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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

Chapter 5

Part B: Spatial Autocorrelation and regression modelling

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autocorrelation
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
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 autocorrelation4
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 autocorrelation5
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 autocorrelation6
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 autocorrelation7
Spatial Autocorrelation
  • Joins counts (50% 1’s)

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spatial autocorrelation8
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 autocorrelation9
Spatial Autocorrelation
  • Joins counts

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spatial autocorrelation10
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 autocorrelation11
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 autocorrelation12
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 autocorrelation13
Spatial Autocorrelation
  • Moran’s I
  • TSA model

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spatial autocorrelation14
Spatial Autocorrelation

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

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spatial autocorrelation15
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 autocorrelation16
Spatial Autocorrelation
  • Moran I Correlogram

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spatial autocorrelation17
Spatial Autocorrelation
  • Geary C
    • Co-variation model uses squared differences rather than products
    • Similar approach is used in geostatistics

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spatial autocorrelation18
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
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 modelling20
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 modelling21
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 modelling22
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 modelling23
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 modelling24
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 modelling25
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 modelling26
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 modelling27
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 modelling28
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 modelling29
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 modelling30
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 modelling31
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 modelling32
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 modelling33
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 modelling34
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 modelling35
Regression modelling
  • SAR models
    • Pure spatial lag:
    • Re-arranging:
    • MRSA model:

Spatial weights matrix

Autoregression parameter

Linear regression added

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regression modelling36
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 modelling37
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 modelling38
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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