高斯混合模型（GMM）实现和可视化

高斯分布公式及图像示例
高斯分布概率密度热力图
高斯混合模型实现代码
高斯混合模型聚簇效果图
参考文献

作者：金良（golden1314521@gmail.com） csdn博客： http://blog.csdn.net/u012176591

需要整理后的代码文件和数据请移步 http://download.csdn.net/detail/u012176591/8748673

1.高斯分布公式及图像示例

定义在D-维连续空间的高斯分布概率密度的表达式
N(x|μ,Σ)=1(2π)D/21|Σ|1/2exp{−12(x−μ)TΣ−1(x−μ)} N({\bf x}|\mu,\Sigma ) = \frac{1}{(2\pi)^{D/2}}\frac{1}{|\Sigma|^{1/2}}\exp\{-\frac{1}{2}({\bf x} - {\bf \mu})^T\Sigma^{-1}({\bf x} - {\bf \mu})\}

其等高线所形成的形状与协方差矩阵Σ\Sigma 密切相关，如下所示，后面的代码中有各个图像的对应的高斯分布的参数。

2. 高斯分布概率密度热力图

代码如下：

fig,axes = plt.subplots(nrows=3,ncols=1,figsize=(4,12))# 标准圆形
mean = [0,0]
cov = [[1,0],[0,1]]
x,y = np.random.multivariate_normal(mean,cov,5000).T
axes[0].plot(x,y,'x')
axes[0].set_xlim(-6,6)
axes[0].set_ylim(-6,6) # 椭圆，椭圆的轴向与坐标平行
mean = [0,0]
cov = [[0.5,0],[0,3]]
x,y = np.random.multivariate_normal(mean,cov,5000).T
axes[1].plot(x,y,'x')
axes[1].set_xlim(-6,6)
axes[1].set_ylim(-6,6) # 椭圆，但是椭圆的轴与坐标轴不一定平行
mean = [0,0]
cov = [[1,2.3],[2.3,1.4]]
x,y = np.random.multivariate_normal(mean,cov,5000).T
axes[2].plot(x,y,'x'); plt.axis('equal')
axes[2].set_xlim(-6,6)
axes[2].set_ylim(-6,6)

我们在下面的高斯混合模型中采用用第三种协方差矩阵，即概率密度的等高线是椭圆，且轴向不一定与坐标轴平行。

下图是高斯密度函数的热图：

以下是作图代码

# 自定义的高维高斯分布概率密度函数
def gaussian(x,mean,cov):    dim = np.shape(cov)[0] #维度covdet = np.linalg.det(cov+np.eye(dim)*0.01) #协方差矩阵的秩covinv = np.linalg.inv(cov+np.eye(dim)*0.01) #协方差矩阵的逆xdiff = x - mean#概率密度prob = 1.0/np.power(2*np.pi,1.0*2/2)/np.sqrt(np.abs(covdet))*np.exp(-1.0/2*np.dot(np.dot(xdiff,covinv),xdiff))return prob#作二维高斯概率密度函数的热力图
mean = [0,0]
cov = [[1,2.3],[2.3,1.4]]
x,y = np.random.multivariate_normal(mean,cov,5000).T
cov = np.cov(x,y) #由真实数据计算得到的协方差矩阵，而不是自己任意设定
n=200
x = np.linspace(-6,6,n)
y = np.linspace(-6,6,n)
xx,yy = np.meshgrid(x, y)
zz = np.zeros((n,n))
for i in range(n):for j in range(n):zz[i][j] = gaussian(np.array([xx[i][j],yy[i][j]]),mean,cov)gci = plt.imshow(zz,origin='lower') # 选项origin='lower' 防止tuixan图像颠倒
plt.xticks([5,100,195],[-5,0,5])
plt.yticks([5,100,195],[-5,0,5])
plt.title(u'高斯函数的热力图',{'fontname':'STFangsong','fontsize':18})

