Slow and Drastic Change Detection in General HMMs Using Particle Filters with Unknown Change Parameters

We study the change detection problem in general HMMs, when change parameters are unknown and the change could be gradual (slow) or sudden (drastic). Drastic changes can be detected easily using the increase in tracking error or the negative log of the observation likelihood conditioned on past observations (OL). But slow changes usually get missed. We propose a statistic for slow change detection called ELL which is the conditional Expectation of the negative Log Likelihood of the state given past observations. We show asymptotic stability (stability under weaker assumptions) of the errors in approximating the ELL for changed observations using a particle filter that is optimal for the unchanged system. It is shown that the upper bound on ELL error is an increasing function of the “rate of change” with increasing derivatives of all orders, and its implications are discussed. We also demonstrate, using the bounds on the errors, the complementary behavior of ELL and OL. Results are shown for simulated examples and for a real abnormal activity detection problem.