Anomaly Search of a Hidden Markov Model

We address the problem of detecting an anomalous process among a large number of processes. At each time t, normal processes are in state zero (normal state), whereas the abnormal process may exist in either state zero (normal state) or state one (abnormal state), with these states remaining hidden. The transitions between states for the abnormal process follow a Markov chain over time. During each time step, observations can be drawn from a selected subset of processes. Each probed process generates an observation based on its hidden state, following a typical distribution under state zero or an abnormal distribution under state one. The objective is to design a sequential search strategy that minimizes the expected detection time, subject to an error probability constraint. In contrast to prior studies on related models that focused on i.i.d. observations, the new model leads to the detection of a hidden Markov model (HMM) of anomaly, introducing significant challenges in both algorithm design and theoretical analysis. We introduce a novel sequential search strat-egy, referred to as the Anomaly Detection under Hidden Markov (ADHM) algorithm, and show that ADHM is asymptotically optimal as the error probability approaches zero. Simulation results demonstrate the superior performance of ADHM over existing methods within a finite regime.