Nonparametric hypothesis testing for a spatial signal

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 increase. 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 FDR (EFDR), which is based on controlling the false discovery rate (FDR) and a concept known as generalized degrees of freedom (GDF). EFDR differs from the standard FDR procedure through its reducing of the number of hypotheses tested. This is done in two ways: first, the model is represented more parisimoniously in the wavelet domain, and 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 or not, it has an added bonus that, if a signal is deemed present, it can indicate its location and magnitude. The EFDR procedure is applied to an air-temperature data set generated from the Climate System Model (CSM) of the National Center for Atmospheric Research (NCAR), where air temperatures in the 1980s are compared to those in the 1990s.