Nonparametric Hypothesis Testing for a Spatial Signal

Abstract

Nonparametric hypothesis testing for a spatial signal can involve a large number of hypotheses. For instance, two satellite images of the same scene, taken before and after an event, could be used to test a hypothesis that the event has no environmental impact. This is equivalent to testing that the mean difference of “after-before” is zero at each of the (typically thousands of) pixels that make up the scene. In such a situation, conventional testing procedures that control the overall type I error deteriorate as the number of hypotheses increases. Powerful testing procedures are needed for this problem of testing for the presence of a spatial signal. In this article we propose a procedure called enhanced false discovery rate (EFDR), which is based on controlling the false discovery rate (FDR) and a concept known as generalized degrees of freedom. EFDR differs from the standard FDR procedure by reducing the number of hypotheses tested. This is done in two ways: first, the model is represented more parsimoniously in the wavelet domain; second, an optimal selection of hypotheses is made using a criterion based on generalized degrees of freedom. Not only does the EFDR procedure tell us whether a spatial signal is present, but, if a signal is deemed present, it can also indicate its location and magnitude. We examine EFDR's operating characteristics and in simulations show that it outperforms the standard FDR and conventional testing procedures. Finally, we apply the EFDR procedure to an air temperature dataset generated from the climate system model of the National Center for Atmospheric Research, in which air temperatures in the 1980s are compared to those in the 1990s. We conclude that temperature change has occurred between the two decades, mostly warming in the central part of the United States and in coastal regions of South America at about 20°S.