目录
1.潜变量模型的学习
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与之对应，无法直接被观测到，需要通过模型和观测变量进行推断的变量就叫做 潜变量。常用的潜变量模型就包括了 GMM（高斯混合模型） 和 HMM（隐马尔可夫模型）。它们能够将将不完全数据（只有观测数据）的边缘分布转换成容易处理的完全数据（观测数据+潜变量）的联合分布。
2.K-Means聚类模型
K-Means聚类属于无监督学习算法，可以看作是一种特殊的，简化的混合高斯模型。它是将n个观测数据点按照一定标准划分到k个聚类中，数据点根据相似度划分。每一个聚类有一个质心，质心是对聚类中所有点的位置求平均值得到的点，每个观测点属于距离它最近的质心所代表的聚类。算法流程为1.先随机选择k个聚类质心。2.将每个观测点按照”距离”划分到离自己最近的质心并划分到此类。3.计算当前划分聚类的新质心。重复第2，3步，直至达到收敛条件。例如两次质心不再发生变化等。
通过K-Means聚类模型，可以对图像进行分割和压缩。
3.GMM模型和参数的估计 **
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GMM模型使用EM算法得到的参数估计
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; 4.EM算法**
利用迭代，在E步中求期望，在M步中最大化。关键是这个Q函数的构造。
5.总结
GMM模型与EM算法十分重要，但自己目前对于公式的推导还存在许多问题不能解决，课下应继续学习，与同学互相探讨。
6.作业代码
def calc_log_likelihood(self, X):
"""Calculate log likelihood of GMM
param: X: A matrix including data samples, num_samples * D
return: log likelihood of current model
"""
log_llh = 0.0
N = X.shape[0]
for n in range(N):
tmp = 0.0
for k in range(self.K):
tmp += self.pi[k] * self.gaussian(X[n], self.mu[k], self.sigma[k])
log_llh += np.log(tmp)
return log_llh
def em_estimator(self, X):
"""Update paramters of GMM
param: X: A matrix including data samples, num_samples * D
return: log likelihood of updated model
"""
log_llh = 0.0
N = X.shape[0]
gama = np.zeros((N, self.K))
for n in range(N):
for k in range(self.K):
gama[n][k] = self.pi[k] * self.gaussian(X[n], self.mu[k], self.sigma[k])
tmp = np.sum(gama, axis=1)
for n in range(N):
gama[n] /= tmp[n]
Nk = np.sum(gama, axis=0)
self.pi = Nk/N
self.mu = list()
for k in range(self.K):
tmp = np.zeros(self.dim)
for n in range(N):
tmp += X[n]*gama[n][k]
tmp /= Nk[k]
self.mu.append(tmp)
self.sigma = list()
for k in range(self.K):
tmp = np.zeros((self.dim, self.dim))
for n in range(N):
tmp += gama[n][k] * np.outer(X[n]-self.mu[k], X[n]-self.mu[k])
tmp /= Nk[k]
self.sigma.append(tmp)
log_llh = self.calc_log_likelihood(X)
return log_llh
作业网址：https://github.com/nwpuaslp/ASR_Course/tree/master/03-GMM-EM
Original: https://blog.csdn.net/weixin_44589825/article/details/125918180
Author: handsomeMB
Title: 第三讲 GMM以及EM算法学习笔记
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