Gaussian Mixture Model (GMM)
A Gaussian Mixture Model (GMM) is a probabilistic model that assumes that the data is generated from a mixture of several Gaussian distributions, each representing a different cluster or group within the data. GMMs are widely used in statistics and machine learning for tasks such as clustering, density estimation, and classification.
Each Gaussian distribution in a GMM is defined by its mean (the center of the distribution) and covariance (which describes the shape and orientation of the distribution). The overall model is a weighted sum of these Gaussian components, where the weights indicate the proportion of the data that belongs to each cluster.
Mathematically, the probability density function of a GMM can be expressed as:
P(x) = Σ (πk * N(x | μk, Σk))
Here, πk represents the weight of the k-th Gaussian component, and N(x | μk, Σk) denotes the probability density of the data point x under the k-th Gaussian with mean μk and covariance Σk.
To fit a GMM to data, algorithms such as the Expectation-Maximization (EM) algorithm are commonly used. The EM algorithm iteratively updates the parameters of the Gaussian components to maximize the likelihood of the observed data.
GMMs are particularly useful in scenarios where the data exhibits cluster-like structures and can be applied in various fields, including finance, image processing, and bioinformatics.