The diagram below shows the hidden and observable states in the weather example. It is assumed that the hidden states (the true weather) are modelled by a simple first order Markov process, and so they are all connected to each other.
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The connections between the hidden states and the observable states represent the probability of generating a particular observed state given that the Markov process is in a particular hidden state. It should thus be clear that all probabilities `entering′ an observable state will sum to 1, since in the above case it would be the sum of Pr(Obs|Sun), Pr(Obs|Cloud) and Pr(Obs|Rain).
In addition to the probabilities defining the Markov process, we therefore have another matrix, termed the confusion matrix, which contains the probabilities of the observable states given a particular hidden state. For the weather example the confusion matrix might be;
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Notice that the sum of each matrix row is 1. 数据挖掘实验室
Summary
We have seen that there are some processes where an observed sequence is probabalistically related to an underlying Markov process. In such cases, the number of observable states may be different to the number of hidden states. 数据挖掘交友
We model such cases using a hidden Markov model (HMM). This is a model containing two sets of states and three sets of probabilities; 数据挖掘论坛
- hidden states : the (TRUE) states of a system that may be described by a Markov process (e.g., the weather).
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- observable states : the states of the process that are `visible′ (e.g., seaweed dampness).
vector : contains the probability of the (hidden) model being in a particular hidden state at time t = 1.
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- state transition matrix : holding the probability of a hidden state given the previous hidden state.
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- confusion matrix : containing the probability of observing a particular observable state given that the hidden model is in a particular hidden state.
Thus a hidden Markov model is a standard Markov process augmented by a set of observable states, and some probabalistic relations between them and the hidden states.
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