GMM聚类

GMM聚类

高斯分布

GMM

一、参数初始化

# GMM 参数初始化
# dataset: [N,D]
# K : cluster的个数（高斯成分的个数）
def init_GMM(dataset,K):N,D = np.shape(dataset)val_max = np.max(dataset,axis=0)val_min = np.min(dataset,axis=0)   centers = np.linspace(val_min,val_max,num=K+2) mus = centers[1:-1,:]   sigmas = np.array([0.5*np.eye(D) for i in range(K)])  ws = 1.0/K * np.ones(K)return mus,sigmas,ws     # (K, D) (K, D, D) (K,)

二、计算 N(X;m,Σ)

# 计算一个高斯pdf
# x(dataset): 数据[N,D]
# sigma 方差[D,D]
# mu 均值[1,D] ；注意这里的mu是数据的均值（不是mus）    
def getPdf(x,mu,sigma,eps=1e-12):N,D = np.shape(x)if D==1:sigma = sigma+epsA = 1.0 / (sigma)det = np.fabs(sigma[0])else:sigma = sigma + eps*np.eye(D)A = np.linalg.inv(sigma)    # np.linalg.inv()这里矩阵求逆det = np.fabs(np.linalg.det(sigma)) # np.linalg.det()矩阵求行列式# 计算系数factor = (2.0 * np.pi)**(D / 2.0) * (det)**(0.5)# 计算 pdfdx = x - mupdf = [(np.exp(-0.5*np.dot(np.dot(dx[i],A),dx[i]))+eps)/ factor for i in range(N)]return pdf      

三、更新参数

def train_GMM_step(dataset,mus,sigmas,ws):N,D = np.shape(dataset)K,D = np.shape(mus)# 计算样本在每个Cluster上的Pdfpdfs = np.zeros([N,K])for k in range(K):      # 这里算出N(x;m...)pdfs[:,k] = getPdf(dataset,mus[k],sigmas[k])# 获取rr = pdfs*np.tile(ws,(N,1))     r_sum = np.tile(np.sum(r,axis=1,keepdims=True),(1,K))  r = r/r_sum                    # 这里为R ik# 进行参数更新for k in range(K):r_k = r[:,k]               # r_k.shape=(N,)N_k = np.sum(r_k)r_k = r_k[:,np.newaxis]    # r_k.shape=(N,1)# 更新mumu = np.sum(dataset*r_k,axis=0)/N_k   #[D,1]# 更新sigmadx = dataset - musigma = np.zeros([D,D])for i in range(N):sigma = sigma + r_k[i,0]*np.outer(dx[i],dx[i])  # np.outer用来求矩阵外积sigma = sigma/N_k# 更新ww = N_k/Nmus[k] = musigmas[k] = sigmaws[k] = wreturn mus,sigmas,ws        # (K, D) (K, D, D) (K,)

四、GMM训练

# GMM训练
def train_GMM(dataset,K,m=10):mus,sigmas,ws = init_GMM(dataset,K)# print(mus,sigmas,ws)for i in range(m):# print("step: ",i)mus,sigmas,ws = train_GMM_step(dataset,mus,sigmas,ws)# print(mus,sigmas,ws)return mus,sigmas,ws

五、计算数据在每个模型上的似然值

def getlogPdfFromeGMM(dataset,mus,sigmas,ws):N,D = np.shape(dataset)K,D = np.shape(mus)weightedlogPdf = np.zeros([N,K])for k in range(K):    temp = getPdf(dataset,mus[k],sigmas[k],eps = 1e-12)  weightedlogPdf[:,k] = np.log(temp) + np.log(ws[k])  # 这里为公式log(w*N())return weightedlogPdf, np.sum(weightedlogPdf,axis=1)

六、利用GMM进行聚类（通过比较似然值）

def clusterByGMM(datas,mus,sigmas,ws):weightedlogPdf,_ = getlogPdfFromeGMM(datas,mus,sigmas,ws)labs = np.argmax(weightedlogPdf,axis=1)return labs  # 得到分类标签

测试：

# 作图
def draw_cluster(dataset,lab,dic_colors):plt.cla()vals_lab = set(lab.tolist())for i,val in enumerate(vals_lab):index = np.where(lab==val)[0]sub_dataset = dataset[index,:]plt.scatter(sub_dataset[:,0],sub_dataset[:,1],s=16.,color=dic_colors[i])# GMM聚类测试
dic_colors = {0:(0.,0.5,0.),1:(0.8,0,0),2:(0.5,0.5,0),3:(0.5,0.5,0.9)}
a = np.random.multivariate_normal([2,2],[[.5,0],[0,.5]],200)
b = np.random.multivariate_normal([0,1],[[.5,0],[0,.5]],200)
c = np.random.multivariate_normal([1,4],[[.5,0],[0,.5]],200)
d = np.random.multivariate_normal([4,4],[[.5,0],[0,.5]],200)
dataset = np.r_[a,b,c,d]
# print(dataset.shape)# 真实标签
lab_true = np.r_[np.zeros(200),np.ones(200),2*np.ones(200),3*np.ones(200)].astype(int)# 绘制原始数据散点图
draw_cluster(dataset,lab_true,dic_colors)

原始数据

# 训练GMM
mus,sigmas,ws = train_GMM(dataset,K=4,m=10)
#print(mus,sigmas,ws)# 进行聚类
labs_GMM = clusterByGMM(dataset,mus,sigmas,ws)
#print(labs_GMM)# 绘制聚类后图片
draw_cluster(dataset,labs_GMM,dic_colors)

GMM聚类效果

GMM与K-means比较

原始数据

K-means聚类效果

GMM聚类效果

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