Wavelet-Domain Hidden Markov Models

In the state estimation problem, the aim is to find a state sequence that best explains the real observations. For this, a new variable 7 is defined in terms of the forward (a) and backward (/ ) variables of the Baum-Welch algorithm  

The new variable, 7t(f) represents the probability of being in state i at time t. The size of the matrix 7 is T x M. One can then find the most likely state at time t using 7t(f)  

Given the dependency of the wavelet coefficients, one still has to find the appropriate framework for modeling their probability density functions. A Gaussian model is not appropriate since the wavelet decomposition tends to produce a large number of small coefficients and a small number of 

The training problem determines the set of model parameters given above for an observed set of wavelet coefficients. In other words, one first obtains the wavelet coefficients for the time series data that we are interested in and then, the model parameters that best explain the observed data are found by using the maximum likelihood principle. The expectation maximization (EM) approach that jointly estimates the model parameters and the hidden state probabilities is used. This is essentially an upward and downward EM method, which is extended from the Baum-Welch method developed for the chain structure HMM [43, 286]. 

In this chapter, several signal characterization and modelling methods have been introduced and discussed. It was shown that the wavelet transforma- 

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JK Romberg, H Choi, and RG Baraniuk. Bayesian tree structured image modeling using wavelet-domain hidden Markov models. IEEE Trans, on Image Processing, 10 1056-1068, 2001. 

W Sun, A Palazoglu, and JA Romagnoli. Detecting abnormal process trends by wavelet-domain hidden Markov models. AIChE J., 749 140-150, 2003. 

A trend analysis strategy is proposed that takes advantage of the wavelet-domain hidden Markov trees (HMTs) for constructing statistical models of wavelets (see Section 6.5). Figure 7.10 depicts the strategy that can be used to detect and classify faulty (abnormal) situations. As before, in the training phase, time series data collected under various conditions are... 

G Fan and XG Xia. Improved hidden Markov models in the wavelet-domain. IEEE Trans, on Signal Processing, 49 115-120, 2001. 

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