Abstract
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.
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We are grateful to the Referees and the Associate Editor for their insightful comments which have led to a significant improvement in the manuscript.
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Research supported by J.C. Bose National Fellowship, Dept. of Science and Technology, Govt. of India.
Appendix: R code
Appendix: R code
We give below two R codes. The first is for the U-statistic based estimate of Bergsma correlation. Input for this first code is bivariate data in the form of two vectors, x and y. The second is a code for computing the \(\tilde{S}_B\) statistic. For this data is to be given in the format of a matrix with T rows (time points) and R columns (locations), and an \(R \times R\) Spatial proximity matrix.
Bergsma’s \({\tilde{\rho }}\)
\(\tilde{S}_B\) statistic
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Kappara, D., Bose, A. & Bhattacharjee, M. An association measure for spatio-temporal time series. Metrika (2023). https://doi.org/10.1007/s00184-023-00939-9
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DOI: https://doi.org/10.1007/s00184-023-00939-9
Keywords
- Bergsma’s correlation
- Spatial association measure
- U-statistic
- Spatial autoregressive model
- Spatial moving average model