Measuring and Testing Multivariate Spatial Autocorrelation in a Weighted Setting: A Kernel Approach

Abstract We propose and illustrate a general framework in which spatial autocorrelation is measured by the Frobenius product of two kernels, a feature kernel and a spatial kernel. The resulting autocorrelation index generalizes Moran's index in the weighted, multivariate setting, where regions, differing in importance, are characterized by multivariate features. Spatial kernels can traditionally be obtained from a matrix of spatial weights, or directly from geographical distances. In the former case, the Markov transition matrix defined by row‐normalized spatial weights must be made compatible with the regional weights, as well as reversible. Equivalently, space is specified by a symmetric exchange matrix containing the joint probabilities to select a pair of regions. Four original weight‐compatible constructions, based upon the binary adjacency matrix, are presented and analyzed. Weighted multidimensional scaling on kernels yields a low‐dimensional visualization of both the feature and the spatial configurations. The expected values of the first four moments of under the null hypothesis of absence of spatial autocorrelation can be exactly computed under a new approach, invariant orthogonal integration, thus permitting to test the significance of beyond the normal approximation, which only involves its expectation and expected variance. Various illustrations are provided, investigating the spatial autocorrelation of political and social features among French departments.