Detailed explanation of EM algorithm in Python

EM algorithm is a commonly used algorithm in statistical learning and is widely used in various fields. As an excellent programming language, Python has great advantages in implementing the EM algorithm. This article will introduce the EM algorithm in Python in detail.

First of all, we need to understand what the EM algorithm is. The EM algorithm is called Expectation-Maximization Algorithm. It is an iterative algorithm that is often used to solve parameter estimation problems containing hidden variables or missing data. The basic idea of ​​EM algorithm is to iteratively solve the maximum likelihood estimation of parameters by continuously estimating unobserved hidden variables or missing data.

Implementing the EM algorithm in Python can be divided into the following four steps:

1. E step

E step compares the observed data with the current The estimation of parameters computes the probability distribution of the latent variable. Essentially, the task of this step is to classify the sample data, cluster the observation data, and obtain the posterior distribution of the latent variables. In actual operations, you can use some clustering algorithms, such as K-means algorithm, GMM, etc.

2. M step

The task of the M step is to re-estimate the parameters through the E-step level classification. At this point, we only need to calculate the maximum likelihood estimate of the parameters in the data distribution of each category and re-update the parameters. This process can be implemented using some optimization algorithms, such as gradient descent and conjugate gradient algorithms.

3. Repeat steps 1 and 2

Next, we need to repeat steps 1 and 2 until the parameters converge and obtain parameters that satisfy the maximum likelihood estimation. This process is the iterative solution step in the EM algorithm.

4. Calculate the likelihood function value

Finally, we need to calculate the likelihood function value. By continuously executing the EM algorithm, the parameters are updated so that the parameter estimates maximize the likelihood function. At this time, we can fix the parameters, calculate the likelihood function value on the current data set, and use it as the objective function of optimization.

Through the above four steps, we can implement the EM algorithm in Python.

The code is as follows:

import numpy as np
import math

class EM:
    def __init__(self, X, k, max_iter=100, eps=1e-6):
        self.X = X
        self.k = k
        self.max_iter = max_iter
        self.eps = eps

    def fit(self):
        n, d = self.X.shape

        # 随机初始化分布概率和均值与协方差矩阵
        weight = np.random.random(self.k)
        weight = weight / weight.sum()
        mean = np.random.rand(self.k, d)
        cov = np.array([np.eye(d)] * self.k)

        llh = 1e-10
        previous_llh = 0

        for i in range(self.max_iter):
            if abs(llh - previous_llh) < self.eps:
                break
            previous_llh = llh

            # 计算隐变量的后验概率，即E步骤
            gamma = np.zeros((n, self.k))
            for j in range(self.k):
                gamma[:,j] = weight[j] * self.__normal_dist(self.X, mean[j], cov[j])
            gamma = gamma / gamma.sum(axis=1, keepdims=True)

            # 更新参数，即M步骤
            Nk = gamma.sum(axis=0)
            weight = Nk / n
            mean = gamma.T @ self.X / Nk.reshape(-1, 1)
            for j in range(self.k):
                x_mu = self.X - mean[j]
                gamma_diag = np.diag(gamma[:,j])
                cov[j] = x_mu.T @ gamma_diag @ x_mu / Nk[j]

            # 计算似然函数值，即求解优化目标函数
            llh = np.log(gamma @ weight).sum()

        return gamma

    def __normal_dist(self, x, mu, cov):
        n = x.shape[1]
        det = np.linalg.det(cov)
        inv = np.linalg.inv(cov)
        norm_const = 1.0 / (math.pow((2*np.pi),float(n)/2) * math.pow(det,1.0/2))
        x_mu = x - mu
        exp_val = math.exp(-0.5 * (x_mu @ inv @ x_mu.T).diagonal())
        return norm_const * exp_val

Among them,

X: observed data

k: number of categories

max_iter: maximum number of iteration steps

eps: Convergence threshold

fit() function: parameter estimation

__normal_dist(): Calculate multivariate Gaussian distribution function

Achieved through the above code, We can easily implement the EM algorithm in Python.

On top of this, the EM algorithm is also applied to various statistical learning problems, such as text clustering, image segmentation, semi-supervised learning, etc. Its flexibility and versatility have made it one of the classic algorithms in statistical learning. Especially for problems such as missing data and noisy data, the EM algorithm can be processed by estimating latent variables, which improves the robustness of the algorithm.

In short, Python is increasingly used in statistical learning, and more attention should be paid to the code implementation and model training of these classic algorithms. As one of the important algorithms, the EM algorithm also has a good optimization implementation in Python. Whether you are learning Python or statistical learning modeling, mastering the implementation of the EM algorithm is an urgent need.

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