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

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 $\delta$ 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 $\delta$ 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 $\delta$ 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.