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
定义在D-维连续空间的高斯分布概率密度的表达式
其等高线所形成的形状与协方差矩阵
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).Taxes[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).Taxes[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).Taxes[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).Tcov = 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=200x = 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})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]*K for 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]*K dim = 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]*K sumlog = [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] += 1 sumlog[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=200 for 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]*Kgammas = [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 = 0oldloglikelyhood = 1while np.abs(loglikelyhood - oldloglikelyhood)> 0.0001: oldloglikelyhood = loglikelyhood # E_step for 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_step for 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)联系客服
