1. 前言
我们之前有介绍过4. EM算法-高斯混合模型GMM详细代码实现，在那片博文里面把GMM说涉及到的过程，可能会遇到的问题，基本讲了。今天我们升级下，主要一起解析下EM算法中GMM（搞事混合模型）带惩罚项的详细代码实现。
2. 原理
由于我们的极大似然公式加上了惩罚项，所以整个推算的过程在几个地方需要修改下。
在带penality的GMM中，我们假设协方差是一个对角矩阵，这样的话，我们计算高斯密度函数的时候，只需要把样本各个维度与对应的\(\mu_k\)和\(\sigma_k\)计算一维高斯分布，再相加即可。不需要通过多维高斯进行计算，也不需要协方差矩阵是半正定的要求。
我们给上面的(1)式加入一个惩罚项，
\[\lambda\sum_{k=1}^K\sum_{j=1}^P\frac{|\mu_k-\bar{x}_j|}{s_j}
\]
其中的\(P\)是样本的维度。\(\bar{x}_j\)表示每个维度的平均值，\(s_j\)表示每个维度的标准差。这个penality是一个L1范式，对\(\mu_k\)进行约束。
加入penality后(1)变为
\[L(\theta,\theta^{(j)})=\sum_{k=1}^Kn_k[log\pi_k-\frac{1}{2}(log(\boldsymbol{\Sigma_k})+\frac{{(x_i-\boldsymbol{\mu}_k})^2}{\boldsymbol{\Sigma}_k})] - \lambda\sum_{k=1}^K\sum_{j=1}^P\frac{|\mu_k-\bar{x}_j|}{s_j}
\]
这里需要注意的一点是，因为penality有一个绝对值，所以在对\(\mu_k\)求导的时候，需要分情况。于是(2)变成了
\[\mu_k=\frac{1}{n_k}\sum_{i=1}^N\gamma_{ik}x_i
\]
\[\mu_k=
\left \{\begin{array}{cc}
\frac{1}{n_k}(\sum_{i=1}^N\gamma_{ik}x_i - \frac{\lambda\sigma^2}{s_j}), & \mu_k >= \bar{x}_j\\
\frac{1}{n_k}(\sum_{i=1}^N\gamma_{ik}x_i + \frac{\lambda\sigma^2}{s_j}), & \mu_k < \bar{x}_j
\end{array}\right.
\]
3. 算法实现
和不带惩罚项的GMM不同的是，我们GMM+LASSO的计算高斯密度函数有所变化。
#计算高斯密度概率函数，样本的高斯概率密度函数，其实就是每个一维mu,sigma的高斯的和
def log_prob(self, X, mu, sigma):
N, D = X.shape
logRes = np.zeros(N)
for i in range(N):
a = norm.logpdf(X[i,:], loc=mu, scale=sigma)
logRes[i] = np.sum(a)
return logRes
在m-step中计算\(\mu_{k+1}\)的公式需要变化，先通过比较\(\mu_{kj}\)和\(means_{kj}\)的大小，来确定绝对值shift的符号。
def m_step(self, step):
gammaNorm = np.array(np.sum(self.gamma, axis=0)).reshape(self.K, 1)
self.alpha = gammaNorm / np.sum(gammaNorm)
for k in range(self.K):
Nk = gammaNorm[k]
if Nk == 0:
continue
for j in range(self.D):
if step >= self.beginPenaltyTime:
# 算出penality的偏移量shift，通过当前维度的mu和样本均值比较，确定shift的符号，相当于把lasso的绝对值拆开了
shift = np.square(self.sigma[k, j]) * self.penalty / (self.std[j] * Nk)
if self.mu[k, j] >= self.means[j]:
shift = shift
else:
shift = -shift
else:
shift = 0
self.mu[k, j] = np.dot(self.gamma[:, k].T, self.X[:, j]) / Nk - shift
self.sigma[k, j] = np.sqrt(np.sum(np.multiply(self.gamma[:, k], np.square(self.X[:, j] - self.mu[k, j]))) / Nk)
最后需要修改loglikelihood的计算公式
def GMM_EM(self):
self.init_paras()
for i in range(self.times):
#m step
self.m_step(i)
# e step
logGammaNorm, self.gamma= self.e_step(self.X)
#loglikelihood
loglike = self.logLikelihood(logGammaNorm)
#penalty
pen = 0
if i >= self.beginPenaltyTime:
for j in range(self.D):
pen += self.penalty * np.sum(abs(self.mu[:,j] - self.means[j])) / self.std[j]
# print("step = %s, alpha = %s, loglike = %s"%(i, [round(p[0], 5) for p in self.alpha.tolist()], round(loglike - pen, 5)))
# if abs(self.loglike - loglike) < self.tol:
# break
# else:
self.loglike = loglike - pen
4. GMM算法实现结果
用我实现的GMM+LASSO算法，对多个penality进行计算，选出loglikelihood最大的k和penality，与sklearn的结果比较。
fileName = amix1-est.dat, k = 2, penalty = 0 alpha = [0.52838, 0.47162], loglike = -693.34677
fileName = amix1-est.dat, k = 2, penalty = 0 alpha = [0.52838, 0.47162], loglike = -693.34677
fileName = amix1-est.dat, k = 2, penalty = 1 alpha = [0.52789, 0.47211], loglike = -695.26835
fileName = amix1-est.dat, k = 2, penalty = 1 alpha = [0.52789, 0.47211], loglike = -695.26835
fileName = amix1-est.dat, k = 2, penalty = 2 alpha = [0.52736, 0.47264], loglike = -697.17009
fileName = amix1-est.dat, k = 2, penalty = 2 alpha = [0.52736, 0.47264], loglike = -697.17009
myself GMM alpha = [0.52838, 0.47162], loglikelihood = -693.34677, bestP = 0
sklearn GMM alpha = [0.53372, 0.46628], loglikelihood = -176.73112
succ = 299/300
succ = 0.9966666666666667
[0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1]
[0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 0 1]
fileName = amix1-tst.dat, loglike = -2389.1852339407087
fileName = amix1-val.dat, loglike = -358.1157431278091
fileName = amix2-est.dat, k = 2, penalty = 0 alpha = [0.56, 0.44], loglike = 53804.54265
fileName = amix2-est.dat, k = 2, penalty = 0 alpha = [0.82, 0.18], loglike = 24902.5522
fileName = amix2-est.dat, k = 2, penalty = 1 alpha = [0.82, 0.18], loglike = 23902.65183
fileName = amix2-est.dat, k = 2, penalty = 1 alpha = [0.56, 0.44], loglike = 52929.96459
fileName = amix2-est.dat, k = 2, penalty = 2 alpha = [0.82, 0.18], loglike = 22907.40397
fileName = amix2-est.dat, k = 2, penalty = 2 alpha = [0.82, 0.18], loglike = 22907.40397
myself GMM alpha = [0.56, 0.44], loglikelihood = 53804.54265, bestP = 0
sklearn GMM alpha = [0.56217, 0.43783], loglikelihood = 11738677.90164
succ = 200/200
succ = 1.0
[0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1]
[0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1]
fileName = amix2-tst.dat, loglike = 51502.878096147084
fileName = amix2-val.dat, loglike = 6071.217012747491
fileName = golub-est.dat, k = 2, penalty = 0 alpha = [0.575, 0.425], loglike = -24790.19895
fileName = golub-est.dat, k = 2, penalty = 0 alpha = [0.525, 0.475], loglike = -24440.82743
fileName = golub-est.dat, k = 2, penalty = 1 alpha = [0.55, 0.45], loglike = -25582.27485
fileName = golub-est.dat, k = 2, penalty = 1 alpha = [0.6, 0.4], loglike = -26137.97508
fileName = golub-est.dat, k = 2, penalty = 2 alpha = [0.55, 0.45], loglike = -26686.02411
fileName = golub-est.dat, k = 2, penalty = 2 alpha = [0.55, 0.45], loglike = -26941.68964
myself GMM alpha = [0.525, 0.475], loglikelihood = -24440.82743, bestP = 0
sklearn GMM alpha = [0.5119, 0.4881], loglikelihood = 13627728.10766
succ = 29/40
succ = 0.725
[0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1]
[0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0]
fileName = golub-tst.dat, loglike = -12949.606698037718
fileName = golub-val.dat, loglike = -11131.35137056415
5. 总结
通过一番改造，实现了GMM+LASSO的代码，如果读者有什么好的改进方法，或者我有什么错误的地方，希望多多指教。
