Claudia Kirch

TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain

A new time series bootstrap scheme, the Time Frequency Toggle (TFT)-Bootstrap, is proposed. Its basic idea is to bootstrap the Fourier coecients of the observed time series, and then back-transforming them to obtain a bootstrap sample in the time domain. Related previous proposals, such as the ‘surrogate data’ approach, resampled only the phase of the Fourier coecients, and thus had only limited validity. By contrast, we show that the appropriate resampling of phase and magnitude in addition to some smoothing of Fourier coecients yields a bootstrap scheme that mimics the correct second-order moment structure for a large class of time series processes. As a main result we obtain a functional limit theorem for the TFT-Bootstrap under a variety of popular ways of frequency domain bootstrapping. Possible applications of the TFT-Bootstrap naturally arise in change-point analysis and unit-root testing where statistics are frequently based on functionals of partial sums. Finally, a small simulation study explores the potential of the TFT-Bootstrap for small samples showing that for the discussed tests in change-point analysis as well as unit-root testing it yields better results than the corresponding asymptotic tests if measured by size and power.

Evaluating stationarity via change-point alternatives with applications to fMRI data

Functionalmagnetic resonance imaging (fMRI) is now a well-established technique for studying the brain. However, in many situations, such as when data are acquired in a resting state, it is difficult to know whether the data are truly stationary or if level shifts have occurred. To this end, change-point detection in sequences of functional data is examined where the functional observations are dependent and where the distributions of change-points from multiple subjects are required. Of particular interest is the case where the change-point is an epidemic change-a change occurs and then the observations return to baseline at a later time. The case where the covariance can be decomposed as a tensor product is considered with particular attention to the power analysis for detection. This is of interest in the application to fMRI, where the estimation of a full covariance structure for the three-dimensional image is not computationally feasible. Using the developed methods, a large study of resting state fMRI data is conducted to determine whether the subjects undertaking the resting scan have nonstationarities present in their time courses. It is found that a sizeable proportion of the subjects studied are not stationary. The change-point distribution for those subjects is empirically determined, as well as its theoretical properties examined.

Changepoints in Times Series of Counts

In this article, we discuss the problem of testing for a changepoint in the structure of an integer‐valued time series. In particular, we consider a test statistic of cumulative sum type for general Poisson autoregressions of order 1. We investigate the asymptotic behaviour of conditional least‐squares estimates of the parameters in the presence of a changepoint. Then, we derive the asymptotic distribution of the test statistic under the hypothesis of no change, allowing for the calculation of critical values. We prove consistency of the test, that is, asymptotic power 1, and consistency of the corresponding changepoint estimate. As an application, we have a look at changepoint detection in daily epileptic seizure counts from a clinical study.

Detecting and estimating changes in dependent functional data

Change point detection in sequences of functional data is examined where the functional observations are dependent. Of particular interest is the case where the change point is an epidemic change (a change occurs and then the observations return to baseline at a later time). The theoretical properties for various tests for at most one change and epidemic changes are derived with a special focus on power analysis. Estimators of the change point location are derived from the test statistics and theoretical properties are investigated.

Bootstrapping confidence intervals for the change-point of time series

We study an at-most-one-change time-series model with an abrupt change in the mean and dependent errors that fulfil certain mixing conditions. We obtain confidence intervals for the unknown change-point via bootstrapping methods. Precisely, we use a block bootstrap of the estimated centred error sequence. Then, we reconstruct a sequence with a change in the mean using the same estimators as before. The difference between the change-point estimator of the resampled sequence and the one of the original sequence can be used as an approximation of the difference between the real change-point and its estimator. This enables us to construct confidence intervals using the empirical distribution of the resampled time series. A simulation study shows that the resampled confidence intervals are usually closer to their target levels and at the same time smaller than the asymptotic intervals. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd

Detection of Changes in Multivariate Time Series With Application to EEG Data

The primary contributions of this article are rigorously developed novel statistical methods for detecting change points in multivariate time series. We extend the class of score type change point statistics considered in 2007 by Hušková, Prášková, and Steinebach to the vector autoregressive (VAR) case and the epidemic change alternative. Our proposed procedures do not require the observed time series to actually follow the VAR model. Instead, following the strategy implicitly employed by practitioners, our approach takes model misspecification into account so that our detection procedure uses the model background merely for feature extraction. We derive the asymptotic distributions of our test statistics and show that our procedure has asymptotic power of 1. The proposed test statistics require the estimation of the inverse of the long-run covariance matrix which is particularly difficult in higher-dimensional settings (i.e., where the dimension of the time series and the dimension of the parameter vector are both large). Thus we robustify the proposed test statistics and investigate their finite sample properties via extensive numerical experiments. Finally, we apply our procedure to electroencephalograms and demonstrate its potential impact in identifying change points in complex brain processes during a cognitive motor task.

Bootstrapping Sequential Change-Point Tests

Abstract In this article we propose some bootstrapping methods to obtain critical values for sequential change-point tests. We consider a change in the mean with i.i.d. errors. Theoretical results show the asymptotic validity of the proposed bootstrap procedures. A simulation study compares the bootstrap and the asymptotic tests and shows that the studentized bootstrap test behave generally better than asymptotic tests if measured by α- resp. β-errors and its run length.

A MOSUM procedure for the estimation of multiple random change points

In this work, we investigate statistical properties of change point estimators based on moving sum statistics, where we allow for random exogenous change points as are e.g. considered in Hidden Markov or regime switching models extending results for testing in a classical (deterministic) multiple change point situations. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates cannot be improved in general by deriving the limit distribution of properly scaled estimators under somewhat stronger assumptions. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples.

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.

Fourier--type tests involving martingale difference processes

ABSTRACT We develop testing procedures which detect if the observed time series is a martingale difference sequence. Furthermore, tests are developed that detect change–points in the conditional expectation of the series given its past. The test statistics are formulated following the approach of Fourier–type conditional expectations first proposed by Bierens (1982) and have the advantage of computational simplicity. The limit behavior of the test statistics is investigated under the null hypothesis as well as under alternatives. Since the asymptotic null distribution contains unknown parameters, a bootstrap procedure is proposed in order to actually perform the test. The performance of the bootstrap version of the test is compared in finite samples with other methods for the same problem. A real–data application is also included.