False discovery rate for functional data

Since Benjamini and Hochberg introduced false discovery rate (FDR) in their seminal paper, this has become a very popular approach to the multiple comparisons problem. An increasingly popular topic within functional data analysis is local inference, i.e. the continuous statistical testing of a null hypothesis along the domain. The principal issue in this topic is the infinite amount of tested hypotheses, which can be seen as an extreme case of the multiple comparisons problem. In this paper, we define and discuss the notion of FDR in a very general functional data setting. Moreover, a continuous version of the Benjamini–Hochberg procedure is introduced along with a definition of adjusted p value function. Some general conditions are stated, under which the functional Benjamini–Hochberg procedure provides control of the functional FDR. Two different simulation studies are presented; the first study has a one-dimensional domain and a comparison with another state-of-the-art method, and the second study has a planar two-dimensional domain. Finally, the proposed method is applied to satellite measurements of Earth temperature. In detail, we aim at identifying the regions of the planet where temperature has significantly increased in the last decades. After adjustment, large areas are still significant.