 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

- 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

www.spatialanalysisonline.com

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

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

www.spatialanalysisonline.com

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

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

Linear regression added

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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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