Testing Independence Between Two Spatial Random Fields

In this article, we consider testing independence between two spatial Gaussian random fields evaluated, respectively, at p and q locations with sample size n, where both p and q are allowed to be larger than n. We impose no spatial stationarity and no parametric structure for the two random fields. Our approach is based on canonical correlation analysis (CCA). But instead of applying CCA directly to the two random fields, which is not feasible for high-dimensional testing considered, we adopt a dimension-reduction approach using a special class of multiresolution spline basis functions. These functions are ordered in terms of their degrees of smoothness. By projecting the data to the function space spanned by a few leading basis functions, the spatial variation of the data can be effectively preserved. The test statistic is constructed from the first sample canonical correlation coefficient in the projected space and is shown to have an asymptotic Tracy–Widom distribution under the null hypothesis. Our proposed method automatically detects the signal between the two random fields and is designed to handle irregularly spaced data directly. In addition, we show that our test is consistent under mild conditions and provide three simulation experiments to demonstrate its powers. Moreover, we apply our method to investigate whether the precipitation in continental East Africa is related to the sea surface temperature (SST) in the Indian Ocean and whether the precipitation in west Australia is related to the SST in the North Atlantic Ocean.