Training HMMS

If the alignment of states to observations is known, then it is a simple matter to use the data to estimate file means and covariances directly. Assuming that we have a set of frames Oj = Oj, oj2, , ojt tiiat are aligned with state j, we find the mean by calculating the simple average 

So long as we have sufficient examples for each state, this provides a robust way to estimate the means and covariances. The transition probabilities can be calculated by simply coimting the number of times for a state i we move to state J and dividing this ly the total number of times we exit ftom state i. 

Viterbi training does, however, have a serious drawback in that it enforces hard decisions during training. If for instance our alignment picks completely wrong boundaries, it will then train on these, and may pick the same wrong boundaries next time in effect the iteration gets in a rut from which it is hard to recover. 

Equations (15.19) and (15.20) are the Baum-Welch re-estimation formulas for the means and covariances of an HMM. A similar but slightly more-complex formula can be derived for the transition probabilities. Of comse, to apply Equations (15.19) and (15.20), the probability of state occupation Lj (t) must be calculated. This is done efficiently using the forward-backward algorithm. Let the forward probability aj(t) for some model M with N states be defined as 

That is, aj t) is the joint probability of observing the first t speech vectors and being in state j at time t. This forward probability can be efficiently calculated by the following recursion  

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We will consider the full issue of how to train an HMM in Section 15.1.8. For now, let us simply assume that we can calculate the transition probabilities and observation probabilities by simply counting occurrences in a labelled database. To see how the tagging operates consider the issue of resolving the classic POS homograph record. This can be a norm or a verb, and a trained HMM would tell us for instance ... 

A trained HMM systems describes a model of speech and so can be use to generate speech. 

Won, K.-J. Priigel-Bennett, A. Krogh, A. (2004). Training HMM structure with genetic algorithm for biological sequence analysis. Bioinformatics, Vol. 20, No. 18, pp. 3613-... 

Fujiwara et al. (1997) proposed an HMM that can detect mitochondrial targeting signals. The HMM was automatically created to best explain the training data. Although it could model the signals in the training data, further analysis using more data is desirable because the model has many numeric parameters. 

The theme of the dinner was Excess and the organiser asked Bee and Bella to do a slave-training show. Bee and Bella wanted to go the whole hog. Can we wear dicks Hmm. . . No, that might be a bit much. Well what about guns Great idea Guns are fine. 

The Hidden Markov Model (HMM) is a powerful statistical tool for modeling a sequence of data elements called the observation vectors. As such, extraction of patterns in time series data can be facilitated by a judicious selection and training of HMMs. In this section, a brief overview will be presented and the interested reader can find more details in numerous tutorials... 

The number of states is usually unknown, but some physical intuition about the system can provide a basis for defining M. Naturally, a small number of states usually results in poor estimation of the data, while a large number of states improves the estimation but leads to extended training times. The quality of the HMM can be gauged by considering the residuals of the model or the correlation coefficients of observed and estimated values of the variables. The residuals are expected to have a Normal distribution (A(0, (T )) if there is no systematic information left in them. Hence, the normality of the residuals can provide useful information about model performance in representing the data. 

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]. 

The first step of the analysis is the training where Hidden Markov Models (HMMs) representing various operating behaviors are trained using labeled historical data from the process. In this section, three broad operat-... 

Once the relevant HMMs are trained, the trend analysis is carried out on the newly observed time series in real-time. The time series is windowed and smoothed before the signal in the window can be represented in the... 

Furthermore, in the multivariable problem, while three to five variables can be handled relatively easily, one reaches a computational bottleneck for larger problems. This can be possibly resolved by considering some of the new developments in HMM training algorithms [254, 71],... 

In a real HMM tagger system, we have to determine these probabilities from data. In general, this data has to be labelled with a POS tag for each word, but in cases where labelled data is scarce, we can use a pre-existing tagger to label more data, or use a bootstrapping technique where we use one HMM tagger to help label the data for another iteration of training. 

Markov models (HMMs) as these are in general easier to train and allow more complexity with regard to noise/covariance terms. 

All of the above assumes that the parameters for a HMM are re-estimated from a single observation sequence, that is a single example of the spoken word. In practice, many examples are needed to get good parameter estimates. However, the use of multiple observation sequences adds no additional complexity to the algorithm. Steps 2 and 3 above are simply repeated for each distinct training sequence. 

While many of these points are valid, there are often solutions which help alleviate any problems. As just explained, the use of d5mamic features helps greatly with the problems of observation independence and discrete states. As we shall see the linearity issue is potentially more of a problem in speech s mthesis. Models such as neural networks which perform classification directly have been proposed [375] and have produced reasonable results. More recently, discriminative training has become the norm in ASR [495], [360] where HMMs as described are used, but where their parameters are trained to maximise discrimination, not data likelihood. 

The context-sensitive models are built in exactly the same way as described in Section 15.1.9, and the resultant decision tree provides a unique mapping from every possible feature combination to HMM model parameters. One significant point of note is that in synthesis the extra prosodic features mean that number of possible unique feature combinations can be many orders of magnitude larger than in ASR. Given that we may be training on less data than in ASR, we see that the sparse data problems can be considerably worse. We will return to this issue in Section 16.4. 

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