HMM model parameters for solving Overview

HMM model parameters can be solved in two cases according to the known conditions.

　　　　The first case is relatively simple, we know that D D of length T observation sequence and corresponding T hidden state sequence, i.e., { ( O . 1 , the I . 1 ) , ( O 2 , the I 2 ) , . . . ( O D , the I D ) } {(O1 of, I1), (the O2, I2), ... (the OD, ID)} is known, then we can easily solved using the maximum likelihood model parameters.

　　　　Suppose a sample from the hidden state Q I transferred to qi Q J frequency count is qj A I J Aij of, then the state transition matrix is obtained:

A = [ A I J ] , which is the A I J = A I J [Sigma S = . 1 N A I S A = [of aij], where aij = AijΣs = 1NAis

 

　　　　Suppose a sample of hidden states Q J QJ and observation state V K frequency count is vk B J K BJK, then the state observation probability matrix:

B = [ B J ( K ) ] , which in B J ( K ) = B J K [Sigma S = . 1 M B J S B = [BJ (K)], where bj (k) = BjkΣs = 1MBjs

 

　　　　Assuming that all of the samples in the initial state hidden Q I frequency count qi is C ( I ) C (I), then the initial probability distribution:

Π = π ( i ) = C ( i ) Σ s = 1 N C ( s ) Π = π (i) = C (s) = Σs 1NC (s)

 

　　　　Visible to solve the model first case is very simple. However, in many cases, we can not be observed sequence of HMM corresponding sample hidden sequence, only D D of length T observation sequence T, i.e., { ( O . 1 ) , ( O 2 ) , . . . ( O D ) } { (O1), (O2), ... (OD)} is known, at this time we can not find a suitable HMM model parameters? This is the focus of our second case, which we discussed in this article. It's the most common solution is to Baum - Welch algorithm, in fact, based on the EM algorithm to solve, but Baum - Welch algorithm era appears, the EM algorithm has not been abstracted, so we still say abalone article Farm - Welch algorithm.