Quickest Detection of Parameter Changes in Stochastic Regression: Nonparametric CUSUM

We consider the problem of detecting abrupt parameter changes in a stochastic regression with unknown noise distribution. The process changes at some unknown point of time. Under general conditions on the regression function and unknown distributions of observations before and after the disruption, the paper develops a nonparametric cumulative sum procedure (CUSUM). Unlike likelihood-based CUSUM algorithms, constructed mostly on log-likelihood ratio statistics, we use a special system of basic statistics in Page’s procedure. By applying a sequential sampling scheme, which measures time in terms of accumulated Kullback–Leibler (K-L) divergence, we come to a system of statistics with the martingale properties close to those of the log-likelihood ratios. The proposed approach suggests also an alternative performance criterion in the analysis of the procedure by replacing the expected detection delay by the corresponding K-L divergence. We show that, under the false alarm probability constraint, the nonparametric CUSUM rule is optimal in the sense that it ensures the logarithmic asymptotic for the detection delay.

[1]  Michèle Basseville,et al.  Detection of abrupt changes , 1993 .

[2]  M. Pollak.  Optimal Detection of a Change in Distribution , 1985 .

[3]  J. Steinebach,et al.  On the detection of changes in autoregressive time series I. Asymptotics , 2007 .

[4]  Moshe Pollak,et al.  Detecting a change in regression: first-order optimality , 1999 .

[5]  P. K. Bhattacharya,et al.  A Nonparametric Control Chart for Detecting Small Disorders , 1981 .

[6]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[7]  P. Meyer.  Probability and potentials , 1966 .

[8]  Tze Leung Lai,et al.  Fixed Accuracy Estimation of an Autoregressive Parameter , 1983 .

[9]  V. Veeravalli,et al.  General Asymptotic Bayesian Theory of Quickest Change Detection , 2005 .

[10]  H. Vincent Poor,et al.  Quickest detection in coupled systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[11]  Rudolf B. Blazek,et al.  Detection of intrusions in information systems by sequential change-point methods , 2005 .

[12]  G. Lorden.  PROCEDURES FOR REACTING TO A CHANGE IN DISTRIBUTION , 1971 .

[13]  Tze Leung Lai,et al.  Information Bounds and Quick Detection of Parameter Changes in Stochastic Systems , 1998, IEEE Trans. Inf. Theory.

[14]  George V. Moustakides,et al.  Optimality of the CUSUM procedure in continuous time , 2003 .

[15]  B. E. Brodsky,et al.  Non-Parametric Statistical Diagnosis , 2000 .

[16]  E. S. Page.  A test for a change in a parameter occurring at an unknown point , 1955 .

[17]  E. S. Page.  On problems in which a change in a parameter occurs at an unknown point , 1957 .

[18]  A. Shiryaev.  Quickest Detection Problems: Fifty Years Later , 2010 .

[19]  O. Hadjiliadis,et al.  Robustness of the N-CUSUM stopping rule in a Wiener disorder problem , 2014, 1410.8765.

[20]  A. Willsky,et al.  A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems , 1976 .

[21]  T. Lai.  Sequential changepoint detection in quality control and dynamical systems , 1995 .

[22]  Qiwei Yao.  Asymptotically optimal ditiction of a change in a linear model , 1993 .

[23]  G. Moustakides.  Optimal stopping times for detecting changes in distributions , 1986 .

[24]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .