Hidden Markov model (II.)--The composition of Hidden Markov model (reproduced)

In the Markov model, each state is an observable sequence, a stochastic process of state about time, and also a visual Markov model (visible Markov model,vmm). The state of the Hidden Markov model (Hidden Markov model,hmm) is invisible, and we can see the probability function of the state and the observed value. In the hidden horse model, the observed value is a stochastic process of state, and the state is a stochastic process of time, so the hidden horse model is a double stochastic process.

Hmm can be used when considering potential events to generate surface events randomly.

For example, show the Hidden Horse model:

There are 4 black boxes, in the dark, each box has 3 kinds of ball without color (red, orange, blue), from the box to take the ball is a certain regularity, now the staff from the dark box to take the ball, went 5 times, we see the frontal observation sequence is: Red Blue blue Orange. This process is a hidden horse model. The box in the dark indicates the state, the number of boxes indicates the number of States, the color of the ball indicates the output value of the state, the number of colors of the ball indicates the number of state output observation states, from one box to another box to indicate the state transition, the observation color of the ball removed from the dark box indicates the output sequence of the state.

Thus the 5 constituent states of the hidden Markov type can be summed up:

(1) The number of States in the Model N (the number of cases in the example);

(2) Each state can be output of different observations m (example of the number of balls in the color);

(3) The state transition matrix a= {AIJ} (in the example AIJ represents the probability of moving from box I to Box J), where the AIJ satisfies the condition:

I. Aij≥0, 1≤i,j≤n

II. aij= P (Qt=sj|qt-1=si),

III. =1

(4) The Emission Matrix B={BJ (k)}, which is the probability distribution matrix of the symbol VK observed from the state SJ. (BJ (k) In the example shows the probability of taking the K-color from the J-Box), where BJ (k) satisfies the condition:

I. BJ (k) ≥0, 1≤j≤n; 1≤k≤m

II. BJ (k) = P (OT=VK|QT=SJ),

Iii. =1

(5) Initial state probability distribution π= {πj}. (The probability of beginning to the first J box in the example), where πj satisfies the condition:

I.πj (k) ≥0, 1≤j≤n

ii.πj= P (Q1=SJ),

Iii. =1

Generally, a hmm is a five-tuple μ={n,m,a,b,π}, in order to be simple, often précis-writers for the ternary group μ={a,b,π}.

Hmm has three basic questions:

(1) Assess the problem: given an observation sequence O=o1o2 ... OT and Model μ={a,b,π}, how to quickly calculate the condition of the given model μ, observe the sequence O=o1o2 ... The probability of OT, i.e. P (o|μ)?

(2) Decoding problem: Given an observation sequence O=o1o2 ... OT and Model μ={a,b,π}, how to quickly select a given model μ under the condition of a certain meaning "optimal" state sequence q=q1q2 ... QT, is the state sequence "best" to interpret the observation sequence?

(3) Learning problem: Given an observation sequence O=o1o2 ... OT, how to adjust the parameter μ={a,b,π}, so that P (o| M) Max?

For the three basic problems of Hmm, the corresponding algorithm is:

(1) Evaluate the problem: forward backward algorithm

(2) Decoding problem: Viterbi algorithm (VITERBI)

(3) Learning problem: Forward backward Algorithm (Baum-welch).

Hidden Markov model (II.)--The composition of Hidden Markov model (reproduced)