了解高斯混合模型

Question

我正在尝试了解 scikit-learn 高斯混合模型实现的结果。看看下面的例子：

#!/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()

在这里，我首先生成由高斯构造的样本分布，然后将高斯混合模型拟合到这些数据。接下来，我想计算一些给定输入的概率。方便的是，scikit 实现提供了 score_samples 方法来做到这一点。现在我试图理解这些结果。我一直认为，我可以从 GMM 拟合中获取高斯参数，并通过对它们求和然后将积分归一化为 1 来构造完全相同的分布。但是，正如您在图中看到的那样，样本来自score_samples 方法与原始数据（蓝色直方图）完美匹配（红线），而手动构建的分布（黑线）则不匹配。我想了解我的想法哪里出错了，为什么我不能通过对 GMM 拟合给出的高斯求和来自己构建分布！？！非常感谢您的任何意见！

Answer 1

以防万一将来有人想知道同样的事情：必须规范化各个组件，而不是总和：

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()