Spatio-temporal Dependence Detection in Model Residuals

Spatio-temporal dependence in atmospheric sciences data must be prioritized in modeling efforts. However, before undertaking complicated spatio-temporal modeling, it is advantageous to quantify if the dependency merits modeling using tests for different hypotheses including lack of dependency or structured patterns of dependency.

One method of testing for lack of dependence is the Space-Time Index (STI) introduced in Griffith [Dynamic Spatial Models, 258-287 (1981)] to detect spatio-temporal correlation for a vector autoregressive (VAR) form. We assessed the usefulness of STI by simulating VAR series with various spatial and temporal correlations. Simulations indicate the power of the test was low, especially as the number of spatial neighbors increased. The STI is also subject to assumptions of normality, restriction to VAR form, and specification of only a null hypothesis of complete independence.

Thus, we introduce a spatio-temporal identification (STrIde) testing method for model residuals. The test statistic is based on the theory of U-processes and is more flexible than STI. Hypothesis testing is done via a Monte Carlo approximation to permutation testing. This allows for many different null hypotheses to be tested under very general conditions. Simulations for null hypotheses of complete independence, time dependence, and space dependence were done. Our simulations indicate a higher power for the test of complete independence than that achieved by STI.

The STrIde statistic is applied to model residuals produced in assessing the probability of exceeding certain daily precipitation thresholds in Indian summer monsoon via a logit-normal mixed model. Results indicate no evidence of spatio-temporal correlation in residuals at the higher thresholds. Consequently, a more complicated model structure appears unnecessary. This implies the spatio-temporal relationship between extreme precipitation events depends additionally on the threshold used to define an extreme event.