Invariant tests for functional data with application to an earthquake impact study

Motivated by an earthquake impact study, this paper develops new tests with several invariant properties for functional data. For multi-sample functional ANOVA (mfANOVA), based on local polynomial regression, exact local and global ANOVA decompositions into within- and between-group of variations are obtained. A local mfANOVA test is developed to examine differences in a local neighborhood, and combining local quantities, a global mfANOVA test statistic is formed. We show that both the local and global mfANOVA test statistics are location, scale, and translation invariant, allow interchanging the order of smoothing and ANOVA projection, and have asymptotic F" id="MathJax-Element-1-Frame" role="presentation" style="font-size: 90%; display: inline-block; position: relative;" tabindex="0">distributions under the null hypotheses with the Gaussian assumption. This paper contributes to the literature by being the first, to our knowledge, to study mfANOVA tests with several invariant properties. Simulation studies are presented to compare the proposed global mfANOVA test with some existing procedures. Application to an earthquake impact study in Taiwan reveals that when an earthquake in 2016 resulted in closed highways, the patterns of traffic flows were significantly different between three time periods, before the earthquake, during, and after the repair period. The information could be useful in planning for disaster preparedness.