让你真正理解HMM(Hidden Markov Model)的算法演示程序

浏览数：27 / 时间：2015年06月08日

HMM, 隐Markov模型, 在人脸, 步态, 语音识别等领域有着广泛的用途.

通过以Javascript语言演示其使用方法, 读者可方便地理解其计算过程(其实,并不难).

理论就不讲解了,直接看计算过程:

?<html>
<head>
<meta charset="UTF-8"/>
<meta author="[email protected]"/>
<meta published="2014-11-28"/>
<meta licence="public"/>
<meta about="Hidden markov model"/>
<title>让你真正理解隐Markov模型的计算示例</title>
</head>
<body>

<h1>HMM Demo</h1>
<hr/>
<div id="info">
<h2>[0] HMM模型参数</h2>
<b>状态S：	{H,	L}</b><br/>
<b>初始状态I：	{0.5, 0.5}</b><br/>
<b>状态转移矩阵A：</b><br/>
<table border="1px">
<tr>
  <td>a_ij</td><td>H</td><td>L</td>
</tr>
<tr>
  <td>H</td><td>0.5</td><td>0.5</td>
</tr>
<tr>
  <td>L</td><td>0.4</td><td>0.6</td>
</tr>
</table>
<b>混淆矩阵B：</b><br/>
<table border="1px">
<tr>
  <td>b_ik</td><td>A</td><td>C</td><td>T</td><td>G</td>
</tr>
<tr>
  <td>H</td><td>0.2</td><td>0.3</td><td>0.3</td><td>0.2</td>
</tr>
<tr>
  <td>L</td><td>0.3</td><td>0.2</td><td>0.2</td><td>0.3</td>
</tr>
</table>
<br/>
<h2>[1] 评估问题：使用上述模型,采用<font color='red'>Forward算法</font>计算发生GGCA观测结果的概率</h2>
<h3>结果矩阵:</h3>
<table border="1px">
<tr>
  <td>概率</td><td>初始</td>
  <td><div id="O1">G</div></td>
  <td><div id="O2">G</div></td>
  <td><div id="O3">C</div></td>
  <td><div id="O4">A</div></td>
</tr>
<tr>
	<td><div id="S0">H</div></td>
	<td>0.5</td>
	<td><div id="11"></div></td>
	<td><div id="12"></div></td>
	<td><div id="13"></div></td>
	<td><div id="14"></div></td>
</tr>
<tr>
	<td><div id="S1">L</div></td>
	<td>0.5</td>
	<td><div id="21"></div></td>
	<td><div id="22"></div></td>
	<td><div id="23"></div></td>
	<td><div id="24"></div></td>
</tr>
</table>

