Quickest Detection of a Random Pulse in White Gaussian Noise

A class of stochastic processes characterized by rapid transitions in their structure is considered and the quickest detection of such transitions is studied in a Bayesian framework. The emphasis is on stochastic processes consisting of a randomly arrived causal pulse (possibly with a set of random parameters such as amplitude and duration) and an additive white Gaussian noise. In this model, the pulse shape and the prior joint density of the arrival time and other random parameters are assumed known. The task of quickest detection in this paper is described mathematically by minimizing the expected detection error. The detection error is represented by a nonlinear function of the distance between the actual transition time and its associated detection time. The assumptions on this function are fairly mild and allow to flexibly design its shape for a desired trade-off between the detection delay and the false alarm rate. Two special cases of such design are well known error measures: mean squared and mean absolute error. The quickest detection problem-a subclass of optimal stopping time problems-is formulated as a stochastic optimal control problem and is resolved using dynamic programming. The optimal detection rule is determined in terms of the solution of an integral equation that cannot be directly solved due to its complexity. This equation is later used to develop a class of suboptimal detection rules and a lower bound on the minimum error. Using this lower bound, it is shown for a numerical example that the suboptimal detector is nearly optimal.

[1]  H. Sussmann.  On the Gap Between Deterministic and Stochastic Ordinary Differential Equations , 1978 .

[2]  G. Carter.  Coherence and time delay estimation , 1987, Proceedings of the IEEE.

[3]  О стохастических моделях и оптимальных методах в задачах скорейшего обнаружения@@@On Stochastic Models and Optimal Methods in the Quickest Detection Problems , 2008 .

[4]  J. Bradley,et al.  Delay estimation by expected value , 1984 .

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

[6]  B. Darkhovski.  Nonparametric methods in change-point problems: a general approach and some concrete algorithms , 1994 .

[7]  R. L. Stratonovich.  A New Representation for Stochastic Integrals and Equations , 1966 .

[8]  Olympia Hadjiliadis,et al.  Quickest detection in a system with correlated noise , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[9]  Aleksey S. Polunchenko,et al.  State-of-the-Art in Sequential Change-Point Detection , 2011, 1109.2938.

[10]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[11]  On the Best 2-CUSUM Stopping Rule for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model@@@On the Best 2-CUSUM Stopping Rule for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model , 2008 .

[12]  Olympia Hadjiliadis,et al.  Optimal and Asymptotically Optimal CUSUM Rules for Change Point Detection in the Brownian Motion Model with Multiple Alternatives , 2006 .

[13]  G. Carter,et al.  The generalized correlation method for estimation of time delay , 1976 .

[14]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

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

[16]  J. Walkup,et al.  Statistical optics , 1986, IEEE Journal of Quantum Electronics.

[17]  Pak-Chung Ching,et al.  Approximate maximum likelihood delay estimation via orthogonal wavelet transform , 1999, IEEE Trans. Signal Process..

[18]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal Stopping and Free-Boundary Problems , 2006 .

[19]  A. Shiryaev,et al.  Quickest detection of drift change for Brownian motion in generalized Bayesian and minimax settings , 2006 .

[20]  G. Moustakides,et al.  State-of-the-Art in Bayesian Changepoint Detection , 2010 .

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

[22]  Sean P. Meyn,et al.  Minimax Robust Quickest Change Detection , 2009, IEEE Transactions on Information Theory.

[23]  Arash Komaee,et al.  Nonlinear Filters for Bayesian Estimation of Pulse Arrival Time in Additive White Gaussian Noise , 2011, IEEE Transactions on Signal Processing.

[24]  H. Vincent Poor,et al.  One Shot Schemes for Decentralized Quickest Change Detection , 2008, IEEE Transactions on Information Theory.

[25]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[26]  A. Shiryaev.  On Optimum Methods in Quickest Detection Problems , 1963 .

[27]  Arash Komaee.  Real-time pulse arrival detection in the presence of white Gaussian noise , 2009, 2009 43rd Annual Conference on Information Sciences and Systems.

[28]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[29]  Karatzas Ioannis.  A note on Bayesian detection of change-points with an expected miss criterion , 2003 .

[30]  G. Carter.  Time delay estimation for passive sonar signal processing , 1981 .