python - GMM - 对数似然不是单调的

Question

昨天我使用期望最大化算法实现了 GMM(高斯混合模型)。

如您所知，它将一些未知分布建模为混合高斯分布，我们需要了解其均值和方差，以及每个高斯分布的权重。

这是代码背后的数学原理(没那么复杂) http://mccormickml.com/2014/08/04/gaussian-mixture-models-tutorial-and-matlab-code/

这是我的代码:

import numpy as np
from scipy.stats import multivariate_normal
import matplotlib.pyplot as plt

#reference for this code is http://mccormickml.com/2014/08/04/gaussian-mixture-models-tutorial-and-matlab-code/

def expectation(data, means, covs, priors): #E-step. returns the updated probabilities
    m = data.shape[0]                       #gets the data, means covariances and priors of all clusters
    numOfClusters = priors.shape[0]

    probabilities = np.zeros((m, numOfClusters))
    for i in range(0, m):
        for j in range(0, numOfClusters):
            sum = 0
            for l in range(0, numOfClusters):
                sum += normalPDF(data[i, :], means[l], covs[l]) * priors[l, 0]
            probabilities[i, j] = normalPDF(data[i, :], means[j], covs[j]) * priors[j, 0] / sum

    return probabilities

def maximization(data, probabilities): #M-step. this updates the means, covariances, and priors of all clusters
    m, n = data.shape
    numOfClusters = probabilities.shape[1]

    means = np.zeros((numOfClusters, n))
    covs = np.zeros((numOfClusters, n, n))
    priors = np.zeros((numOfClusters, 1))

    for i in range(0, numOfClusters):
        priors[i, 0] = np.sum(probabilities[:, i]) / m #update priors

        for j in range(0, m): #update means
            means[i] += probabilities[j, i] * data[j, :]

            vec = np.reshape(data[j, :] - means[i, :], (n, 1))
            covs[i] += probabilities[j, i] * np.dot(vec, vec.T) #update covs

        means[i] /= np.sum(probabilities[:, i])
        covs[i] /= np.sum(probabilities[:, i])

    return [means, covs, priors]

def normalPDF(x, mean, covariance): #this is simply multivariate normal pdf
    n = len(x)

    mean = np.reshape(mean, (n, ))
    x = np.reshape(x, (n, ))

    var = multivariate_normal(mean=mean, cov=covariance,)
    return var.pdf(x)


def initClusters(numOfClusters, data): #initialize all the gaussian clusters (means, covariances, priors
    m, n = data.shape

    means = np.zeros((numOfClusters, n))
    covs = np.zeros((numOfClusters, n, n))
    priors = np.zeros((numOfClusters, 1))

    initialCovariance = np.cov(data.T)

    for i in range(0, numOfClusters):
        means[i] = np.random.rand(n) #the initial mean for each gaussian is chosen randomly
        covs[i] = initialCovariance #the initial covariance of each cluster is the covariance of the data
        priors[i, 0] = 1.0 / numOfClusters #the initial priors are uniformly distributed.

    return [means, covs, priors]

def logLikelihood(data, probabilities): #data is our data. probabilities[i, j] = k means probability example i belongs in cluster j is 0 < k < 1
    m = data.shape[0] #num of examples

    examplesByCluster = np.zeros((m, 1))
    for i in range(0, m):
        examplesByCluster[i, 0] = np.argmax(probabilities[i, :])
    examplesByCluster = examplesByCluster.astype(int) #examplesByCluster[i] = j means that example i belongs in cluster j

    result = 0
    for i in range(0, m):
        result += np.log(probabilities[i, examplesByCluster[i, 0]]) #example i belongs in cluster examplesByCluster[i, 0]

    return result

m = 2000 #num of training examples
n = 8 #num of features for each example

data = np.random.rand(m, n)
numOfClusters = 2 #num of gaussians
numIter = 30 #num of iterations of EM
cost = np.zeros((numIter, 1))

[means, covs, priors] = initClusters(numOfClusters, data)

for i in range(0, numIter):
    probabilities = expectation(data, means, covs, priors)
    [means, covs, priors] = maximization(data, probabilities)

    cost[i, 0] = logLikelihood(data, probabilities)

plt.plot(cost)
plt.show()

问题是对数似然表现得很奇怪。我希望它是单调增加的。但事实并非如此。

例如，对于具有 3 个高斯聚类的 8 个特征的 2000 个示例，对数似然看起来像这样(30 次迭代)-

所以这很糟糕。但是在我运行的其他测试中，例如一个测试有 2 个特征和 2 个集群的 15 个示例，对数似然是这样的 -

更好，但仍不完美。

为什么会发生这种情况，我该如何解决？

Answer 1

问题在于最大化步骤。

代码使用means计算covs。然而，这是在同一个循环中完成的，在将 means 除以概率总和之前。

这会导致估计的协方差激增。

这里有一个建议的修复:

def maximization(data, probabilities): #M-step. this updates the means, covariances, and priors of all clusters
    m, n = data.shape
    numOfClusters = probabilities.shape[1]

    means = np.zeros((numOfClusters, n))
    covs = np.zeros((numOfClusters, n, n))
    priors = np.zeros((numOfClusters, 1))

    for i in range(0, numOfClusters):
        priors[i, 0] = np.sum(probabilities[:, i]) / m   #update priors

        for j in range(0, m): #update means
            means[i] += probabilities[j, i] * data[j, :]

        means[i] /= np.sum(probabilities[:, i])

    for i in range(0, numOfClusters):
        for j in range(0, m): #update means
            vec = np.reshape(data[j, :] - means[i, :], (n, 1))
            covs[i] += probabilities[j, i] * np.multiply(vec, vec.T) #update covs

        covs[i] /= np.sum(probabilities[:, i])

    return [means, covs, priors]

以及由此产生的成本函数(200 个数据点，4 个特征):

编辑: 我确信这个错误是代码中的唯一问题，但是运行一些额外的示例，我有时仍然会看到非单调行为(尽管比以前更不稳定)。所以这似乎只是问题的一部分。

编辑2: 协方差计算中还有另一个问题:向量乘法应该是逐元素的，而不是点积——记住结果应该是一个向量。结果现在似乎一直在单调递增。