EM algorithm

https://en.wikipedia.org/wiki/EM_algorithm_and_GMM_model

EM (expectation maximization step)

• Algorithm:
  1. Given a set of incomplete data, consider a set of starting parameters.
  2. Expectation step (E – step): Using the observed available data of the dataset, estimate (guess) the values of the missing data.
  3. Maximization step (M – step): Complete data generated after the expectation (E) step is used in order to update the parameters.
  4. Repeat step 2 and step 3 until convergence.

• Use case:
  • discovering the values of latent variables
  • fill the missing data in a sample
  • basis of unsupervised learning of clusters
  • purpose of estimating the parameters of Hidden Markov Model (HMM)

The Expectation-Maximization (EM) algorithm is a powerful iterative method used for finding maximum likelihood estimates (MLE) or maximum a posteriori (MAP) estimates of parameters in statistical models when there are latent variables or missing data. It's particularly useful in situations where direct optimization of the likelihood function is not feasible due to the presence of these latent variables.

How EM Algorithm Works:

1. Expectation Step (E-step):
   • In the E-step, the algorithm computes the expected value of the latent variables given the observed data and the current estimate of the parameters. This step involves calculating the posterior distribution of the latent variables.
   • The E-step essentially involves calculating the posterior probabilities of the latent variables given the observed data and the current parameter estimates.

2. Maximization Step (M-step):
   • In the M-step, the algorithm updates the parameters to maximize the expected log-likelihood obtained from the E-step. This step involves finding the parameter values that maximize the expected log-likelihood.
   • The M-step essentially involves updating the parameter estimates using the expected values of the latent variables obtained from the E-step.

3. Iterative Process:
   • The E-step and M-step are repeated iteratively until convergence. In each iteration, the algorithm computes better estimates of the parameters until they stabilize.

Advantages of EM Algorithm:

1. Handles Missing Data: EM algorithm can handle situations where there are missing or incomplete data by estimating the missing values as part of the parameter estimation process.

2. Flexible: EM algorithm is quite flexible and can be applied to a wide range of statistical models, including mixture models, hidden Markov models, and more.

3. Converges to Local Optima: EM algorithm converges to a local maximum of the likelihood function, making it useful for finding parameter estimates in complex models.

Disadvantages of EM Algorithm:

1. Sensitive to Initializations: EM algorithm's convergence may depend on the initial values chosen for the parameters, and it may converge to different local optima based on the starting point.

2. Slow Convergence: The convergence of the EM algorithm can be slow, particularly in high-dimensional parameter spaces or when the likelihood function has many local maxima.

Applications of EM Algorithm:

1. Clustering: EM algorithm is commonly used for clustering data when the clusters have different distributions or when there are missing data.

2. Gene Expression Analysis: EM algorithm can be used to estimate gene expression levels from microarray data, where missing data may arise due to experimental noise.

3. Natural Language Processing: EM algorithm is used in various NLP tasks, such as part-of-speech tagging and speech recognition, where latent variables are present.

Example:

A classic example of the EM algorithm is the Gaussian Mixture Model (GMM), where the goal is to estimate the parameters of multiple Gaussian distributions that best fit the observed data. In this case, the latent variables represent the cluster assignments of the data points to the different Gaussian distributions.

Here's a simple Python code example using scikit-learn to fit a Gaussian Mixture Model using the EM algorithm:

from sklearn.mixture import GaussianMixture

# Initialize Gaussian Mixture Model
gmm = GaussianMixture(n_components=3)  # Number of components/clusters

# Fit the model to the data
gmm.fit(X)  # X is the observed data

# Get the estimated parameters
means = gmm.means_
covariances = gmm.covariances_
weights = gmm.weights_

In this example, X is the observed data, and the GaussianMixture class from scikit-learn fits a Gaussian Mixture Model to the data using the EM algorithm. After fitting the model, we can extract the estimated parameters such as the means, covariances, and weights of the Gaussian components.