Pini and Vantini (2017) introduced the interval-wise testing procedure which performs local inference for functional data defined on an interval domain, where the output is an adjusted p-value function that controls for type I errors. We extend this idea to a general setting where domain is a Riemannian manifolds. This requires new methodology such as how to define adjustment sets on product manifolds and how to approximate the test statistic when the domain has non-zero curvature. We propose to use permutation tests for inference and apply the procedure in three settings: a simulation on a "chameleon-shaped" manifold and two applications related to climate change where the manifolds are a complex subset of $S^2$ and $S^2 \times S^1$, respectively. We note the tradeoff between type I and type II errors: increasing the adjustment set reduces the type I error but also results in smaller areas of significance. However, some areas still remain significant even at maximal adjustment.

翻译：Pini和Vantini（2017）提出了区间检验程序，用于对区间域上定义的函数型数据进行局部推断，其输出为能控制第一类错误的调整后p值函数。我们将这一思想推广到域为黎曼流形的一般情形。这需要引入新方法，例如如何在乘积流形上定义调整集，以及如何在域具有非零曲率时近似检验统计量。我们建议使用置换检验进行推断，并在三种场景中应用该程序：在"变色龙形"流形上的模拟研究，以及两个与气候变化相关的实际应用——其中流形分别为$S^2$的复杂子集和$S^2 \times S^1$。我们注意到第一类错误与第二类错误之间的权衡：扩大调整集会降低第一类错误，但也会导致显著区域减小。然而，即使在最大调整条件下，某些区域仍保持显著性。