Experiences of hidden Markov models in statistical learning methods

First, we will introduce the hidden Markov model, which is a probability model of time series. The concept book has its original meaning: an unobserved state random sequence randomly generated by a hidden Markov chain, each State generates an observed random sequence. Combined with the example in the book, this state can be a box. The observed values are the red or white balls in the box.

The following are some definitions:

Q: the set of all possible states q = {Q1, q2,..., qN} n is the number of possible states.

V: all possible observations v = {V1, V2,... VM} m are possible observations.

I: The state sequence O with a length of T is the corresponding observation sequence.

I = (I1, I2,... it), O = (O1, O2,..., ot)

A is the state transfer probability matrix:

A = [AIJ] nxn (1)

Where,

AIJ = P (IT + 1 = Qj | it = Qi), I = 1, 2,... n; j = 1, 2,..., n (2)

It indicates the probability that the Qj is transferred to the State at the t + 1 moment under the condition of T and State Qi.

B is the observed probability matrix:

B = [bij] nxm (3)

Where,

BJ (K) = P (Ot = VK | it = Qj), k = 1, 2,... m; j = 1, 2,..., n (4)

It indicates the probability of generating a observed VK under the condition of T and status Qj.

Is the initial test state probability vector:

(5)

Where,

(6)

It is the probability that T = 1 is in the State Qi.

Therefore, the hidden Markov model is determined by the preceding three elements:

(7)

The following example describes the three elements. (Used directly)

Experiences of hidden Markov models in statistical learning methods