Optimality of CUSUM rule approximations in change-point detection problems: application to nonlinear state--space systems

The well-known cumulative sum (CUSUM) sequential rule for abrupt model change detection in stochastic dynamic systems relies on the knowledge of the probability density functions of the system output variables conditional on their past values and on the system functioning mode at each time step. This paper shows how to build an asymptotically optimal detection rule under the common average run length (ARL) constraint when these densities are not available but can be consistently estimated. This is the case for nonlinear state--space systems observed through output variables: for such systems, a new class of particle filters based on convolution kernels allows to get consistent estimates of the conditional densities, leading to an optimal CUSUM-like filter detection rule (FDR). Index Terms--Average run length (ARL) constraint, convolution kernel filter, cumulative sum (CUSUM) rule, model change detection, particle filter, state--space systems.