如何在python中实现EM-GMM？

Posted 2023-03-12

【中文标题】如何在python中实现EM-GMM？

【英文标题】：How can implement EM-GMM in python?

【发布时间】：2020-12-04 10:11:10

【问题描述】：

我使用此帖子GMMs and Maximum Likelihood Optimization Using NumPy 为GMM 实施了EM algorithm 未成功如下：

import numpy as np

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, iterations):
    weights = np.ones((k, 1)) / k # shape=(k, 1)
    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

当我在一维时间序列数据集上运行算法时，对于 k 等于 3，它返回如下输出：

array([[0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    3.05053810e-003, 2.36989898e-025, 2.36989898e-025,
    1.32797395e-136, 6.91134950e-031, 5.47347807e-001,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 2.25849208e-064, 0.00000000e+000,
    1.61228562e-303, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 3.94387272e-242,
    1.13078186e+000, 2.53108878e-001, 5.33548114e-001,
    9.14920432e-001, 2.07015697e-013, 4.45250680e-038,
    1.43000602e+000, 1.28781615e+000, 1.44821615e+000,
    1.18186109e+000, 3.21610659e-002, 3.21610659e-002,
    3.21610659e-002, 3.21610659e-002, 3.21610659e-002,
    2.47382844e-039, 0.00000000e+000, 2.09150855e-200,
    0.00000000e+000, 0.00000000e+000],
   [5.93203066e-002, 1.01647068e+000, 5.99299162e-001,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 2.14690238e-010,
    2.49337135e-191, 5.10499986e-001, 9.32658804e-001,
    1.21148135e+000, 1.13315278e+000, 2.50324069e-237,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 1.73966953e-125, 2.53559290e-275,
    1.42960975e-065, 7.57552338e-001],
   [0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
    3.05053810e-003, 2.36989898e-025, 2.36989898e-025,
    1.32797395e-136, 6.91134950e-031, 5.47347807e-001,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 1.44637007e+000, 1.44637007e+000,
    1.44637007e+000, 2.25849208e-064, 0.00000000e+000,
    1.61228562e-303, 0.00000000e+000, 0.00000000e+000,
    0.00000000e+000, 0.00000000e+000, 3.94387272e-242,
    1.13078186e+000, 2.53108878e-001, 5.33548114e-001,
    9.14920432e-001, 2.07015697e-013, 4.45250680e-038,
    1.43000602e+000, 1.28781615e+000, 1.44821615e+000,
    1.18186109e+000, 3.21610659e-002, 3.21610659e-002,
    3.21610659e-002, 3.21610659e-002, 3.21610659e-002,
    2.47382844e-039, 0.00000000e+000, 2.09150855e-200,
    0.00000000e+000, 0.00000000e+000]])

我认为这是错误的，因为输出是两个向量，其中一个代表means 值，另一个代表variances 值。让我对实现产生怀疑的模糊点是它会返回0.00000000e+000，因为它可以看到大多数输出​​，并且不需要真正可视化这些输出。顺便说一句，输入数据是时间序列数据。我已经检查了所有内容并多次跟踪，但没有出现任何错误。

这是我的输入数据：

[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      ]

我想知道是否有通过numpy 或SciKit-learn 实现它的优雅方式。任何帮助将不胜感激。

更新 以下是当前输出和预期输出：

【问题讨论】：

您能分享一下您认为这是错误的原因吗？可视化可能会有所帮助，但即使没有，似乎也缺少一些解释

@dia 输出是两个向量，其中一个代表means 值，另一个代表variances 值。让我对实现产生怀疑的模糊点是它会返回0.00000000e+000，因为它可以看到大多数输出​​，并且不需要真正可视化这些输出。顺便说一句，输入数据是时间序列数据。

可以可视化。我有一个关于平均值的链接帖子。你为什么不参考它。

@dia 你的意思是post 或GMM/EM on time series cluster。您是否有任何报价和解决方案可以在输出或解释我当前的结果时获得正确的结果？

