Testing spatial autocorrelation in weighted networks: the modes permutation test
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Testing spatial autocorrelation in weighted networks: the modes permutation test Franc¸ois Bavaud
Received: 17 February 2012 / Accepted: 21 March 2013 / Published online: 9 April 2013 Springer-Verlag Berlin Heidelberg 2013
Abstract In a weighted spatial network, as specified by an exchange matrix, the variances of the spatial values are inversely proportional to the size of the regions. Spatial values are no more exchangeable under independence, thus weakening the rationale for ordinary permutation and bootstrap tests of spatial autocorrelation. We propose an alternative permutation test for spatial autocorrelation, based upon exchangeable spatial modes, constructed as linear orthogonal combinations of spatial values. The coefficients obtain as eigenvectors of the standardized exchange matrix appearing in spectral clustering and generalize to the weighted case the concept of spatial filtering for connectivity matrices. Also, two proposals aimed at transforming an accessibility matrix into an exchange matrix with a priori fixed margins are presented. Two examples (inter-regional migratory flows and binary adjacency networks) illustrate the formalism, rooted in the theory of spectral decomposition for reversible Markov chains. Keywords Bootstrap Local variance Markov and semi-Markov processes Moran’s I Permutation test Spatial autocorrelation Spatial filtering Weighted networks JEL Classification
C12 C15 C31
1 Introduction Permutation tests of spatial autocorrelation are justified under exchangeability, that is the premise that the observed scores follow a permutation-invariant joint F. Bavaud (&) Department of Computer Science and Mathematical Methods, Department of Geography, University of Lausanne, Lausanne, Switzerland e-mail: [email protected]
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distribution. Yet, in the frequently encountered case of geographical data collected on regions differing in importance, the variance of a regional score is expected to decrease with the size of the region, in the same way that the variance of an average is inversely proportional to the size of the sample in elementary statistics: heteroscedasticity holds in effect, already under spatial independence, thus weakening the rationale of the celebrated spatial autocorrelation permutation test (e.g. Cliff and Ord 1973; Besag and Diggle 1977) in the case of a weighted network. This paper presents an alternative permutation test for spatial autocorrelation, whose validity extends to the weighted case. The procedure relies upon spatial modes, that is linear orthogonal combinations of spatial values, originally based upon the eigenvectors of the standardized connectivity or adjacency matrix (Tiefelsdorf and Boots 1995; Griffth 2000). In contrast to regional scores, the variance of the spatial modes turn out to be constant under spatial independence, thereby justifying the modes permutation test for spatial autocorrelation. Section 2.1 presents the definition of the local variance and Moran’s I in the arguably most general setup for spatial
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