理解高斯混合模型

Question

我正在试图理解scikit学习高斯混合的结果
模型实现。请看以下示例：

#!/opt/local/bin/python
import numpy as np
import matplotlib.pyplot as plt
from sklearn.mixture import GaussianMixture

# Define simple gaussian
def gauss_function(x, amp, x0, sigma):
    return amp * np.exp(-(x - x0) ** 2. / (2. * sigma ** 2.))

# Generate sample from three gaussian distributions
samples = np.random.normal(-0.5, 0.2, 2000)
samples = np.append(samples, np.random.normal(-0.1, 0.07, 5000))
samples = np.append(samples, np.random.normal(0.2, 0.13, 10000))

# Fit GMM
gmm = GaussianMixture(n_components=3, covariance_type="full", tol=0.001)
gmm = gmm.fit(X=np.expand_dims(samples, 1))

# Evaluate GMM
gmm_x = np.linspace(-2, 1.5, 5000)
gmm_y = np.exp(gmm.score_samples(gmm_x.reshape(-1, 1)))

# Construct function manually as sum of gaussians
gmm_y_sum = np.full_like(gmm_x, fill_value=0, dtype=np.float32)
for m, c, w in zip(gmm.means_.ravel(), gmm.covariances_.ravel(), 
               gmm.weights_.ravel()):
    gmm_y_sum += gauss_function(x=gmm_x, amp=w, x0=m, sigma=np.sqrt(c))

# Normalize so that integral is 1    
gmm_y_sum /= np.trapz(gmm_y_sum, gmm_x)

# Make regular histogram
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=[8, 5])
ax.hist(samples, bins=50, normed=True, alpha=0.5, color="#0070FF")
ax.plot(gmm_x, gmm_y, color="crimson", lw=4, label="GMM")
ax.plot(gmm_x, gmm_y_sum, color="black", lw=4, label="Gauss_sum")

# Annotate diagram
ax.set_ylabel("Probability density")
ax.set_xlabel("Arbitrary units")

# Draw legend
plt.legend()
plt.show()

这里我首先生成一个由高斯分布构造的样本分布，然后用高斯混合模型拟合这些数据。接下来，我要计算
某些给定输入的概率。方便的是，scikit实现提供“score\u samples”方法来实现这一点。现在我在努力
了解这些结果。我一直在想，我可以从GMM得到的高斯参数拟合和构造非常相似
通过对它们求和，然后将积分正规化为1。但是，正如您在绘图中看到的，从
score\u samples方法与原始数据（蓝色）完全吻合（红线）
直方图），手动构造的分布（黑线）没有。我想知道我的想法哪里出错了，为什么我不能
通过对GMM给出的高斯分布求和来构造分布合身！？！非常感谢您的意见！

Answer 1

以防将来有人对同样的事情感到疑惑：一个人有
要使单个组件（而不是总和）标准化，请执行以下操作：

import numpy as np
import matplotlib.pyplot as plt
from sklearn.mixture import GaussianMixture

# Define simple gaussian
def gauss_function(x, amp, x0, sigma):
    return amp * np.exp(-(x - x0) ** 2. / (2. * sigma ** 2.))

# Generate sample from three gaussian distributions
samples = np.random.normal(-0.5, 0.2, 2000)
samples = np.append(samples, np.random.normal(-0.1, 0.07, 5000))
samples = np.append(samples, np.random.normal(0.2, 0.13, 10000))

# Fit GMM
gmm = GaussianMixture(n_components=3, covariance_type="full", tol=0.001)
gmm = gmm.fit(X=np.expand_dims(samples, 1))

# Evaluate GMM
gmm_x = np.linspace(-2, 1.5, 5000)
gmm_y = np.exp(gmm.score_samples(gmm_x.reshape(-1, 1)))

# Construct function manually as sum of gaussians
gmm_y_sum = np.full_like(gmm_x, fill_value=0, dtype=np.float32)
for m, c, w in zip(gmm.means_.ravel(), gmm.covariances_.ravel(), gmm.weights_.ravel()):
    gauss = gauss_function(x=gmm_x, amp=1, x0=m, sigma=np.sqrt(c))
    gmm_y_sum += gauss / np.trapz(gauss, gmm_x) * w

# Make regular histogram
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=[8, 5])
ax.hist(samples, bins=50, normed=True, alpha=0.5, color="#0070FF")
ax.plot(gmm_x, gmm_y, color="crimson", lw=4, label="GMM")
ax.plot(gmm_x, gmm_y_sum, color="black", lw=4, label="Gauss_sum", linestyle="dashed")

# Annotate diagram
ax.set_ylabel("Probability density")
ax.set_xlabel("Arbitrary units")

# Make legend
plt.legend()

plt.show()