We investigate several rank-based change-point procedures for the covariance operator in a sequence of observed functions, called FKWC change-point procedures. Our methods allow the user to test for one change-point, to test for an epidemic period, or to detect an unknown amount of change-points in the data. Our methodology combines functional data depth values with the traditional Kruskal Wallis test statistic. By taking this approach we have no need to estimate the covariance operator, which makes our methods computationally cheap. For example, our procedure can identify multiple change-points in $O(n\log n)$ time. Our procedure is fully non-parametric and is robust to outliers through the use of data depth ranks. We show that when $n$ is large, our methods have simple behaviour under the null hypothesis.We also show that the FKWC change-point procedures are $n^{-1/2}$-consistent. In addition to asymptotic results, we provide a finite sample accuracy result for our at-most-one change-point estimator. In simulation, we compare our methods against several others. We also present an application of our methods to intraday asset returns and f-MRI scans.
翻译:我们的方法允许用户测试一个变化点,测试一个流行病期,或者检测数据中的未知变化点。我们的方法将功能性数据深度值与传统的Kruskal Wallis测试统计结合起来。我们采用这种方法不需要估计共变点操作器,这使得计算方法变得低廉。例如,我们的程序可以确定一个变化点(n\log n)时间的多个变化点。我们的程序是完全非参数性的,并且通过使用数据深度等级对外端进行强力。我们表明,当美元大时,我们的方法在无效假设下的行为很简单。我们还表明,FKWC的变更点程序是$n ⁇ -1/2} 美元。我们除了得出乐观的结果外,我们还为我们最接近的变更点测量器提供了有限的抽样准确性结果。在模拟中,我们比较了我们内部资产回报的方法。我们目前还对照了其他方法进行扫描。