Sequential Detection and Identification of a Change in the Distribution of a Markov-Modulated Random Sequence

The problem of detection and identification of an unobservable change in the distribution of a random sequence is studied via a hidden Markov model (HMM) approach. The formulation is Bayesian, on-line, discrete-time, allowing both single- and multiple- disorder cases, dealing with both independent and identically distributed (i.i.d.) and dependent observations scenarios, allowing for statistical dependencies between the change-time and change-type in both the observation sequence and the risk structure, and allowing for general discrete-time disorder distributions. Several of these factors provide useful new generalizations of the sequential analysis theory for change detection and/or hypothesis testing, taken individually. In this paper, a unifying framework is provided that handles each of these considerations not only individually, but also concurrently. Optimality results and optimal decision characterizations are given as well as detailed examples that illustrate the myriad of sequential change detection and identification problems that fall within this new framework

Approximation of discrete phase-type distributions

The analysis of discrete stochastic models such as generally distributed stochastic Petri nets can be done using state space-based methods. The behavior of the model is described by a Markov chain that can be solved mathematically. The phase-type distributions that are used to describe non-Markovian distributions have to be approximated. An approach for the fast and accurate approximation of discrete phase-type distributions is presented. This can be a step towards a practical state space-based simulation method, whereas formerly this approach often had to be discarded as unfeasible due to high memory and runtime costs. Discrete phases also fit in well with current research on proxel-based simulation. They can represent infinite support distribution functions with considerably fewer Markov chain states than proxels. Our hope is that such a combination of both approaches will lead to a competitive simulation algorithm. 1

Combining Proxels and Discrete Phases

The analysis of discrete stochastic models such as generally distributed stochastic Petri nets can be done by using state spacebased methods. They describe the behavior of a model by a Markov chain that can be solved mathematically. Formerly this approach often had to be discarded as unfeasible due to high memory and runtime costs. The recently developed Proxel-based algorithm is a state space-based simulation method, and has already performed well in several application fields. Experiments suggest, that the selective use of discrete phase approximations could further improve the method, because they can often represent infinite support distribution functions with considerably fewer Markov chain states than proxels. By replacing certain on-the-fly proxel approximations by predetermined phase-type approximations, the total runtime and memory requirement of the simulation method could be drastically reduced for some test models. An efficient algorithm for the approximation of discrete phase-type distributions based on common optimization methods was recently introduced. The formal inclusion of discrete phases into the proxel paradigm is another step toward a practically usable state spacebased simulation method. Our hope is that such a combination of both approaches will lead to a competitive simulation algorithm. 1