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 © 2003 Luc Anselin, All Rights Reserved Outline Terminology Null and Alternative Hypothesis Spatial Autocorrelation Tests © 2003 Luc Anselin, All Rights Reserved Terminology Concepts 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 © 2003 Luc Anselin, All Rights Reserved Spatial Data Types and 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 © 2003 Luc Anselin, All Rights Reserved Spatial Data Types and Autocorrelation (2) 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” © 2003 Luc Anselin, All Rights Reserved Null and Alternative Hypothesis © 2003 Luc Anselin, All Rights Reserved Spatial Randomness 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 © 2003 Luc Anselin, All Rights Reserved Random or Clustered? © 2003 Luc Anselin, All Rights Reserved Random or Clustered? © 2003 Luc Anselin, All Rights Reserved Observed (left) and Randomized (right) 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 © 2003 Luc Anselin, All Rights Reserved Autocorrelation and Diffusion 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 © 2003 Luc Anselin, All Rights Reserved True vs. Apparent Contagion 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 © 2003 Luc Anselin, All Rights Reserved Spatial Autocorrelation Tests Spatial Autocorrelation Statistics 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 | © 2003 Luc Anselin, All Rights Reserved Spatial Lag Chart 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 © 2003 Luc Anselin, All Rights Reserved Ww_hoval 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 .