<div id="prob">
</div>
<br/>
<div id="output">
<b>计算过程:</b><br/>
</div>
<hr/>

<h2>[2] 解码问题: 给定观测序列GGCACTGAA,问产生该序列概率最大的状态路径是什么?</h2>
<b>把HMM模型换算为以2为底的对数值,则有</b><br/>
<b>状态S：	{H,	L}</b><br/>
<b>初始状态I：	{-1, -1}</b><br/>
<b>状态转移矩阵A：</b><br/>
<table border="1px">
<tr>
  <td>a_ij</td><td>H</td><td>L</td>
</tr>
<tr>
  <td>H</td><td>-1</td><td>-1</td>
</tr>
<tr>
  <td>L</td><td>-1.322</td><td>-0.737</td>
</tr>
</table>
<b>混淆矩阵B：</b><br/>
<table border="1px">
<tr>
  <td>b_ik</td><td>A</td><td>C</td><td>T</td><td>G</td>
</tr>
<tr>
  <td>H</td><td>-2.322</td><td>-1.737</td><td>-1.737</td><td>-2.322</td>
</tr>
<tr>
  <td>L</td><td>-1.737</td><td>-2.322</td><td>-2.322</td><td>-1.737</td>
</tr>
</table>
<h3>结果矩阵:</h3>
<table border="1px">
<tr>
  <td>概率</td><td>初始</td>
  <td><div id="D1">G</div></td>
  <td><div id="D2">G</div></td>
  <td><div id="D3">C</div></td>
  <td><div id="D4">A</div></td>
  <td><div id="D5">C</div></td>
  <td><div id="D6">T</div></td>
  <td><div id="D7">G</div></td>
  <td><div id="D8">A</div></td>
  <td><div id="D9">A</div></td>
</tr>
<tr>
	<td><div id="S0">H</div></td>
	<td>-1.0</td>
	<td><div id="V11"></div></td>
	<td><div id="V12"></div></td>
	<td><div id="V13"></div></td>
	<td><div id="V14"></div></td>
	<td><div id="V15"></div></td>
	<td><div id="V16"></div></td>
	<td><div id="V17"></div></td>
	<td><div id="V18"></div></td>
	<td><div id="V19"></div></td>
</tr>
<tr>
	<td><div id="S1">L</div></td>
	<td>-1.0</td>
	<td><div id="V21"></div></td>
	<td><div id="V22"></div></td>
	<td><div id="V23"></div></td>
	<td><div id="V24"></div></td>
	<td><div id="V25"></div></td>
	<td><div id="V26"></div></td>
	<td><div id="V27"></div></td>
	<td><div id="V28"></div></td>
	<td><div id="V29"></div></td>
</tr>
</table>
<div id="result1"></div>
计算过程:<br/>
<div id="output1"><div>

<script type="text/javascript">
//状态集合
var S = ['H', 'L'];
//观测集合
var O = ['A', 'C', 'T', 'G'];
//初始状态发生H和L的概率矩阵
var imat = [0.5, 0.5];
//不同状态间转换的概率矩阵---状态转换矩阵
var amat = {'HH': 0.5, 'HL':0.5, 'LH':0.4, 'LL': 0.6};
//由状态产生特定观测的概率矩阵---混淆矩阵
var bmat = {'HA': 0.2, 'HC': 0.3, 'HG': 0.3, 'HT': 0.2, 
            'LA': 0.3, 'LC': 0.2, 'LG': 0.2, 'LT': 0.3};
//测试用的观测结果
var dest = 'GGCA';

//设置概率计算矩阵
var pmat = new Array(S.length);
for(var i = 0; i < S.length; i++) {
	pmat[i] = new Array();
	for(var j = 0; j < dest.length; j++)
		pmat[i].push(0);
}

var out = document.getElementById("output");
var tmp;

//计算评估问题算法---前向算法
//分为两个部分计算
//计算结果矩阵pmat第1列---由初始状态矩阵及混淆矩阵所决定
for(var i = 0; i < S.length; i++) {
	pmat[i][0] = imat[i] * bmat[S[i] + dest[0]];
	document.getElementById(String(i + 1) + "1").innerHTML = pmat[i][0];
}

//计算后续列的概率---由前一状态概率 * 状态间转换概率amat * 状态到观测概率矩阵bmat所决定
for(var i = 1; i < dest.length; i++) {
	for(var rowA = 0; rowA < S.length; rowA++) {
		//输出
		out.innerHTML += "<b>" + String(rowA + 1) + "行" + String(i + 1) + "列</b><br/>";
		for(var rowB = 0; rowB < S.length; rowB++) {
			if (rowA == rowB) {
				tmp = pmat[rowA][i-1] * amat[S[rowA] + S[rowA]] * bmat[S[rowA] + dest[i]];
				pmat[rowA][i] += tmp;
				//输出
				out.innerHTML += S[rowA] + S[rowB] + dest[i] + ": " + String(pmat[rowA][i-1]) + "*" + String(amat[S[rowA] + S[rowA]]) + "*" + String(bmat[S[rowA] + dest[i]]) + "=" + String(tmp) + "<br/>";
			}
			else {
				tmp = pmat[rowB][i-1] * amat[S[rowB] + S[rowA]] * bmat[S[rowA] + dest[i]];
				pmat[rowA][i] += tmp;
				//输出
				out.innerHTML += S[rowB] + S[rowA] + dest[i] + ": " + String(pmat[rowB][i-1]) + "*" + String(amat[S[rowB] + S[rowA]]) + "*" + String(bmat[S[rowB] + dest[i]]) + "=" + String(tmp) + "<br/>";
			}
		}
		//输出
		out.innerHTML += "和为: <b>" + String(pmat[rowA][i]) + "</b><br/>";
		document.getElementById(String(rowA + 1) + String(i + 1)).innerHTML = pmat[rowA][i];
	}
}