一开始我会得到这些我觉得不合适的cmets，但现在我似乎得到了它们。请允许我改写一下。您必须让其他人更容易理解和解决您的问题，否则您必须自己解决。

【参考方案1】：

# Expectation step
likelihood = PDF(data, means, np.sqrt(variances))

为什么我们要通过sqrt 或variances？ pdf 函数接受方差。所以这应该是PDF(data, means, variances)。

另一个问题，

# Maximization step
b = likelihood * weights # shape=(k, n)
b /= np.sum(b, axis=1)[:, np.newaxis] + eps

上面的第二行应该是b /= np.sum(b, axis=0)[:, np.newaxis] + eps

也在variances的初始化中，

variances = np.random.random_sample(size=k)[:, np.newaxis] # shape=(k, 1)

为什么我们随机初始化方差？我们有data 和means，为什么不像vars = np.expand_dims(np.mean(np.square(data - means), axis=1), -1) 那样计算当前估计的方差？

通过这些更改，这是我的实现，

import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
plt.style.use('seaborn')

eps=1e-8


def pdf(data, means, vars):
    denom = np.sqrt(2 * np.pi * vars) + eps
    numer = np.exp(-0.5 * np.square(data - means) / (vars + eps))
    return numer /denom


def em_gmm(data, k, n_iter, init_strategy='k_means'):
    weights = np.ones((k, 1), dtype=np.float32) / k
    if init_strategy == 'k_means':
        from sklearn.cluster import KMeans
        km = KMeans(k).fit(data[:, None])
        means = km.cluster_centers_
    else:
        means = np.random.choice(data, k)[:, np.newaxis]
    data = np.repeat(data[np.newaxis, :], k, 0)
    vars = np.expand_dims(np.mean(np.square(data - means), axis=1), -1)
    for step in range(n_iter):
        p = pdf(data, means, vars)
        b = p * weights
        denom = np.expand_dims(np.sum(b, axis=0), 0) + eps
        b = b / denom
        means_n = np.sum(b * data, axis=1)
        means_d = np.sum(b, axis=1) + eps
        means = np.expand_dims(means_n / means_d, -1)
        vars = np.sum(b * np.square(data - means), axis=1) / means_d
        vars = np.expand_dims(vars, -1)
        weights = np.expand_dims(np.mean(b, axis=1), -1)

    return means, vars


def main():
    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, vars = em_gmm(s, k, n_iter)
    y = 0
    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, [0.0] * len(s), color='navy', s=40, marker=2, label='Series data')
    for i, (m, v) in enumerate(zip(means, vars)):
        sns.lineplot(bins, pdf(bins, m, v), color=colors[i], label=f'Cluster i + 1')
    plt.legend()
    plt.plot()

    plt.show()
    pass

这是我的结果。

【讨论】：

感谢您的意见。我想知道为什么侧高斯曲线的顶点如此sharp，为什么PDF超过1.0？

PDF 是密度函数。它的值可以超过 1.0。 （例如 dirac-delta pdf）但曲线下的面积（总概率）必须为 1.0

如果您觉得我的回答可以接受，请点赞我的回答。

`我想知道为什么侧高斯曲线的顶点如此尖锐`？为了绘制高斯曲线，我们对区间进行采样，并在没有任何平滑的情况下绘制 vaues 的线图。因此锐度

我明白你的观点！再次感谢您富有洞察力的意见。

【参考方案2】：

正如我在评论中提到的，我看到的关键点是 means 初始化。按照sklearn Gaussian Mixture的默认实现，我没有随机初始化，而是切换到了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

这似乎更一致地产生了所需的输出：

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]]

可以看到这些结果有多么不同，在某些情况下，结果均值是恒定的，这意味着 inizalization 选择了 3 个相似的值，并且在迭代时没有太大变化。在EM_GMM 中添加一些打印语句将阐明这一点。

【讨论】：

我发现这个实现有一些问题。我把它放在下面。

感谢您的实施。有一个严重的错误......这不起作用。在这一行b /= np.sum(b, axis=1)[:, np.newaxis] + eps 你应该使用axis=0

@ivallesp 我知道错误的解决方案，我无法删除它，因为已经给予赏金，我没有时间正确审查它，我将此问题标记给版主，但没有做任何事情.我认为马里奥应该接受 koshy george 的回答。