Detection and Detectability of Changes in a Multi-parameter Exponential Distribution

Abstract

The security of safety-critical systems requires special algorithmic tools detecting suddenly arriving faults, attacks, or intrusions. It is assumed that such anomalies lead to serious degradation of the system safety only if these anomalies are detected with the detection delay greater than the required time-to-alert. If the anomalies are detected with the delay smaller than or equal to the required time-to-alert, the monitored system can be reconfigurable/adaptable without compromising safety. The goal of this paper is to study the reliable sequential detection of transient changes in a multi-parameter exponential distribution. The sequentially observed data are represented by a sequence of independent random vectors with the exponentially distributed components. The parameter vector consists of the expected values of exponentially distributed random variables (components of the vectors). This parameter vector changes at an unknown time (changepoint). It is necessary to reliably detect this changepoint. The considered optimality criterion minimizes the worst-case probability of missed detection provided that the worst-case probability of false alarm during a certain period is upper bounded. The statistical test discussed in the paper is optimal with respect to this criterion in a subclass of truncated sequential probability ratio tests. Special attention is paid to the problem of change detectability. The maximum/minimum contrast vectors of post-change parameters are defined w.r.t. the vector of pre-change parameters by using a quadratic maximization/minimization problem. An application of the obtained results to the detection of spectral changes is also considered.