//输出状态S产生观测序列的概率
var sm = 0;
for(var i = 0; i < S.length; i++)
  sm += pmat[i][dest.length-1];
document.getElementById("prob").innerHTML = "评估结果: HMM(S(H, L), O(A, T, C, G), imat, amat, bmat)产生观测结果" + dest + "的概率为: <font color='red'>" + String(sm) + "</font><br/>";

//------------------------
//取得最佳路径问题---解码问题
//测试用的观测结果
var dest = 'GGCACTGAA';
//把数值矩阵转换为对数
for(var i = 0; i < S.length; i++)
  	imat[i] = Math.log(imat[i]) / Math.LN2;
for(var i = 0; i < S.length; i++)
	for(var j = 0; j < S.length; j++) {
  		amat[S[i] + S[j]] = Math.log(amat[S[i] + S[j]]) / Math.LN2;
	}
for(var i = 0; i < S.length; i++)
	for(var j = 0; j < O.length; j++) {
  		bmat[S[i] + O[j]] = Math.log(bmat[S[i] + O[j]]) / Math.LN2;
  		console.log( S[i] + "->" + O[j] + "=" + bmat[S[i] + O[j]] );
  	}
  		
//初始化概率计算矩阵pmat
var pmat = new Array(S.length);
for(var i = 0; i < S.length; i++) {
	pmat[i] = new Array();
	for(var j = 0; j < dest.length; j++)
		pmat[i].push(0);
}

var out = document.getElementById("output1");
//计算解码问题算法---Viterbi算法
//分为两个部分计算
//计算结果矩阵pmat第1列---由初始状态矩阵及混淆矩阵所决定
var link = new Array();
var maxval = -1e15, maxid = -1;
for(var i = 0; i < S.length; i++) {
	pmat[i][0] = imat[i] + bmat[S[i] + dest[0]];
	if (pmat[i][0] > maxval) {
		maxval = pmat[i][0];
		maxid = i;
	}
	document.getElementById("V" + String(i + 1) + "1").innerHTML = pmat[i][0];
}
//存储第1个最大概率点
link.push(S[maxid]);

//计算后续列的概率---由前一状态最大概率 * 前一状态到其他状态的转换概率amat * 当前状态对观测值的发生概率
//记录当前可能产生观测结果的最大概率
for(var i = 1; i < dest.length; i++) {
	var thisO = dest[i];
    //计算由上次状态lastS出发,发生状态转换到S[rowA]产生观测值dest[i]的最大概率
	for(var rowA = 0; rowA < S.length; rowA++) {
		var thisS = S[rowA];
		var maxval = -1e15;
		//由thisS产生thisO的概率
		var pp = bmat[thisS + thisO]; 
		for(var rowB = 0; rowB < S.length; rowB++) {
			var lastS = S[rowB];
			//由lastS到thisS的转移概率
			var tp = amat[lastS + thisS];
			//上次的历史概率
			var lp = pmat[rowB][i-1];
			//总概率
			var totalP = pp + tp + lp;
			if (totalP > maxval) {
				maxval = totalP;
				document.getElementById("V" + String(rowA + 1) + String(i + 1)).innerHTML = String(totalP);
			}
			out.innerHTML += "O" + String(i + 1) + ": " + lastS + "->" + thisS + "= " + 
				String(pp) + "(" + thisS + "->" + thisO + ") +" + String(tp) + "(T: " + lastS + thisS + ") +" 
					+ String(lp) + "(" + lastS + ", " + String(i-1)  + ") ==>" + String(totalP) + "<br/>";
		}
		pmat[rowA][i] = maxval;
	}
	if (pmat[0][i] > pmat[1][i])
		link.push(S[0]);
	else
		link.push(S[1]);
	out.innerHTML += "最佳: " + link[link.length - 1] + "<br/>";
}
var out = document.getElementById("result1");
out.innerHTML = "最佳状态序列:";
for(var i = 0; i < link.length; i++)
	out.innerHTML += link[i];
</script>
</body>
</html>