３.高斯混合模型实现代码：

下面是几个功能函数，在主函数中被调用

# 计算概率密度，
# 参数皆为array类型，过程中参数不变
def gaussian(x,mean,cov):dim = np.shape(cov)[0] #维度#之所以加入单位矩阵是为了防止行列式为0的情况covdet = np.linalg.det(cov+np.eye(dim)*0.01) #协方差矩阵的行列式covinv = np.linalg.inv(cov+np.eye(dim)*0.01) #协方差矩阵的逆xdiff = x - mean#概率密度prob = 1.0/np.power(2*np.pi,1.0*dim/2)/np.sqrt(np.abs(covdet))*np.exp(-1.0/2*np.dot(np.dot(xdiff,covinv),xdiff))return prob#获取初始协方差矩阵
def getconvs(data,K):convs = [0]*Kfor i in range(K):# 初始的协方差矩阵源自于原始数据的协方差矩阵，且每个簇的初始协方差矩阵相同convs[i] = np.cov(data.T)  return convsdef isdistinct(means,criter=0.03): #检测初始中心点是否靠得过近K = len(means)for i in range(K):for j in range(i+1,K):if criter > np.linalg.norm(means[i]-means[j]):return 0       return True
#获取初始聚簇中心
def getmeans(data,K,criter):means = [0]*Kdim  = np.shape(data)[1]minmax = [] #各个维度的极大极小值for i in range(dim):minmax.append(np.array([min(data[:,i]),max(data[:,i])]))while True:#生成初始点的坐标for i in range(K):means[i] = []for j in range(dim):means[i].append(np.random.random()*(minmax[j][1]-minmax[j][0])+minmax[j][0])  means[i] = np.array(means[i])if isdistinct(means,criter):break  return means# k-means算法的实现函数。
#用K-means算法输出的聚类中心，作为高斯混合模型的输入
def kmeans(data,K):N = np.shape(data)[0]#样本数目dim = np.shape(data)[1] #维度means = getmeans(data,K,criter=15)means_old = [np.zeros(dim) for k in range(K)]while np.sum([np.linalg.norm(means_old[k]-means[k]) for k in range(K)]) > 0.01:means_old = cp.deepcopy(means)numlog = [0]*Ksumlog = [np.zeros(dim) for k in range(K)]for n in range(N):distlog = [np.linalg.norm(data[n]-means[k]) for k in range(K)]toK = distlog.index(np.min(distlog))numlog[toK] += 1sumlog[toK] += data[n]for k in range(K):means[k] = 1.0/numlog[k]*sumlog[k]return means#对程序结果进行可视化，注意这里的K只能取2，否则该函数运行出错
def visualresult(data,gammas,K):N = np.shape(data)[0]#样本数目dim = np.shape(data)[1] #维度minmax = [] #各个维度的极大极小值xy = []n=200for i in range(dim):delta = 0.05*(np.max(data[:,i])-np.min(data[:,i]))xy.append(np.linspace(np.min(data[:,i])-delta,np.max(data[:,i])+delta,n))xx,yy = np.meshgrid(xy[0], xy[1])zz = np.zeros((n,n))for i in range(n):for j in range(n):zz[i][j] = np.sum(gaussian(np.array([xx[i][j],yy[i][j]]),means[k],convs[k]) for k in range(K))gci = plt.imshow(zz,origin='lower',alpha = 0.8) # 选项origin='lower' 防止tuixan图像颠倒plt.xticks([0,len(xy[0])-1],[xy[0][0],xy[0][-1]])plt.yticks([0,len(xy[1])-1],[xy[1][0],xy[1][-1]])for i in range(N):if gammas[i][0] >0.5:plt.plot((data[i][0]-np.min(data[:,0]))/(xy[0][1]-xy[0][0]),(data[i][1]-np.min(data[:,1]))/(xy[1][1]-xy[1][0]),'r.')else:plt.plot((data[i][0]-np.min(data[:,0]))/(xy[0][1]-xy[0][0]),(data[i][1]-np.min(data[:,1]))/(xy[1][1]-xy[1][0]),'k.')deltax = xy[0][1]-xy[0][0]deltay = xy[1][1]-xy[1][0]plt.plot((means[0][0]-xy[0][0])/deltax,(means[0][1]-xy[1][0])/deltay,'*r',markersize=15)plt.plot((means[1][0]-xy[0][0])/deltax,(means[1][1]-xy[1][0])/deltay,'*k',markersize=15)plt.title(u'高斯混合模型图',{'fontname':'STFangsong','fontsize':18})

高斯混合模型的主函数

N = np.shape(data)[0]#样本数目
dim = np.shape(data)[1] #维度
K = 2 # 聚簇的个数means = kmeans(data,K)convs = getconvs(data,K)pis = [1.0/K]*K
gammas = [np.zeros(K) for i in range(N)] #*N 注意不能用 *N，否则N个array只指向一个地址loglikelyhood = 0
oldloglikelyhood = 1while np.abs(loglikelyhood - oldloglikelyhood)> 0.0001:oldloglikelyhood = loglikelyhood# E_stepfor n in range(N):respons = [pis[k]*gaussian(data[n],means[k],convs[k]) for k in range(K)]sumrespons = np.sum(respons)for k in range(K):gammas[n][k] = respons[k]/sumrespons# M_stepfor k in range(K):nk = np.sum([gammas[n][k] for n in range(N)])means[k] = 1.0/nk * np.sum([gammas[n][k]*data[n] for n in range(N)],axis=0)xdiffs = data - means[k]convs[k] = 1.0/nk * np.sum([gammas[n][k]*xdiffs[n].reshape(dim,1)*xdiffs[n] for n in range(N)],axis=0)pis[k] = 1.0*nk/N# 计算似然函数值loglikelyhood =np.sum( [np.log(np.sum([pis[k]*gaussian(data[n],means[k],convs[k]) for k in range(K)])) for n in range(N) ])#print means#print loglikelyhood#print '=='*10visualresult(data,gammas,K)

4.高斯混合模型聚簇效果图

5.参考文献：

K-means聚类和EM思想
http://www.cnblogs.com/jerrylead/archive/2011/04/06/2006910.html
（EM算法）The EM Algorithm
http://www.cnblogs.com/jerrylead/archive/2011/04/06/2006936.html
从决策树学习谈到贝叶斯分类算法、EM、HMM
http://blog.csdn.net/v_july_v/article/details/7577684
EM算法学习(Expectation Maximization Algorithm)
http://www.vjianke.com/XUHV3.clip
EM算法——理论与应用
http://blog.sina.com.cn/s/blog_a8fead9b01014p6k.html
EM算法一个简单的例子
http://blog.sina.com.cn/s/blog_a7da5cda010158b3.html
高斯混合模型（GMM）
http://www.cnblogs.com/mindpuzzle/archive/2013/04/24/3036447.html
高斯混合模型
http://www.cnblogs.com/zhangchaoyang/articles/2624882.html
CS229 Lecture notes,Andrew Ng讲义
http://cs229.stanford.edu/notes/cs229-notes7b.pdf
https://github.com/lzhang10/maxent/blob/master/doc/manual.pdf
http://nlp.stanford.edu/software/classifier.shtml
https://github.com/juandavm/em4gmm
用最大期望算法求解高斯混合模型，dat文件夹里的压缩文件是数据
http://insideourminds.net/python-unsupervised-learning-using-em-algorithm-implementation/
Python实现的期望最大化算法，数据是鸢尾花数据