University of Twente Student Theses


Obtaining bounds on the number of hidden Markov states in a tracking context

Severijn, S.A. (2020) Obtaining bounds on the number of hidden Markov states in a tracking context.

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Abstract:For the classification of moving objects based on their trajectories, one wants to be able to build tracking models which are as accurate as possible. For this purpose, double hidden Markov models are used. The goal of this research is to find a method or a combination of methods to determine the optimal number of hidden Markov states for a double hidden Markov model. The method most looked into in this thesis, called the autocovariance method in this report, relates the number of Markov states of the considered model to the orders of a vector autoregressive moving average model with the same autocovariance structure as the considered model. This method provides a theoretical lower bound on the number of hidden Markov states, although this bound is not guaranteed in practice due to the method of determining the autoregressive and moving average orders in this research. In this report, the existing lower bounds are generalised for a wider class of models and for differenced time series, useful in the case of cointegrated time series. The bounds of the existing literature and introduced in this report become weak if no assumptions are made on the correlation of the time series with the Markov Process. More promising bounds are available if there is no correlation. From synthetic experiments, it appeared that these bounds perform best for processes where the hidden Markov chain stays, on average, in the same hidden Markov state for longer periods of time, for higher dimensional problems, for lower orders of the number of hidden Markov states, for a higher number of observations, for problems where the Markov states are sufficiently distinctive and when the process does not contain Markov states in which the process is (highly) nonstationary. For differenced time series, the number of different Markov states effectively is the original number of Markov states squared, which can often not all be distinguished. Therefore, this method is in these cases often not useful in practice, perhaps except in the case when one wants to find out if one needs multiple hidden Markov states at all. It is preferable to use the non-differenced method if possible. The autocovariance method is computationally inexpensive and therefore suitable to be incorporated in the model building process, although it is advisible to use the estimate of the method in combination with other information, such as estimates from information criteria which can be used as upper bounds.
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics MSc (60348)
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