Hidden Markov Model (Hidden Markov models) Series II

Hidden Markov Model (Hidden Markov models) Series II

• Introduction (Introduction)
• Generating patterns)
• Implicit patterns)
• Hidden Markov Model (Hidden Markov models)
• Forward Algorithm (Forward Algorithm)
• Viterbi Algorithm)
• Forward and backwardAlgorithm(Forward-backward Algorithm)
• Summary

 

 

Implicit patterns)

What should we do when the Markov process is not powerful enough?

In some cases, the Markov process is insufficient to describe the pattern we want to discover. Back to the previous weather example, can a retired person intuitively observe the weather, but there are some seaweed. The saying tells us that the state of seaweed is probably related to the weather. In this case, we have two State sets, one that can be observed (the State of seaweed) and one that is hidden (the State of the weather ). We hope to find an algorithm to predict weather conditions based on seaweed conditions and Markov assumptions.

A more practical example is speech recognition. The voice we hear is the result of the joint action of vocal cords, throat, and other pronunciation organs. These factors interact with each other and jointly determine the sound of each word, while a speech recognition system detects the sound (an observed state) it is produced by various physical changes within the human body (hidden state, meaning that a person really wants to express.

Some speech recognition devices regard the internal pronunciation mechanism as a hidden state sequence, and regard the final sound as a sequence of observed states very similar to the hidden state sequence. In these two examples, the number of hidden states and the number of States that can be observed may be different. Four wet seaweed (dry, dryish, damp, soggy) can also be observed in a three-state weather system (Sunny, cloudy, and rainy ). In speech recognition, a simple speech may only need 80 elements to describe, but an internal pronunciation mechanism can produce less than 80 or more different voices.

In the above cases, we can see that the state sequence and the hidden state sequence are probability-related. So we can model this type of process as another hidden Markov process and a set of states that are relevant to the probability of the Markov process and can be observed.

Shows the relationship between the hidden state and the observed state in the weather example. We assume that the hidden state is a simple first-order Markov process, and they can both switch between each other.

There is a probability relationship between the hidden state and the observed state. That is to say, a hidden state H is considered as a certain observed state O1 with a probability, assume it is P (O1 | H ). If three States can be observed, it is clear thatP (O1 | h) + P (O2 | h) + P (O3 | H) = 1. Here I mean different from the original one.P (O1 | H1) + P (O1 | H2) + P (O1 | H3) = 1But this is different from the example below.

In this way, we can also obtain another matrix, called a confusion matrix. The content of this matrix is the probability that a hidden state is observed as a set of different States that can be observed. In the weather example, this matrix is as follows:

Note that the sum of each row in the graph is 1,But the sum of the values in each column is not 1. Here I think the original text may have an error, or the hidden state may have something else.

 

Summary

We have seenSome processes are related to the probability of a hidden Markov process.. In this case, the number of observed states and hidden states may be different. We can model this processHidden Markov Model (HMM).This model contains two State sets and three probability sets..

• Hidden State: A Hidden Markov Process
• Observed status: such as name
• Initial vector: hidden probability of initial state
• State Transition matrix: hidden state transition probability
• Confusion matrix: the probability that the hidden state is observed as the various States that can be observed

We can think that the hidden Markov model is obtained by adding a set of observed states to an inobserved Markov process and some probability relationships between the Process and the set.