# Sequential Nonparametric Detection of High-Dimensional Signals under Dependent Noise

Prause, Annabel; Steland, Ansgar (Thesis advisor); von Sachs, Rainer (Thesis advisor); Gombay, Edit (Thesis advisor)

*Aachen / Publikationsserver der RWTH Aachen University (2015)* [Dissertation / PhD Thesis]

*Page(s): IX, 143 S. : graph. Darst.*

Abstract

This thesis deals with sequential nonparametric detection schemes for high-dimensionalsignals, i.e. with the identification of a change-point in a given data set of a discretelysampled signal, where we allow for dependent noise as error terms. Thus, the thesisgeneralizes a work of Pawlak and Steland, in which detection algorithms for univariatesignals were investigated, in two different ways: For one thing we consider vector-valuedsignals with univariate time component, for another thing we consider real-valued signalswith multivariate time component.Chapter 2 introduces appropriate function spaces, the so-called Skorohod spaces.These do not only contain continuous functions, but also functions with certain discontinuitiesand are hence a convenient framework for the proposed detection processes of thefollowing chapters.Chapter 3 completes the introductory part of the work by giving a short summary ofthe most important results of the work of Pawlak and Steland.Finally, in Chapter 4 we develop these results for vector-valued signals. This meansthat we first establish an appropriate multivariate detection process for which we thendetermine its asymptotic distribution under the null hypothesis as well as under the alternative.If the null hypothesis holds true, the observed signal and the given referencesignal coincide; if, however, the alternative holds true, the observed signal and the referencesignal differ.Chapter 5 serves as an introduction to the multivariate Riemann-Stieltjes integralplus two definitions of functions of bounded variation in several variables. We need thesenotions for the generalization of the results to signals with multivariate time componentwhich we do in Chapter 6. Here, we model the noise by dependent random fields. Moreover,we establish the asymptotic distribution under the null hypothesis and under thealternative.An important parameter that appears in the limit distributions is the asymptotic varianceof the random field which is, in general, unknown and hence needs to be estimated.Thus, in Chapter 7 we propose an appropriate consistent estimator for it. However, forsome dependence structures of the underlying random field the smallest possible rootmean square error (RMSE) is still quite high for what reason we introduce a furtherestimator which leads to a smaller RMSE in some models.In Chapter 8 we perform an extensive simulation study. First, we investigate thebehaviour of the two estimators of Chapter 7 for different dependence models of therandom field. Next, we consider the whole detection process of Chapter 6. We analysethe power of the algorithm as well as the type one and type two errors whereby for theasymptotic variance we partly use the true value and partly the estimators of Chapter 7.The thesis concludes with Chapter 9 where we consider further generalizations of theresults of Chapter 6 which should be subject for future research.

### Identifier

- URN: urn:nbn:de:hbz:82-rwth-2015-013522
- REPORT NUMBER: RWTH-2015-01352