Testing Independence Between Two Spatial Random Fields
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INTRODUCTION In climate and atmospheric research, an important problem is to test whether two series of spatial random fields (or images) are independent, and if not, how they are related.
Shih-Hao Huang, Department of Mathematics, National Central University, Taoyuan, Taiwan. Hsin-Cheng Huang (B), Institute of Statistical Science, Academia Sinica, Taipei, Taiwan (E-mail: [email protected]). Ruey S. Tsay, Booth School of Business, University of Chicago, Chicago, USA. Guangming Pan, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore. © 2020 International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics https://doi.org/10.1007/s13253-020-00421-3
S.- H. Huang et al.
For example, Montroy (1997) and Montroy et al. (1998) study the relationship between the precipitation in America and the sea surface temperature (SST) in the Pacific Ocean. Rajeevan and Sridhar (2008) investigate the teleconnection between the precipitation in India and the SST in the Atlantic Ocean. Rana et al. (2018) study the teleconnection between the precipitation in Central southwest Asia and the SST in the Pacific Ocean. Hewitt et al. (2018) explore the relationship between the precipitation in Colorado and the SST in the Pacific Ocean. One commonly used approach to study the dependency between two random fields is the canonical correlation analysis (CCA, see, e.g., Hotelling 1936). The CCA seeks important patterns, called canonical weights, that successively explain the maximum amount of correlation between the two random fields. In the CCA framework, the leading sample canonical correlation can be used to test the independence between the two random fields (Johnstone 2008; Bao et al. 2019). However, this test is not applicable when the dimension of either image is larger than the sample size. To tackle this difficulty, several variants of CCA have been developed, such as shrinkage CCA (van der Merwe and Zidek 1980; Brieman and Friedman 1997), sparse CCA (Witten et al. 2009), regularized CCA (Yang and Pan 2015), and EOF-reduced CCA (see, e.g., von Storch and Zwiers 1999). However, these approaches do not account for the spatial structure inherent in images. In this article, we take a basis-function approach to capture signals between the two spatial random fields. This approach is quite popular in recent years for spatial modeling due to its computational advantage (see, e.g., Cressie and Johannesson 2008; Nychka et al. 2015). We develop a novel method that incorporates spatial structure in CCA by first projecting the images to a nested sequence of subspaces spanned by the multiresolution spline basis functions (Tzeng and Huang 2018) and then applying CCA to the projected data on a selected subspace corresponding to a suitable spatial resolution. This procedure allows us to transform a high-dimensional problem into a low-dimensional one without losing much information of the signal, thereby significantly increasing the testing power. The key to preserv
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