Mean Shift Detection for State Space Models

In this paper we develop and validate a procedure for testing against a shift in mean in the observa-tions and hidden state sequence of state space models with Gaussian noise. State space models are popular for modelling stochastic networks as they allow to take into account that observations of the true state of a sys-tem may be corrupted by measurement noise (usually, a Gaussian noise process is assumed). Although state space models are very general, they are still relatively tractable in that the true system state can be estimated efficiently by a recursive procedure known as Kalman filtering. State space models can be regarded as a special type of hidden Markov model. As such, they are a flexible modelling tool that has been found useful, for example, for modelling road networks (Stathopoulos and Karlaftis, 2003) to account for uncertainty in the measurement of travel times. For instance, we may assume that travel times have to be estimated from flow and occupancy data. An increase in the unobserved mean travel time can be caused by traffic congestion; a shift in the mean value of the observations on the other hand could indicate a bias of the sensors. State space models can also be used to model communication networks: Suppose that the current state of a channel (e.g. measured by the probability of packet loss) is not observed directly, but has to be inferred from the received package flow. A change in the mean value of the hidden state sequence, or a change in the mean value of the received package flow can deteriorate the performance of the network if it remains unrecognised. This motivates us to investigate procedures for testing against a shift in mean in the observations and hidden state sequence of state space models. The objective is to detect a change as quickly as possible while keeping the ratio of false alarms at a pre-specified low level. Since the observations are generally not independent, in (Basseville and Nikiforov, 1993) a cumulative sum (CUSUM) procedure is applied to the (independent) sequence of innovations, which is obtained as a byproduct from Kalman filter estimation of the hidden states. That is, a log-likelihood ratio (LLR) test statistic is used and an alarm is raised as soon as this test statistic exceeds a certain threshold that is assumed to be given. Change point detection for state space models has also been considered in (Lai and Shan, 1999) for the case where the size of the mean shift is unknown, in which case a generalized LLR test can be applied. In this paper we tackle the question of how the threshold of the sequential LLR test can be chosen when the shift size is assumed to be known. In practice, the latter assumption can be dealt with by realizing that typically there will be a minimum change size that is of interest from an engineering perspective, and that can thus be used as input for the model. Based on this assumption we can identify the appropriate level of the threshold based on approximations of the false alarm probability – essentially the probability that a random walk process exceeds a given threshold on an interval. A persistent change in the mean value of the observations results in a dynamic change in the mean value of the innovations, which are therefore not identically distributed after the change point. However, it follows from the stability properties of the Kalman filter that under weak conditions the magnitude of the shift converges to a constant. This allows for large-deviations (LD) approximations as well as approximations motivated by a functional central limit theorem (CLT). LD approximations to the false alarm probability have been considered in (Bucklew, 1985; Ellens et al., 2013; Kuhn et al., 2014) for testing i.i.d. and vector autoregressive moving average (VARMA) models. CLT approximations were motivated, for example, in Siegmund (1985). We compare the numerical performance of the tests under both types of limiting regimes with respect to the false alarm probability and the detection delay.