Boris Brodsky

Nonparametric methods in change-point problems

Preface. Introduction: Goals and problems of change point detection. 1. Preliminary considerations. 2. State-of-the-art review. 3. A posteriori change point problems. 4. Sequential change point detection problems. 5. Disorder detection of random fields. 6. Applications of nonparametric change point detection methods. 7. Proofs, new results and technical details. References.

Minimax Methods for Multihypothesis Sequential Testing and Change-Point Detection Problems

Abstract In this paper a unified methodological approach to sequential testing of many composite hypotheses and multi-decision change-point detection for composite alternatives is proposed. New performance measures for methods of hypotheses testing and change-point detection are introduced. Theoretical lower bounds for these performance measures are proved that do not depend on methods of sequential testing and detection. Minimax tests are proposed for which these lower bounds are attained asympototically as decision thresholds tend to infinity. Results of Monte Carlo experiments are given.

Asymptotically Optimal Methods of Early Change-Point Detection

Abstract In this article the problem of early change-point detection is considered. The main difference from traditional sequential change-point detection problems consists in the nature of changes: we consider both gradual and abrupt changes and wish to detect instants of the beginning of these changes as soon as possible on condition that false alarms are few. We prove the theoretical informational inequalities for the main performance characteristics of early change-point detection methods that help us to consider the asymptotically optimal methods. Our criterion of asymptotic optimality does not coincide with traditional criteria of Shiryaev (1963), Lorden (1971), and Pollak (1985) but allows us to consider both univariate and multivariate nonstationary stochastic models. Besides the theoretical analysis of univariate and multivariate models with changes, we present results of the Monte Carlo study of the proposed methods.

Sequential Detection of Change-Points in Linear Models

Abstract A new method for sequential detection of change-points in multivariate linear models is proposed. The main performance characteristics of this method are analyzed theoretically for finite sample volumes. Comparison with other well known methods for sequential detection of structural changes in linear models is carried out via Monte Carlo tests. Practical applications for the analysis of stability of the German quarterly model of demand for money (1961–1995) and the Russian monthly model of inflation (1994–2005) are considered.

Sequential Detection and Estimation of Change-Points

Abstract The problem of sequential detection and estimation of a change-point is considered. The ‘large parameter’ approach for solving this problem is proposed. A two-stage method is designed according to this approach, which includes the nonparametric version of the CUSUM procedure for sequential change-point detection and the modified Kolmogorov-Smirnov test for the retrospective estimation of the detected change-point. The asymptotic effectiveness of this method is proved for dependent observations (Theorems 2.1 and 3.1) when the large parameter infinitely increases. Monte Carlo study of this method for the Gaussian case (changes in mean and dispersion) is performed. The a priori theoretical lower bounds are proved for new performance measures in sequential change-point detection and retrospective estimation (Theorems 5.1 and 5.2) and the asymptotic optimality of the proposed method is demonstrated.

Sequential detection of switches in models with changing structures

Models with a changing structure, and in particular with switches in a model’s parameters, are often used in applications. In this paper we consider sequential detection of such fully stochastic switches of parameters and propose the nonparametric and asymptotically optimal method for detection of switches. We prove that type 1 and type 2 errors of detection converge to zero exponentially for dependent sequences of observations satisfying Cramer’s and -mixing conditions as the number of observations increases to infinity. The asymptotic optimality result is proved for the main performance criterion of detection (convergence of estimates to the true values of parameters). Results of experimental testing of the proposed method and its comparison with other detection techniques are given.

Discussion on “Is Average Run Length to False Alarm Always an Informative Criterion?” by Yajun Mei

Abstract Criteria of optimality in change-point detection are discussed.

Applications of Nonparametric Change-Point Detection Methods

The nonparametric methods of change-point detection developed in this book have been tested many times by statistical simulation. However, the most interesting point, in our opinion, is running the methods on real statistical data. We had the opportunity to verify our results on three practical problems: (1) Computer analysis of historical texts; (2) Computer-aided space data processing: the analysis of information obtained from an orbiting satellite; (3) Computer analysis of geophysical information.

A Posteriori Change-Point Problems

In this chapter, a posteriori change-point problems are considered. As was already mentioned in Chapter 2, the a posteriori change-point problem can be formulated in the following way: given a realisation of a random sequence X, the hypothesis of its stochastic homogeneity has to be proved. If this hypothesis is rejected, then estimates of change-points have to be obtained. Following the abovementioned general approach to disorder detection (see 2.4), almost everywhere in this chapter a posteriori change-point problems are considered in a standard situation when an unknown shift of the mean value of a random sequence X occurs (other changing characteristics of distributions are considered as nuisance parameters).

Disorder Detection of Random Fields

In solving many problems relating to the recognition of complex signals, in the synthesis of artificial intelligence systems and in experimental data processing, the problem often arises to detect homogeneous regions of a random field. The crux of this problem is as follows. Suppose that a domain of definition J of a random field is divided into k disjoint sets I i , i = 1,..., k of statistical homogeneity in characteristics of this random field. The problem is to reconstruct the boundaries of the sets I i using observations of a random field in the domain J.