计算结果如下,方便大家检验:

HMM Demo

[0] HMM模型参数

状态S： {H, L}
初始状态I： {0.5, 0.5}
状态转移矩阵A：
a_ij H L
H 0.5 0.5
L 0.4 0.6
混淆矩阵B：
b_ik A C T G
H 0.2 0.3 0.3 0.2
L 0.3 0.2 0.2 0.3

[1] 评估问题：使用上述模型,采用Forward算法计算发生GGCA观测结果的概率

结果矩阵:

概率 初始
G
G
C
A
H
 0.5
0.15
0.0345
0.008415
0.0013767000000000002
L
 0.5
0.1
0.027
0.00669
0.00246645
评估结果: HMM(S(H, L), O(A, T, C, G), imat, amat, bmat)产生观测结果GGCA的概率为: 0.00384315

计算过程:
1行2列
HHG: 0.15*0.5*0.3=0.0225
LHG: 0.1*0.4*0.2=0.012000000000000002
和为: 0.0345
2行2列
HLG: 0.15*0.5*0.3=0.015
LLG: 0.1*0.6*0.2=0.012
和为: 0.027
1行3列
HHC: 0.0345*0.5*0.3=0.005175
LHC: 0.027*0.4*0.2=0.0032400000000000003
和为: 0.008415
2行3列
HLC: 0.0345*0.5*0.3=0.0034500000000000004
LLC: 0.027*0.6*0.2=0.00324
和为: 0.00669
1行4列
HHA: 0.008415*0.5*0.2=0.0008415000000000001
LHA: 0.00669*0.4*0.3=0.0005352
和为: 0.0013767000000000002
2行4列
HLA: 0.008415*0.5*0.2=0.0012622500000000001
LLA: 0.00669*0.6*0.3=0.0012041999999999997
和为: 0.00246645

[2] 解码问题: 给定观测序列GGCACTGAA,问产生该序列概率最大的状态路径是什么?

把HMM模型换算为以2为底的对数值,则有
状态S： {H, L}
初始状态I： {-1, -1}
状态转移矩阵A：
a_ij H L
H -1 -1
L -1.322 -0.737
混淆矩阵B：
b_ik A C T G
H -2.322 -1.737 -1.737 -2.322
L -1.737 -2.322 -2.322 -1.737

结果矩阵:

