正如我在评论中提到的，我看到的关键点是“手段”`初始化。遵循[sk]的默认实现混合物, 我没有随机初始化，而是切换到KMeans。
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
plt.style.use('seaborn')
eps=1e-8
def PDF(data, means, variances):
return 1/(np.sqrt(2 * np.pi * variances) + eps) * np.exp(-1/2 * (np.square(data - means) / (variances + eps)))
def EM_GMM(data, k=3, iterations=100, init_strategy='kmeans'):
weights = np.ones((k, 1)) / k # shape=(k, 1)
if init_strategy=='kmeans':
from sklearn.cluster import KMeans
km = KMeans(k).fit(data[:, None])
means = km.cluster_centers_ # shape=(k, 1)
else: # init_strategy=='random'
means = np.random.choice(data, k)[:, np.newaxis] # shape=(k, 1)
variances = np.random.random_sample(size=k)[:, np.newaxis] # shape=(k, 1)
data = np.repeat(data[np.newaxis, :], k, 0) # shape=(k, n)
for step in range(iterations):
# Expectation step
likelihood = PDF(data, means, np.sqrt(variances)) # shape=(k, n)
# Maximization step
b = likelihood * weights # shape=(k, n)
b /= np.sum(b, axis=1)[:, np.newaxis] + eps
# updage means, variances, and weights
means = np.sum(b * data, axis=1)[:, np.newaxis] / (np.sum(b, axis=1)[:, np.newaxis] + eps)
variances = np.sum(b * np.square(data - means), axis=1)[:, np.newaxis] / (np.sum(b, axis=1)[:, np.newaxis] + eps)
weights = np.mean(b, axis=1)[:, np.newaxis]
return means, variances
This seems to yield the desired output much more consistently:
s = np.array([25.31 , 24.31 , 24.12 , 43.46 , 41.48666667,
41.48666667, 37.54 , 41.175 , 44.81 , 44.44571429,
44.44571429, 44.44571429, 44.44571429, 44.44571429, 44.44571429,
44.44571429, 44.44571429, 44.44571429, 44.44571429, 44.44571429,
44.44571429, 44.44571429, 39.71 , 26.69 , 34.15 ,
24.94 , 24.75 , 24.56 , 24.38 , 35.25 ,
44.62 , 44.94 , 44.815 , 44.69 , 42.31 ,
40.81 , 44.38 , 44.56 , 44.44 , 44.25 ,
43.66666667, 43.66666667, 43.66666667, 43.66666667, 43.66666667,
40.75 , 32.31 , 36.08 , 30.135 , 24.19 ])
k=3
n_iter=100
means, variances = EM_GMM(s, k, n_iter)
print(means,variances)
[[44.42596231]
[24.509301 ]
[35.4137508 ]]
[[0.07568723]
[0.10583743]
[0.52125856]]
# Plotting the results
colors = ['green', 'red', 'blue', 'yellow']
bins = np.linspace(np.min(s)-2, np.max(s)+2, 100)
plt.figure(figsize=(10,7))
plt.xlabel('$x$')
plt.ylabel('pdf')
sns.scatterplot(s, [0.05] * len(s), color='navy', s=40, marker=2, label='Series data')
for i, (m, v) in enumerate(zip(means, variances)):
sns.lineplot(bins, PDF(bins, m, v), color=colors[i], label=f'Cluster {i+1}')
plt.legend()
plt.plot()
最后我们可以看到，纯随机初始化会产生不同的结果；让我们看看结果“means”：
for _ in range(5):
print(EM_GMM(s, k, n_iter, init_strategy='random')[0], '\n')
[[44.42596231]
[44.42596231]
[44.42596231]]
[[44.42596231]
[24.509301 ]
[30.1349997 ]]
[[44.42596231]
[35.4137508 ]
[44.42596231]]
[[44.42596231]
[30.1349997 ]
[44.42596231]]
[[44.42596231]
[44.42596231]
[44.42596231]]
在某些情况下，结果是如何不同的呢是常量，意味着选择了3个相似的值而没有 迭代时会有很大的变化。在EM\GMM中添加一些打印语句我会澄清的。
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