Multi-Sensor Slope Change Detection

We develop a mixture procedure for multi-sensor systems to monitor parallel streams of data for a change-point that causes a gradual degradation to a subset of data streams. Observations are assumed initially to be normal random variables with known constant means and variances. After a change-point the observations in a subset of the streams of data have increasing or decreasing mean values. The subset and the slope changes are unknown. Our procedure uses a mixture statistics which assumes that each sensor is affected with probability $p_0$. Analytic expressions are obtained for the average run length (ARL) and the expected detection delay (EDD) of the mixture procedure, which are demonstrated to be quite accurate numerically. We establish asymptotic optimality of the mixture procedure. Numerical examples demonstrate the good performance of the proposed procedure. We also discuss an adaptive mixture procedure using empirical Bayes. This paper extends our earlier work on detecting an abrupt change-point that causes a mean-shift, by tackling the challenges posed by the non-stationarity of the slope-change problem.