Gaussian Mixture Model

A Gaussian mixture model (GMM) is a statistical model that represents a distribution as a mixture of multiple normal (Gaussian) distributions. It is a probabilistic model that assumes that the data is generated from a mixture of several underlying normal distributions, where each distribution is characterized by its mean and covariance matrix.

A GMM is typically fit to a dataset using the expectation-maximization (EM) algorithm, which iteratively estimates the parameters of the model (means, covariances, and mixing coefficients) by maximizing the likelihood of the data given the model.

Here is an example of fitting a GMM to a dataset using the sklearn.mixture library in Python:

import numpy as np
from sklearn.mixture import GaussianMixture

# Define the number of components (normal distributions) in the mixture
n_components = 3

# Fit the GMM to the data
gmm = GaussianMixture(n_components=n_components)
gmm.fit(X)

# Print the model parameters
print(f'Means: {gmm.means_}')
print(f'Covariances: {gmm.covariances_}')
print(f'Mixing coefficients: {gmm.weights_}')

In this example, X is the dataset, and n_components is the number of normal distributions in the mixture. The GaussianMixture class is initialized with the number of components, and the fit method is used to fit the model to the data. The model parameters (means, covariances, and mixing coefficients) are accessed using the means_, covariances_, and weights_ attributes of the gmm object, respectively.

To make predictions using the fitted GMM, you can use the predict_proba method, which returns the probability of each data point belonging to each component. You can also use the sample method to generate random samples from the GMM.

I hope this helps! Let me know if you have any questions.