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‎15_EM/EM.py‎

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from numpy import *
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def initData(k,u,sigma,dataNum):
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'''
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初始化高斯混合模型的数据
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k:比例系数
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u:均值
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sigma:标准差
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dataNum:数据个数
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'''
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dataSet=zeros(dataNum,dtype=float)
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#高斯分布个数
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n=len(k)
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for i in range(dataNum):
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#产生0-1的随机数
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rand=random.random()
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sK=0
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index=0
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while index<n:
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sK+=k[index]
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if rand<sK:
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dataSet[i]=random.normal(u[index],sigma[index])
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break
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else:
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index+=1
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return dataSet
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def normalFun(x,u,sigma):
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'''
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计算均值为u,标准差为sigma的正太分布函数的密度函数值
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'''
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return (1.0/sqrt(2*pi)*sigma)*(exp(-(x-u)**2/(2*sigma**2)))
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def em(dataSet,k,u,sigma,step=10):
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'''
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高斯混合模型
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'''
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n=len(k)
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dataNum=len(dataArr)
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gamaArr=zeros((n,dataNum))
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for s in range(step):
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#E步，确定Q函数
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for i in range(n):
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for j in range(dataNum):
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wSum=sum([k[t]*normalFun(dataSet[j],u[t],sigma[t]) for t in range(n)])
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gamaArr[i][j]=k[i]*normalFun(dataSet[j],u[i],sigma[i])/float(wSum)
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#M步
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#更新u
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for i in range(n):
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u[i]=sum(gamaArr[i]*dataSet)/sum(gamaArr[i])
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#更新sigma
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for i in range(n):
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sigma[i]=sqrt(sum(gamaArr[i]*(dataSet-u[i])**2)/sum(gamaArr[i]))
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#更新k
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for i in range(n):
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k[i]=sum(gamaArr[i])/dataNum
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return [k,u,sigma]
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if __name__=='__main__':
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#参数的准确值
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k=[0.3,0.4,0.3]
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u=[2,4,3]
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sigma=[1,1,4]
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#样本数
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dataNum=5000
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dataArr=initData(k,u,sigma,dataNum)
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k0=[0.3,0.3,0.4]
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u0=[1,2,2]
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sigma0=[1,1,1]
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step=100
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k1,u1,sigma1=em(dataArr,k0,u0,sigma0,step)
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print("参数实际值：")
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print("k:",k)
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print("u:",u)
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print("sigma:",sigma)
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print("参数估计值：")
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print("k1:",k1)
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print("u1:",u1)
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print("sigma1:",sigma1)
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‎15_EM/README.md‎

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# EM算法(期望最大算法)
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## 概述
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EM算法是常用的估计参数隐变量的利器，它是一种迭代式的方法，其基本想法是：若参数已知，则可根据训练数据推断出最优隐变量的值；若最优隐变量的值已知，则可对参数做最大似然估计
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## Jensen不等式
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若f是定义域为实数的函数，如果对于所有实数x，f''(x)>=0，那么f是凸函数。当x是向量时，如果其hession矩阵是H是半正定的(H>=0)，那么f是凸函数。如果f''(x)>0或者H>0，那么f是严格的凸函数。
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Jensen不等式:
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如果f是凸函数，X是随机变量，那么E[f(X)]>=f(EX)
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如果f是严格凸函数，那么E[f(X)]=f(EX)当且仅当p(x=E[X])=1
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如图：
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/Jensen%E4%B8%8D%E7%AD%89%E5%BC%8F.PNG)
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## EM算法
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95E%E6%AD%A5.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95M%E6%AD%A51.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95M%E6%AD%A52.PNG)
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## EM算法在高斯混合模型中的应用
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B1.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B2.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B3.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B4.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B5.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B6.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B7.PNG)
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## 高斯混合模型算法实现
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```python
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from numpy import *
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def initData(k,u,sigma,dataNum):
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'''
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初始化高斯混合模型的数据
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k:比例系数
54+
u:均值
55+
sigma:标准差
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dataNum:数据个数
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'''
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dataSet=zeros(dataNum,dtype=float)
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#高斯分布个数
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n=len(k)
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for i in range(dataNum):
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#产生0-1的随机数
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rand=random.random()
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sK=0
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index=0
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while index<n:
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sK+=k[index]
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if rand<sK:
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dataSet[i]=random.normal(u[index],sigma[index])
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break
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else:
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index+=1
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return dataSet
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def normalFun(x,u,sigma):
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'''
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计算均值为u,标准差为sigma的正太分布函数的密度函数值
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'''
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return (1.0/sqrt(2*pi)*sigma)*(exp(-(x-u)**2/(2*sigma**2)))
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def em(dataSet,k,u,sigma,step=10):
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'''
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高斯混合模型
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'''
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n=len(k)
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dataNum=len(dataArr)
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gamaArr=zeros((n,dataNum))
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for s in range(step):
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#E步，确定Q函数
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for i in range(n):
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for j in range(dataNum):
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wSum=sum([k[t]*normalFun(dataSet[j],u[t],sigma[t]) for t in range(n)])
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gamaArr[i][j]=k[i]*normalFun(dataSet[j],u[i],sigma[i])/float(wSum)
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#M步
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#更新u
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for i in range(n):
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u[i]=sum(gamaArr[i]*dataSet)/sum(gamaArr[i])
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#更新sigma
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for i in range(n):
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sigma[i]=sqrt(sum(gamaArr[i]*(dataSet-u[i])**2)/sum(gamaArr[i]))
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#更新k
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for i in range(n):
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k[i]=sum(gamaArr[i])/dataNum
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return [k,u,sigma]
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```
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‎docs/15_EM算法(期望最大算法).md‎

