Change Point Estimation and Detection in Hidden Markov Models(2/2)

In the modern information age, with the advance of sensor network technology, massivevolumes of sensor data are being generated and explored at an unprecedented pace and scopein science, bioinformatics, genetics, engineering, finance, economics and homeland securityapplications, where sensors may broadly include wireless sensor networks, in-situ sensorinfrastructures and remote sensors. In many real-world applications such as biosurveillance,quality control, key infrastructure or internet traffic monitoring, we have an ultimateobjective to develop multi-sensor network systems that do not simply collect \raw" data,but rather help us to learn valuable knowledge or make intelligent decisions quickly, e.g.,anomaly detection, signal detection.To make a linkage between the off line data set and the streaming data set, in this project,we first consider the problem of change point estimation with fixed sample size. It is knownthat one setting of change point estimation is equivalent to determine the number of hiddenstates for a HMM. This is also called model selection problem. However, model selectionproblem in HMM, which has not yet been satisfactorily solved, especially for the highlyused Gaussian HMM with heterogeneous covariance. Here we will propose a consistent methodfor determining the number of hidden states of HMM based on the marginal likelihood, whichis obtained by integrating out both the parameters and hidden states. Moreover, other thanusing Bayesian model selection method, we will try to prove the consistency property ofBayesian information criterion (BIC) in HMM, which is a long-standing open problem in modelselection literature. Next, we will extend these results to Markov switching models, bothin means and volatilities.In the second part of this research proposal, we will investigate quickest detectionproblems in HMM sensor networks. Instead of proposing some ad hoc methodologies, we areinterested in developing fundamental information bounds in certain scenarios and findingschemes that achieve these bounds asymptotically. Given the difficulty and complexity ofthe general problem, in this research proposal, we will focus on some simple but usefulsituations, and the results will shed light on more complicated real-world problems. Tobe more specific, we will study the following matters: First, we will investigate the casewhen both pre- and post-change parameters, 0 and 1 , are completely specified. Second,we will investigate the scenario when the magnitude of the local post-change is specified,but we do not know the directions of changes. Third, we will investigate the scenario whenboth direction and magnitude of the change are unknown. In other words, the post-changeparameter 1 of the HMM is unknown. Last, we will investigate the robust quickest detectionunder the special scenario when one hidden state, say, { 0} n X , is dominant, and we donot have enough observations to estimate the parameters under other non-dominant hiddenstates.The objective of this proposal is to offer theoretical foundation and a host of efficientscalable methodologies for change point estimation and quickest detection in hidden Markovmodels.Bayesian Information Criterion (BIC), consistency, change-point, CUSUM, Kullback-Leibler(KL) divergence, marginal likelihood, model selection, normalizing constant, score test,sensor networks, sequential detection.