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.

自从Benjamini和Hochberg在他们的开创性论文中引入错误发现率（FDR）以来，这已成为解决多重比较问题的一种非常流行的方法。在功能数据分析中，一个越来越流行的话题是局部推理，即沿域对虚无假设进行连续的统计检验。本主题中的主要问题是检验假设的数量无穷，这可以看作是多重比较问题的极端情况。在本文中，我们定义并讨论了非常通用的功能数据设置中的FDR概念。此外，引入了Benjamini-Hochberg过程的连续版本以及调整后的p的定义。值函数。陈述了一些一般条件，在这些条件下，功能性Benjamini-Hochberg程序可控制功能性FDR。提出了两种不同的仿真研究。第一项研究具有一维域，并且与另一种现有技术进行了比较，第二项研究具有平面二维域。最后，将所提出的方法应用于卫星对地球温度的测量。详细地说，我们旨在确定过去几十年来温度显着升高的星球区域。调整后，大面积面积仍然很大。