Spatial Autocorrelation Concepts

Sergio Rey

2/28/23

Outline

  • Concepts and Issues
  • Null and Alternative Hypotheses
  • Spatial Autocorrelation Tests

Concepts and Issues

Spatial Dependence

There is no question with respect to emergent geospatial science. The important harbingers were Geary’s article on spatial autocorrelation, Dacey’s paper about two- and K-color maps, and that of Bachi on geographic series.

– Berry, Griifth, Tiefelsdorf (2008)

Spatial Dependence

Working Concept

  • what happens at one place depends on events in nearby places

  • all things are related but nearby things are more related than distant things (Tobler)

  • central focus in lattice data analysis

Goodchild 1991

  • a world without positive spatial dependence would be an impossible world

  • impossible to describe

  • impossible to live in

  • hell is a place with no spatial dependence

Spatial Dependence

Categorizing

  • Type: Substantive versus nuisance

  • Direction: Positive versus negative

Issues

  • Time versus space

  • Inference

Substantive Spatial Dependence

Process Based

  • Part of the process under study

  • Leaving it out

    • Incomplete understanding

    • Biased inferences

Nuisance Spatial Dependence

Not Process Based

  • Artifact of data collection

  • Process boundaries not matching data boundaries

  • Scattering across pixels

  • GIS induced

Boundary

Boundary Mismatch

  • Even if \(A\) and \(B\) are independent

  • \(A'\) and \(B'\) will be dependent

Nusiance vs. Substantive Dependence

Issues

  • Not always easy to differentiate from substantive

  • Different implications for each type

  • Specification strategies (Econometrics)

  • Both can be operating jointly

Space versus Time

Temporal Dependence

  • Past influences the future

  • Recursive

  • One dimension

Space versus Time

Spatial Dependence

  • Multi-directional

  • Simultaneous

Terminology

Related Concepts

  • Spatial Dependence

  • Spatial Autocorrelation

  • Spatial Association

Spatial Dependence

Distributional Characteristic

  • Multivariate density function

  • difficult/impossible to verify empirically

Dependent Distribution

  • does not factor in marginal densities

Spatial Autocorrelation

  • Auto = same variable

  • Correlation = scaled covariance

  • Spatial - geographic pattern to the correlation

Spatial Autocorrelation

Measurement of Moment of Distribution

  • off-diagonal elements of variance-covariance matrix

  • autocovariance

  • \(C[y_i,y_j] \ne 0 \ \forall i\ne j\)

  • can be estimated

Spatial Autocorrelation Coefficient

  • significance test on coefficient = 0

Spatial Autocorrelation

Joint multivariate distribution function \[f(y) = \frac{ \exp\left[ -\frac{1}{2} (y-\mu)' \Sigma^{-1} (y-\mu) \right]} {\sqrt{(2\pi)^n|\Sigma|}}\]

Variance-Covariance Matrix

\[\Sigma= \left[ \begin{array}{rrrr} \sigma_{1,1}&\sigma_{1,2}&\ldots&\sigma_{1,n}\\ \sigma_{2,1}&\sigma_{2,2}&\ldots&\sigma_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ \sigma_{n,1}&\sigma_{n,2}&\ldots&\sigma_{n,n} \end{array} \right]\]

  • covariance: \(\sigma_{i,j} = E[(y_i - \mu_i)(y_j-\mu_j) ]\)

  • symmetry: \(\sigma_{i,j} =\sigma_{i,j}\)

  • variance: \(\sigma_{i,i} = E[(y_i - \mu_i)(y_i-\mu_i) ]\)

Correlation

\[\rho_{ij} = \frac{\sigma_{ij}}{\sqrt{\sigma_{i}^2}\sqrt{\sigma_{j}^2}}\] \[-1.0 \le \rho_{ij} \le 1.0\]

Data Types and Autocorrelation

Point Data

  • focus on geometric pattern

  • random vs. nonrandom

  • clustered vs. uniform

Geostatistics

  • 2-D modeling of spatial covariance (pairs of observations in function of distance)

  • kriging, spatial prediction

Data Types and Autocorrelation

Lattice Data

  • areal units: states, counties, census tracts, watersheds

  • points: centroids of areal units

  • focus on the spatial nonrandomness of attribute values

Spatial Association

Not a Rigorously Defined Term

  • Usually the same as spatial autocorrelation

  • often used in non-technical discussion

  • avoid unless meaning is clear

Spatial Dependence

Good News (for geographers)