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# EM算法(期望最大算法)
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## 概述
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EM算法是常用的估计参数隐变量的利器，它是一种迭代式的方法，其基本想法是：若参数已知，则可根据训练数据推断出最优隐变量的值；若最优隐变量的值已知，则可对参数做最大似然估计
6+
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## Jensen不等式
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若f是定义域为实数的函数，如果对于所有实数x，f''(x)>=0，那么f是凸函数。当x是向量时，如果其hession矩阵是H是半正定的(H>=0)，那么f是凸函数。如果f''(x)>0或者H>0，那么f是严格的凸函数。
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Jensen不等式:
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如果f是凸函数，X是随机变量，那么E[f(X)]>=f(EX)
14+
15+
如果f是严格凸函数，那么E[f(X)]=f(EX)当且仅当p(x=E[X])=1
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如图：
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/Jensen%E4%B8%8D%E7%AD%89%E5%BC%8F.PNG)
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## EM算法
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95E%E6%AD%A5.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95M%E6%AD%A51.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/EM%E7%AE%97%E6%B3%95M%E6%AD%A52.PNG)
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## EM算法在高斯混合模型中的应用
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B1.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B2.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B3.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B4.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B5.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B6.PNG)
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![](https://github.com/TonyJent/myMachineLearning/blob/master/images/15_EM/%E9%AB%98%E6%96%AF%E6%B7%B7%E5%90%88%E6%A8%A1%E5%9E%8B7.PNG)
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## 高斯混合模型算法实现
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```python
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from numpy import *
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def initData(k,u,sigma,dataNum):
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'''
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初始化高斯混合模型的数据
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k:比例系数
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u:均值
55+
sigma:标准差
56+
dataNum:数据个数
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'''
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dataSet=zeros(dataNum,dtype=float)
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#高斯分布个数
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n=len(k)
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for i in range(dataNum):
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#产生0-1的随机数
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rand=random.random()
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sK=0
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index=0
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while index<n:
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sK+=k[index]
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if rand<sK:
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dataSet[i]=random.normal(u[index],sigma[index])
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break
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else:
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index+=1
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return dataSet
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def normalFun(x,u,sigma):
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'''
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计算均值为u,标准差为sigma的正太分布函数的密度函数值
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'''
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return (1.0/sqrt(2*pi)*sigma)*(exp(-(x-u)**2/(2*sigma**2)))
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def em(dataSet,k,u,sigma,step=10):
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'''
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高斯混合模型
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'''
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n=len(k)
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dataNum=len(dataArr)
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gamaArr=zeros((n,dataNum))
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for s in range(step):
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#E步，确定Q函数
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for i in range(n):
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for j in range(dataNum):
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wSum=sum([k[t]*normalFun(dataSet[j],u[t],sigma[t]) for t in range(n)])
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gamaArr[i][j]=k[i]*normalFun(dataSet[j],u[i],sigma[i])/float(wSum)
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#M步
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#更新u
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for i in range(n):
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u[i]=sum(gamaArr[i]*dataSet)/sum(gamaArr[i])
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#更新sigma
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for i in range(n):
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sigma[i]=sqrt(sum(gamaArr[i]*(dataSet-u[i])**2)/sum(gamaArr[i]))
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#更新k
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for i in range(n):
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k[i]=sum(gamaArr[i])/dataNum
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return [k,u,sigma]
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```
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