Parameter estimate of a hidden Markov chain.

摘要

Hidden Markov Models (HMM) are of considerable interest for science and for various applications. They consist of a Markov chain with "hidden" states and emissions which are statistically dependent on the states but can be observed. The model is parameterized by two conditional probability matrices, the transition and emission matrices.; Among the algorithms used for the parameter estimate of HMM, Baum-Welch (B-W) algorithm is by far the most popular algorithm. However, it has some well-known shortcomings. For example, it is only guaranteed to find a local maximum, with a strong dependency on the initial parameters chosen. The literature notes an "overfitting" problem, in which the B-W estimate gives high likelihood to a given observation sequence, but low likelihood to other observation sequences of the same hidden Markov chain. As a consequence of these this researcher has shown that usually the B-W estimator is inconsistent for the simplest possible case of two hidden states and two emission states, and with a generic choice of parameters and generic observation sequences.; The dissertation also provides an algorithm for computation of the least square error (LSE) estimate of the hidden Markov chain. The LSE estimate is consistent by definition, needs only one time computation, and again with the simplest possible case of HMM described above this researcher has demonstrated the estimates are remarkably closer to the actual parameters, with much better results than what could typically be obtained using the B-W algorithm. A possible reason for the LSE estimation not being very popular regarding HMM, in spite of its much superior quality of its estimates, could be the computational complexity it requires. Although a straightforward computation could require the complexity that increases exponentially with respect to the sequence length, this researcher has shown that a polynomial complexity (with still exponential complexity in the state size) can be achieved using an algorithm proposed in this dissertation, making the LSE estimation quite feasible in some applications such as the ones related to precipitation, heart rate monitoring, and so on.