  • Space matters

  • Suggestive of underlying process

Bad news

  • invalidates random sampling assumption

  • necessitates new methods = spatial statistics and spatial econometrics

Spatial Dependence: Implications

The specific process we are simulating is as follows:\[\begin{aligned} \label{eq:simdgp}y&=&X\beta + \epsilon \\ \nonumber\epsilon &=& \lambda W \epsilon + \nu \end{aligned}\] where \(\nu^{\sim}N(0,\sigma^{2}I)\), \(\lambda\) is a spatial autocorrelation parameter (scalar) and \(W\) is a spatial weights matrix. If \(\lambda=0\) then the \(i.i.d.\) assumption holds, otherwise there is spatial dependence.

\(\beta=40, \ \sigma^2=16, \ x=[1,1,\ldots]\)

\(\lambda=[0.0, 0.25, 0.50, 0.75], \ n=25\)

Spatial Dependence: Implications

For each D.G.P. we are going to generate 500 samples of size \(n=25\) for our map. You can think of this as generating 500 maps using the same D.G.P.. For each sample we will then do the following:

  1. Estimate \(\mu\) with \(\bar{y}\)

  2. Test the hypothesis that \(\mu=40\)

Implications

Monte Carlo Results
\(\lambda\) 0.00 0.25 0.50 0.75
\(\hat{\mu}\) 39.947 39.931 39.901 39.814
\(\sigma_{\bar{x}}\) 0.816 1.090 1.641 3.304
\(p\) 0.056 0.148 0.278 0.492

Null and Alternative Hypotheses

Spatial Randomness

Null Hypothesis

  • observed spatial pattern of values is equally likely as any other spatial pattern

  • values at one location do no depend on values at other (neighboring) locations

  • under spatial randomness, the location of values may be altered without affecting the information content of the data

Spatial Autocorrelation on a Grid

Negative, Random, Positive

Positive Spatial Autocorrelation

Clustering

  • like values tend to be in similar locations

Neighbor similarity

  • more alike than they would be under spatial randomness

Compatible with Diffusion

  • but not necessarily caused by diffusion

Positive Spatial Autocorrelation

Negative Spatial Autocorrelation

Checkerboard pattern

  • anti-clustering

Neighbor dissimilarity

  • more dissimilar than they would be under spatial randomness

Compatible with Competition

  • but not necessarily caused by competition

Negative Spatial Autocorrelation

Autocorrelation and Diffusion

One does not necessarily imply the other

  • diffusion tends to yield positive spatial autocorrelation but the reverse is not necessary

  • positive spatial correlation may be due to structural factors, without contagion or diffusion

True vs. Apparent Contagion

What is the Cause behind the clustering?

  • True contagion

    • result of a contagious process, social interaction, dynamic process

  • Apparent contagion

    • spatial heterogeneity

    • stratification

  • Cannot be distinguished in a pure cross section

  • Equifinality or Identification Problem

Spatial Autocorrelation Tests

Clustering

Global characeristic

  • property of overall pattern = all observations

  • are like values more grouped in space than random

  • test by means of a global spatial autocorrelation statistic

  • no location of the clusters determined

Clusters

Local characeristic

  • where are the like values more grouped in space than random?

  • property of local pattern = location-specific

  • test by means of a local spatial autocorrelation statistic

  • local clusters may be compatible with global spatial randomness

Spatial Autocorrelation Statistic

Structure

  • Formal Test of Match between Value Similarity and Locational Similarity

  • Statistic Summarizes Both Aspects

  • Significance

    • how likely is it (p-value) that the computed statistic would take this (extreme) value in a spatially random pattern

Attribute Similarity

  • Summary of the similarity or dissimilarity of a variable at different locations

    • variable \(y\) at locations \(i,j\) with \(i\ne j\)

  • Measures of similarity

    • cross product: \(y_i y_j\)

  • Measures of dissimilarity

    • squared differences: \((y_i - y_j)^2\)

    • absolute differences: \(|y_i - y_j|\)

Locational Similarity

  • Formalizing the notion of Neighbor

    • when two spatial units a-priori are likely to interact

  • Spatial Weights

    • not necessarility geographical

    • many approaches

Summary

Spatial Dependence

  • Core of Lattice Analysis

  • Spatial Autocorrelation More Complex than Temporal Autocorrelation

  • Combine Value and Locational Similarities