Hidden Markov Models with time-continuous output behavior
Evelyn Dittmer · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2009
In this thesis a set of procedures for the analysis of time series was developed. The models introduced here are based on the concept of Hidden Markov models. A hidden Markov model (HMM) consists of two stochastic processes. However, only one of these is observable. The HMM variants presented herein have been developed with regard to a prospective application to biomolecular time series. Therefore, the investigated time series are realizations of processes that can be characterized as follows: They jump between metastable states. The correlation times are small with respect to the exit times from a metastable state. On a time scale, chosen in such a way that the process is Markovian, these jumps occur instantaneously. That is: Transitions between metastable states in equilibrium take place in one or very few time steps. By means of the presented methods in particular, the question how metastable states can be distinguished by kinetic patterns is addressed. The local dynamics are modeled either by a space-discrete Markov jump process or by a continuous diffusion process. Both concepts are discussed by means of several examples. Particularly we focussed on the issue of generator estimation. The combination of the generator estimation with the concept of the hidden Markov model is new. It allows for analyzing time-continuous processes with standard HMM techniques. Especially the time-continuity of the model makes the analysis of time series with varying time lags possible. Furthermore, a model with Markov jump output process has the advantage that the process is determined by the generator matrix only. Hence no additional assumption about the distribution of the data is required. Beyond this in the examples we observed that even if the box-discretization is rather coarse- grained, the local dynamics still can be expressed satisfactorily by an HMM- MJP. In the scope of this thesis HMM-MJP was applied to small systems, generated by Smoluchowski dynamics, by a discrete generator or by an HMM itself. However, the algorithms are applicable to the high-dimensional case. How HMM-MJP performs in the application to larger systems - such as the simulation of biomolecules - has to be clarified in further investigations. Difficulties arising with larger systems are on the one hand computational costs and on the other hand cumulations of small entries in the generator matrix. Too many small entries close to zero can lead to numerical instabilities. One approach to handle these problems is the restriction of the state space as described in the examples 7.2.5 and 7.2.6.