Impact of Compensation Coefficients on Active Sequential Change-Point Detection

Under a general setting of active sequential change-point detection problems, there are $p$ local streams in a system but we are only able to take observations from $q$ out of these $p$ local streams at each time instant due to the sampling control constraint. At some unknown change time, an undesired event occurs to the system and changes the local distributions from $f$ to $g$ for a subset of $s$ unknown local streams. The objective is how to adaptively sample local streams and decide when to raise a global alarm, so that we can detect the correct change as quickly as possible subject to the false alarm constraint. One efficient algorithm is the TRAS algorithm proposed in Liu et al. (2015) which incorporates an idea of compensation coefficients for unobserved data streams. However, it is unclear how to choose the compensation coefficients suitably from theoretical point of view so as to balance the trade-off between the detection delay and false alarm. In this article, we investigate the impact of compensation coefficients on TRAS algorithm. Our main contributions are two-folded. On the one hand, under the general setting, we prove that if compensation coefficient is larger than $I(f,g)q/(p-q)$ , where $I(f,g)$ is the Kullback-Leibler divergence, then the TRAS algorithm is suboptimal in the sense of having too large detection delays. On the other hand, under the special case of $q=s=1$ , if compensation coefficient is small enough, then the TRAS algorithm is efficient to detect when the change occurs at time $\nu =0$ . While it remains an open problem to develop general asymptotic optimality theorems, our results shed lights how to tune compensation coefficients suitably in real world application, and extensive numerical studies are conducted to validate our results.