Gaussian mixture model

1. Sablon:Matematika A Gaussian Mixture Model (GMM) is a probabilistic model used to represent the presence of subpopulations within an overall population. It assumes that the data is generated from a mixture of several Gaussian distributions, each representing a cluster or subpopulation.

1. Mixture of Gaussians: The model is defined as a weighted sum of multiple Gaussian components, each with its own mean and variance. - Mathematically, the probability density function is: $$p(x)=\sum _{k=1}^K{\pi }_k𝒩(x\mid {\mu }_k,{\mathrm{\Sigma }}_k)$$ where ${\pi }_k$ are the mixture weights, $𝒩(x\mid {\mu }_k,{\mathrm{\Sigma }}_k)$ is the Gaussian distribution with mean ${\mu }_k$ and covariance ${\mathrm{\Sigma }}_k$, and $K$ is the number of components.

2. Unsupervised Learning: GMM is commonly used in unsupervised learning to discover clusters in the data. Unlike $k$-means clustering, GMM provides a probability for each point belonging to each cluster, offering a softer classification.

3. Expectation-Maximization (EM) Algorithm: To estimate the parameters (${}_k$, ${}_k$, ${}_k$), GMM uses the EM algorithm: - Expectation Step (E-Step): Calculate the probability that each data point belongs to each component. - Maximization Step (M-Step): Update the parameters of each component to maximize the likelihood of the data given these probabilities.

4. Applications: GMMs are used in a variety of fields such as clustering, density estimation, and anomaly detection. They are particularly useful when the data distribution is complex and can be better represented by a combination of Gaussian distributions rather than a single one.