基本隐马尔可夫模型，维特比算法

Question

I am fairly new to Hidden Markov Models and I am trying to wrap my head around a pretty basic part of the theory.

I would like to use a HMM as a classifier, so, given a time series of data I have two classes: background and signal.

How are the emission probabilities estimated for each class? Does the Viterbi algorithm need a template of the background and signal to estimate prob(data|state)? Or have I completely missed the point?

Answer 1

To do classification with Viterbi you need to already know the model parameters.
Background and Signal are your two hidden states. With the model parameters and observed data you want to use Viterbi to calculate the most likely sequence of hidden states.

To quote the hmmlearn documentation :

The HMM is a generative probabilistic model, in which a sequence of observable X variables is generated by a sequence of internal hidden states Z. The hidden states are not be observed directly. The transitions between hidden states are assumed to have the form of a (first-order) Markov chain. They can be specified by the start probability vector π and a transition probability matrix A. The emission probability of an observable can be any distribution with parameters θ conditioned on the current hidden state. The HMM is completely determined by π, A and θ

.

There are three fundamental problems for HMMs:

 Given the model parameters and observed data, estimate the optimal sequence of hidden states. Given the model parameters and observed data, calculate the likelihood of the data. Given just the observed data, estimate the model parameters. 

The first and the second problem can be solved by the dynamic programming algorithms known as the Viterbi algorithm and the Forward-Backward algorithm, respectively. The last one can be solved by an iterative Expectation-Maximization (EM) algorithm, known as the Baum-Welch algorithm.

Answer 2

So we've got two states for our Hidden Markov model, noise and signal. We also must have something we observe, which could be ones and zeroes. Basically, ones are signal and zeroes are noise, but you get a few zeros mixed in with your signal and a few ones with the noise. So you need to know

Probablity of 0,1 when in the "noise" state 
Probability of 0,1 when in the "signal" state  
Probability of transitioning to "signal" when in the "noise" state. 
Probability of transitioning to "noise" when in the "signal" state.

So we keep track of the probability of each state for each time slot and, crucially, the most likely route we got there (based on the transition probabilities). Then we assume that the most probable state at the end of the time series is our actual final state, and trace backwards.

Answer 3

The Viterbi Algorithm requires to know the HMM.

The HMM can be estimated with a Maximum-Likely-Estimation (MLE) and it's called the Baum–Welch algorithm .

If you have trouble with the Viterbi Algorithm there's a working implementation here