概率 初始
G
G
C
A
C
T
G
A
A
H
 -1.0
-2.7369655941662066
-5.473931188332413
-8.210896782498619
-11.53282487738598
-14.006756065718395
-17.328684160605757
-19.539580943104376
-22.861509037991738
-25.65736832121151
L
 -1.0
-3.321928094887362
-6.058893689053569
-8.795859283219775
-10.947862376664826
-14.006756065718395
-16.480687254050807
-19.539580943104376
-22.013512131436787
-24.4874433197692
最佳状态序列:HHHLLLLLL
计算过程:
O2: H->H= -1.7369655941662063(H->G) +-1(T: HH) +-2.7369655941662066(H, 0) ==>-5.473931188332413
O2: L->H= -1.7369655941662063(H->G) +-1.3219280948873622(T: LH) +-3.321928094887362(L, 0) ==>-6.380821783940931
O2: H->L= -2.321928094887362(L->G) +-1(T: HL) +-2.7369655941662066(H, 0) ==>-6.058893689053569
O2: L->L= -2.321928094887362(L->G) +-0.7369655941662062(T: LL) +-3.321928094887362(L, 0) ==>-6.380821783940931
最佳: H
O3: H->H= -1.7369655941662063(H->C) +-1(T: HH) +-5.473931188332413(H, 1) ==>-8.210896782498619
O3: L->H= -1.7369655941662063(H->C) +-1.3219280948873622(T: LH) +-6.058893689053569(L, 1) ==>-9.117787378107138
O3: H->L= -2.321928094887362(L->C) +-1(T: HL) +-5.473931188332413(H, 1) ==>-8.795859283219775
O3: L->L= -2.321928094887362(L->C) +-0.7369655941662062(T: LL) +-6.058893689053569(L, 1) ==>-9.117787378107138
最佳: H
O4: H->H= -2.321928094887362(H->A) +-1(T: HH) +-8.210896782498619(H, 2) ==>-11.53282487738598
O4: L->H= -2.321928094887362(H->A) +-1.3219280948873622(T: LH) +-8.795859283219775(L, 2) ==>-12.4397154729945
O4: H->L= -1.7369655941662063(L->A) +-1(T: HL) +-8.210896782498619(H, 2) ==>-10.947862376664826
O4: L->L= -1.7369655941662063(L->A) +-0.7369655941662062(T: LL) +-8.795859283219775(L, 2) ==>-11.269790471552188
最佳: L
O5: H->H= -1.7369655941662063(H->C) +-1(T: HH) +-11.53282487738598(H, 3) ==>-14.269790471552188
O5: L->H= -1.7369655941662063(H->C) +-1.3219280948873622(T: LH) +-10.947862376664826(L, 3) ==>-14.006756065718395
O5: H->L= -2.321928094887362(L->C) +-1(T: HL) +-11.53282487738598(H, 3) ==>-14.854752972273342
O5: L->L= -2.321928094887362(L->C) +-0.7369655941662062(T: LL) +-10.947862376664826(L, 3) ==>-14.006756065718395
最佳: L
O6: H->H= -2.321928094887362(H->T) +-1(T: HH) +-14.006756065718395(H, 4) ==>-17.328684160605757
O6: L->H= -2.321928094887362(H->T) +-1.3219280948873622(T: LH) +-14.006756065718395(L, 4) ==>-17.65061225549312
O6: H->L= -1.7369655941662063(L->T) +-1(T: HL) +-14.006756065718395(H, 4) ==>-16.743721659884603
O6: L->L= -1.7369655941662063(L->T) +-0.7369655941662062(T: LL) +-14.006756065718395(L, 4) ==>-16.480687254050807
最佳: L
O7: H->H= -1.7369655941662063(H->G) +-1(T: HH) +-17.328684160605757(H, 5) ==>-20.065649754771965
O7: L->H= -1.7369655941662063(H->G) +-1.3219280948873622(T: LH) +-16.480687254050807(L, 5) ==>-19.539580943104376
O7: H->L= -2.321928094887362(L->G) +-1(T: HL) +-17.328684160605757(H, 5) ==>-20.65061225549312
O7: L->L= -2.321928094887362(L->G) +-0.7369655941662062(T: LL) +-16.480687254050807(L, 5) ==>-19.539580943104376
最佳: L
O8: H->H= -2.321928094887362(H->A) +-1(T: HH) +-19.539580943104376(H, 6) ==>-22.861509037991738
O8: L->H= -2.321928094887362(H->A) +-1.3219280948873622(T: LH) +-19.539580943104376(L, 6) ==>-23.1834371328791
O8: H->L= -1.7369655941662063(L->A) +-1(T: HL) +-19.539580943104376(H, 6) ==>-22.276546537270583
O8: L->L= -1.7369655941662063(L->A) +-0.7369655941662062(T: LL) +-19.539580943104376(L, 6) ==>-22.013512131436787
最佳: L
O9: H->H= -2.321928094887362(H->A) +-1(T: HH) +-22.861509037991738(H, 7) ==>-26.1834371328791
O9: L->H= -2.321928094887362(H->A) +-1.3219280948873622(T: LH) +-22.013512131436787(L, 7) ==>-25.65736832121151
O9: H->L= -1.7369655941662063(L->A) +-1(T: HL) +-22.861509037991738(H, 7) ==>-25.598474632157945
O9: L->L= -1.7369655941662063(L->A) +-0.7369655941662062(T: LL) +-22.013512131436787(L, 7) ==>-24.4874433197692
最佳: L

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