An association measure for spatio-temporal time series

Spatial association measures for univariate static spatial data are widely used. Suppose the data is in the form of a collection of spatial vectors, say \(X_{rt}\) where \(r=1, \ldots , R\) are the regions and \(t=1, \ldots , T\) are the time points, in the same temporal domain of interest. Using Bergsma’s correlation coefficient \(\rho \), we construct a measure of similarity between the regions’ series. Due to the special properties of \(\rho \), unlike other spatial association measures which test for spatial randomness, our statistic can account for spatial pairwise independence. We have derived the asymptotic distribution of our statistic under null (independence of the regions) and alternate cases (the regions are dependent) when, across t the vector time series are assumed to be independent and identically distributed. The alternate scenario of spatial dependence is explored using simulations from the spatial autoregressive and moving average models. Finally, we provide application to modelling and testing for the presence of spatial association in COVID-19 incidence data, by using our statistic on the residuals obtained after model fitting.