Application of Hidden Markov model and Viterbi algorithm

Jostree Reprint Please specify the source http://www.cnblogs.com/jostree/p/4335810.html

An example:

Wei Xiaobao use dice to play, he has two kinds of dice a normal dice, there is a kind of uneven dice, to carry out thousands.

He had a 2/5 chance of being a player when he started the game.

For normal dice A, the probability of each point appearing is 1/6.

For non-uniform dice b,5,6 Two probability of occurrence is 3/10, the remaining is 1/10.

The random law of the thousand is as follows:

We observed the result of throwing: ob={1,3,4,5,5,6,6,3,2,6}

Please tell me when Wei Xiaobao is out.

We can model this $x_i$ represents the type of dice $i$ to throw, $y _i$ represents the number of points $i$ thrown, $\lambda$ represents each probability parameter.

Then the probability of $t$ using the $i$ dice $\delta_t (i) $ equals

\begin{equation} \delta_t (i) =\max_{x_1,\dots,x_{t-1}}p (X_1,\DOTS,X_{T-1},X_T=I,Y_1,\DOTS,Y_T|\LAMBDA) \end{ Equation}

In fact $\delta_{t+1} (i) $ can be torn down by $\delta_t (i) $:

\begin{eqnarray} \delta_{t+1} (i) &=& \max_{x_1,\dots,x_{t}}p (x_1,\dots,x_{t},x_{t+1}=i,y_1,\dots,y_{t+1}| \LAMBDA) \ \ &=& \max_j \delta_t (j) \alpha_{ji}\beta_i (Y_{t+1}) \end{eqnarray}

where $\alpha_{ji}$ represents the probability of transferring from the $j$ dice to the $i$ dice.

$\beta_i (Y_{t+1}) $ indicates the probability of casting a point $y_{t+1}$ using the I-dice.

This allows the use of dynamic programming algorithms for successive recursive computations.

The results are as follows:

T	$y _t$	$\delta_t (A) $	$\psi_t (A) $	$\delta_t (B) $	$\psi_t (B) $
1	1	0.1	A	0.04	A
2	3	0.0133333	A	0.0036	B
3	4	0.00177778	A	0.000324	B
4	5	0.000237037	A	0.000106667	A
5	5	3.16049e-05	A	2.88e-05	B
6	6	4.21399e-06	A	7.776e-06	B
7	6	5.61866e-07	A	2.09952e-06	B
8	3	7.49154e-08	A	1.88957e-07	B
9	2	9.98872e-09	A	1.70061e-08	B
10	6	1.33183e-09	A	4.59165e-09	B

Thus from $\psi_t (A) $ and $\psi_t (B) $ select $\delta_t () $ larger dice, thus predicting the type of dice sequence for: aaaaabbbbb

The code looks like this:

1#include <stdlib.h>2#include <stdio.h>3#include <string.h>4#include <string>5#include <iostream>6 using namespacestd;7 Doubleinitp[2] = {0.6,0.4};//the initial probability of the dice A, b8 Doubletransfermatrix[2][2] = {{0.8,0.2}, {0.1,0.9}};//probability of transfer between dice9 Doubleemissionp[2][6]={{1/6.0,1/6.0,1/6.0,1/6.0,1/6.0,1/6.0},//the probability of firing the dice aTen{0.1,0.1,0.1,0.1,0.3,0.3}};//probability of firing of dice B One Doubledp[Ten][2];//Dp[i][j] Step i, the maximum probability of using the J Dice A Doubledps[Ten][2];//Dps[i][j] step I, when using the J dice, get the maximum probability when using the dice type, 0->a, 1->b - intob[Ten] = {1,3,4,5,5,6,6,3,2,6};//observed points - BOOLdicearray[Ten];//predicting dice use sequences the voidViterbi () - { -Memset (DP,0,sizeof(DP)); -memset (DpS,0,sizeof(DpS)); +memset (Dicearray,0,sizeof(Dicearray)); -dp[0][0] = initp[0]* emissionp[0][ob[0]-1]; +dp[0][1] = initp[1]* emissionp[1][ob[0]-1]; A      for(inti =1; I <Ten; i++)//number of throws at     { -          for(intj =0; J <2; J + +)//Current status -         { -              for(intK =0; K <2; k++)//Previous State -             { -                 DoubleTEMPP = dp[i-1][K] * TRANSFERMATRIX[K][J] * emissionp[j][ob[i]-1] ; in                 if(Dp[i][j] <tempp) -                 { toDP[I][J] =tempp; +DPS[I][J] =K; -                 } the             } *         } $         if(dp[i][0] < dp[i][1] )Panax Notoginseng         { -Dicearray[i] = dps[i][1]; the         } +         Else A         { theDicearray[i] = dps[i][0]; +         } -     } $ } $ intMainintargcChar*argv[]) - { - Viterbi (); thecout<<"probabilities and dice in each state of each step:"<<Endl; -      for(inti =0; I <Ten; i++ )Wuyi     { the          for(intj =0; J <2; J + + ) -         { Wucout<<dp[i][j]<<" "<<dpS[i][j]<<"    "; -         } Aboutcout<<Endl; $     } -cout<<"predict the type of dice, 0->a, 1->b:"<<Endl; -      for(inti =0; I <Ten; i++ ) -     { Acout<<dicearray[i]<<" "; +     } thecout<<Endl; -}

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Hidden Markov model and application of Viterbi algorithm