On the Use of Neural Networks in the Generalized Likelihood Ratio Test for Detecting Abrupt Changes in Signals

With the advent of efficient algorithms and fast computers for training neural networks, it is now feasible to employ neural network predictors in the generalized likelihood ratio (GLR) test for the purpose of detecting abrupt nonstationary changes in the dynamics of a time series. We examine some of the special issues involved and present some simulation results validating the new hybrid algorithm. 1.0 Introduction It’s been almost 25 years since the generalized likelihood ratio (GLR) test was first applied to the detection of abrupt nonstationary changes in a signal. And yet, because of the computational cost of the GLR test, sophisticated nonlinear models such as neural networks have never been employed in this regard, even for off-line analysis. However, recent advances in both training algorithms and computer speed make both off-line and on-line implementations feasible for the first time. In this paper we take a first look at some of the unique issues that arise in employing neural networks in the GLR test. The basic signal segmentation problem is defined as follows: given a single realization of a piecewise ergodic random process, segment the data into contiguous stationary regions. We do not know how many stationary regions are contained in the data, nor the change points between regions. Because of this, the problem cannot be easily formulated as a global test between known hypotheses. The only possible approach then is to perform sequential detection of changepoints, which makes the off-line segmentation problem identical to the on-line case, reducing the problem to one of detecting changepoints between successive pairs of stationary regions. This problem was first addressed by Page (1955), who studied the case of a detecting a change between two known models for the data at an unknown change point. However, when both models and change point are unknown, the GLR test is required, and was first introduced for this purpose by Willsky and Jones (1976) and later Appel and Brandt (1982). We first review the GLR test and then examine the issues involved in using neural networks to model the signal dynamics. 2.0 The Generalized Likelihood Ratio Test For simplicity, we consider a scalar time series, xn, where the superscript n is the current time index. We wish to determine if a non-stationary change occurred at some time intermediate T, . The GLR test is a NeymanPearson test for deciding between two hypotheses: the null hypothesis, H0, that no change occurred and the posited hypothesis, H1, that a change did occur at some intermediate time. It differs from the standard likelihood ratio test in that the pdf’s are unknown and must be estimated directly from the data within their respective regions, as defined by the hypothesized change point time. However, because the changepoint time itself is not known, a search must be done for the most likely changepoint. The resulting optimized likelihood forms a decision function which is compared with a preset threshold at each time step n. When it exceeds the threshold a change is detected, the exact transition time is determined, and the algorithm is then reset and started anew. A stationary random process is completely characterized by its multivariate (across-time) pdf. Therefore, to simplify notation we define to be a vector of the history of the time series between times a and b. We assume that the time series both before and after the changepoint can be described by a parametric family of pdf’s, , characterized by the parameter set θ. However, we do not know the parameters of the pdf before the change, θ0, or after the change, θ1, nor the changepoint time, T. Nevertheless, we can still form the log-likelihood 1 T n ≤ < Xa b x ...x [ ] † = b a >