A Darling–Erdős-type CUSUM-procedure for functional data

Abstract

This article considers testing for mean-level shifts in functional data. The class of the famous Darling-Erdős-type cumulative sums (CUSUM) procedures is extended to functional time series under short range dependence conditions which are satisfied by functional analogues of many popular time series models including the linear functional AR and the non-linear functional ARCH. We follow a data driven, projection-based approach where the lower-dimensional subspace is determined by (long run) functional principal components which are eigenfunctions of the long run covariance operator. This second-order structure is generally unknown and estimation is crucial -it plays an even more important role than in the classical univariate setup because it generates the finitedimensional subspaces. We discuss suitable estimates and demonstrate empirically that altogether this change-point procedure performs well under moderate temporal dependence.Moreover, Darling-Erdős-type change-point estimates based on (long run) functional principal components as well as the corresponding »fully-functional« counterparts are provided and the testing procedure is finally applied to publicly accessible electricity data from a German power company.

Sequential block bootstrap in a Hilbert space with application to change point analysis

“…For functional data, Berkes et al () have developed estimators and tests for a change point in the mean, which is extended by Hörmann & Kokoszka (), Aston & Kirch (), Horvath, Kokoszka, & Rice (), and Torgovitski () to weakly dependent data. They reduce the dimension of the observations with the help of functional principal components, whereas—motivated by Theorems 1 and 2—we consider the test statistic based on full functional data.…”

“…In the simulation study we have reproduced the implementation mode of Torgovitski (), using the R‐package fda. The are created from Brownian bridges, which are then transformed to functional data objects by the R‐function Data2fd, using 25 B‐spline functions.…”

“…Table shows the empirical power of the test. We consider the same alternative as Torgovitski (), that is and . The power is very good and decreases only slightly as the dependence grows.…”

“…An alternative to our test is an adaption of the fpca‐based method of Berkes et al () to weakly dependent data. The behaviour of the test in the case of a FAR(1)'model was investigated in Torgovitski () using d principal components and a Bartlett‐type estimator of the long‐run covariance matrix with bandwidth h . We have copied the results of the simulation study of Torgovitski () and added them to Tables and .…”