What is the EM algorithm in Python?

The EM algorithm in Python is an iterative method based on maximum likelihood estimation, which is commonly used for parameter estimation problems in unsupervised learning. This article will introduce the definition, basic principles, application scenarios and Python implementation of the EM algorithm.

1. Definition of EM algorithm

EM algorithm is the abbreviation of Expectation-maximization Algorithm. It is an iterative algorithm designed to solve the maximum likelihood estimate given the observed data.

In the EM algorithm, it is necessary to assume that the sample data comes from a certain probability distribution, and the parameters of the distribution are unknown and need to be estimated through the EM algorithm. The EM algorithm assumes that the unknown parameters can be divided into two categories, one is observable variables and the other is unobservable variables. Through iteration, the expected value of the unobservable variable is used as the estimated value of the parameter, and then the solution is solved again until convergence.

2. Basic principles of EM algorithm

1. E step (Expectation)

In the E step, it is necessary to calculate based on the current parameter estimates To find out the probability distribution of hidden variables, that is to find the conditional distribution of each hidden variable, which is the expected value of the hidden variable. This expected value is calculated based on the current parameter estimates.

2. M step (Maximization)

In the M step, the current parameter values ​​need to be re-estimated based on the expected value of the latent variable calculated in the E step. This estimate is calculated based on the expected value of the latent variable calculated in step E.

3. Update parameter values

Through the iteration of the E step and the M step, a set of parameter estimates will eventually be obtained. If the estimate converges, the algorithm ends, otherwise the iteration continues. Each iteration optimizes parameter values ​​until the optimal parameter estimate is found.

3. Application scenarios of EM algorithm

EM algorithm is widely used in the field of unsupervised learning, such as cluster analysis, model selection and hidden Markov model, etc., and has strong robustness It has the advantages of high flexibility and iterative efficiency.

For example, in clustering problems, the EM algorithm can be used for parameter estimation of Gaussian mixture models, that is, the observed data distribution is modeled as a mixture model of multiple Gaussian distributions, and the samples are grouped so that each group The data within them obey the same probability distribution. In the EM algorithm, the problem is solved by grouping the data in the E step and updating the parameters of the Gaussian distribution in the M step.

In addition, in image processing, the EM algorithm is often used in tasks such as image segmentation and image denoising.

4. Implementing EM algorithm in Python

In Python, there are many functions that can use the EM algorithm for parameter estimation, such as the EM algorithm implementation in the SciPy library and Gaussian in the scikit-learn library. Mixed model GMM, variational autoencoder VAE in TensorFlow library, etc.

The following is an introduction using the EM algorithm implementation of the SciPy library as an example. First, you need to import it into Pyhton as follows:

import scipy.stats as st
import numpy as np

Then, define the probability density function of a Gaussian mixture model as the optimization objective function of the EM algorithm:

def gmm_pdf(data, weights, means, covs):
    n_samples, n_features = data.shape
    pdf = np.zeros((n_samples,))
    for i in range(len(weights)):
        pdf += weights[i]*st.multivariate_normal.pdf(data, mean=means[i], cov=covs[i])
    return pdf

Next, define the function of the EM algorithm :

def EM(data, n_components, max_iter):
    n_samples, n_features = data.shape
    weights = np.ones((n_components,))/n_components
    means = data[np.random.choice(n_samples, n_components, replace=False)]
    covs = [np.eye(n_features) for _ in range(n_components)]

    for i in range(max_iter):
        # E步骤
        probabilities = np.zeros((n_samples, n_components))
        for j in range(n_components):
            probabilities[:,j] = weights[j]*st.multivariate_normal.pdf(data, mean=means[j], cov=covs[j])
        probabilities = (probabilities.T/probabilities.sum(axis=1)).T

        # M步骤
        weights = probabilities.mean(axis=0)
        means = np.dot(probabilities.T, data)/probabilities.sum(axis=0)[:,np.newaxis]
        for j in range(n_components):
            diff = data - means[j]
            covs[j] = np.dot(probabilities[:,j]*diff.T, diff)/probabilities[:,j].sum()

    return weights, means, covs

Finally, the following code can be used to test the EM algorithm:

# 生成数据
np.random.seed(1234)
n_samples = 100
x1 = np.random.multivariate_normal([0,0], [[1,0],[0,1]], int(n_samples/2))
x2 = np.random.multivariate_normal([3,5], [[1,0],[0,2]], int(n_samples/2))
data = np.vstack((x1,x2))

# 运行EM算法
weights, means, covs = EM(data, 2, 100)

# 输出结果
print('weights:', weights)
print('means:', means)
print('covs:', covs)

References:

[1] Xu, R. & Wunsch, D. C. (2005). Survey of clustering algorithms. IEEE Transactions on Neural Networks, 16(3), 645-678.

[2] Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent dirichlet allocation. Journal of Machine Learning Research, 3(4-5), 993-1022.

The above is the detailed content of What is the EM algorithm in Python?. For more information, please follow other related articles on the PHP Chinese website!