Spatial Autocorrelation (1) Basic Concepts

Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu

Terminology Null and Alternative Hypothesis Spatial Autocorrelation Tests

Spatial Dependence property of joint (multivariate) density functions difficult or impossible to verify in practice Spatial Autocorrelation moment of a joint (multivariate) density can be estimated/tested • autocorrelation coefficient, autocovariance Spatial Association usually same as spatial autocorrelation

Point Patterns interest in absence of spatial randomness of locations • clusters, dispersion Continuous Surfaces (geostatistics) interest in modeling spatial covariance between pairs of observations as it changes with distance • spatial prediction, kriging

Lattice Data discrete areal units • counties, census tracts or points as representation of areal units • centroids of counties interest in absence of spatial randomness of attributes • similarity between “neighbors”

Null Hypothesis: No Spatial Autocorrelation spatial randomness values observed at a location do not depend on values observed at neighboring locations observed spatial pattern of values is equally likely as any other spatial pattern the location of values may be altered without affecting the information content of the data

Randomization polyid 1 became 14 polyid 2 became 20 polyid 3 became 48 ... Alternative Hypotheses of SA

Positive Spatial Autocorrelation like values tend to cluster in space neighbors are similar compatible with diffusion • but not necessarily diffusion Negative Spatial Autocorrelation neighbors are dissimilar checkerboard pattern

Positive Spatial Autocorrelation Does NOT Imply Diffusion diffusion tends to yield positive spatial autocorrelation, but the reverse is not necessary spatial randomness is not compatible with diffusion/contagion Apparent Contagion cluster is the result of spatial heterogeneity

Distinguishing True from Apparent Contagion not possible in cross-section without further information extra information: time domain, theory, priors Contagion is Dynamic space-time analysis

Formal Test of Match Between Locational Similarity and Value (Attribute) Similarity Locational Similarity spatial weights W Types of Value Similarity cross-product: xi.xj 2 squared difference: (xi - xj) absolute difference: | xi - xj |

Spatial Lag Visualization value at i compared to weighted average

of neighbors: xi relative to (Wx)i similar values = positive SA dissimilar values = negative SA Spatial Lag Pie Chart xi and (Wx)i as proportions of “pie” • x > 0 only Spatial Lag Bar Chart xi and (Wx)i as bars

Hoval

spatial lag pie chart blue = housing value in i, red = average housing value for neighbors spatial lag bar chart blue = crime at i, red = spatial lag, average crime for neighbors