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Data Assimilation for Hidden non-Markov Models
Ma, Wenjun
Ma, Wenjun
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Abstract
This dissertation works on the state and parameter estimation for non-Markov state space models. Many people have contributed to the development and improvement of particle filtering methods for Markov models. It is well known that generic particle filtering methods suffer from degeneracy problem. A very common way to deal with this problem is resampling. However, in the resampling procedure, samples with large weights will be selected multiple times and samples with very small weights will be neglected at each time step. Therefore, resampling will lead to sample impoverishment after a few steps, which will further lead to inaccurate estimation for probability density estimation. A sampling method called block sampling introduced in \cite{DJ11} may reduce degeneracy of particle filters. Block sampling generates samples for multiple steps as a block of the target distribution rather than sampling only for one current step of the target. This idea is not only applicable to Markov models but also can be extended to non-Markov models to reduce the variance of particle weights at each step. For non-Markov state space models' state estimation, we will first extend sequential importance resampling, implicit particle filter, auxiliary particle filter and particle filter with block sampling to non-Markov models. Similar to particle filters for hidden Markov models, SIR, IPF, APF suffer from degeneracy after extension to hidden non-Markov models. For block sampling, forward filtering backward sampling framework will no longer be available due to the memory effects of the hidden process. Hence, we have to seek the solution in a different way. We observe that the design of the sampling method is crucial to block sampling. Block sampling does not guarantee the improvement of the performance of particle filter in practice. Another important application of particle filter is estimating unknown parameters in state and observation equations. A commonly used approach is maximization likelihood estimation (MLE) approach. Instead of applying MLE method directly, a well known method is expectation maximization (EM) method. This method is essentially equivalent to MLE. The difference is that EM is a general method of finding the MLE of unknown parameters of an underlying distribution from an incomplete data set. For applying EM method with particle filter to non-Markov models, one needs to smooth the particles. However, this process is infeasible for hidden non-Markov models. This dissertation will use Particle Gibbs with ancestor sampling (pGAS) method to estimate unknown parameters in state and observation equations for non-Markov models. SMC method has its limitation, that is degeneracy along with time. MCMC method does not have degeneracy issue. However, it is difficult to construct a high-dimensional proposal density for the chain. Particle MCMC method is a combination of SMC and MCMC. It aims at constructing a Markov chain whose invariant distribution is the target. SMC is used to design the high dimensional proposal density for the Markov move. Conditional SMC with ancestor sampling is used to guide trajectories moving toward the target distribution. With the idea of Gibbs sampling, an alternative update of unknown parameters conditioned on states and states conditioned on parameters will lead to an approximation of the joint distribution of parameters and states.
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Date
2018-01-01
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University of Kansas
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This item contains archived web content.
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Keywords
Applied mathematics, block sampling, expectation maximization, particle filters