Abstract
Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.
Highlights
Spatial autocorrelation is fundamental to spatial science (Getis [1])
We show that the spatial autocorrelation measured by Geary’s c is positive if spatially smoother graph Laplacian eigenvectors are dominant
(= κn ), which is its upper bound; and (iii) if a equals 0, all graph Laplacian eigenvectors contribute to y∗ and the corresponding Geary’s c equals 1
Summary
Spatial autocorrelation is fundamental to spatial science (Getis [1]). This is because it describes the similarity between signals on adjacent vertices. As stated, Geary’s c is a spatial generalization of the von Neumann [3] ratio: η=. We briefly describe this in the Appendix A.14. As an example of L, we show L† , which denotes the graph Laplacian of G† = (V† , E† ). (i) Harvey [12] (Equation (2.13)) represents Parseval’s identity in the Fourier representation of time series, which can be regarded as a special case of (11). N n−1 ∑i =1 ( yi − ȳ ) , and denote the graph Fourier transform of z by β:. This paper reconsiders Geary’s c using concepts from spectral graph theory. We illustrate that the spatial autocorrelation measured by Geary’s c is positive
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