Automatic detection of significant areas for functional data with directional error control

In this paper, we propose a large-scale multiple testing procedure to find the significant sub-areas between two samples of curves automatically. The procedure is optimal in that it controls the directional false discovery rate at any specified level on a continuum asymptotically. By introducing a nonparametric Gaussian process regression model for the two-sided multiple test, the procedure is computationally inexpensive. It can cope with problems with multidimensional covariates and accommodate different sampling designs across the samples. We further propose the significant curve/surface, giving an insight on dynamic significant differences between two curves. Simulation studies demonstrate that the proposed procedure enjoys superior performance with strong power and good directional error control. The procedure is also illustrated with the application to two executive function studies in hemiplegia.