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Gaussian Mixture Models

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Soft clustering with probability distributions: how GMMs work, how they're fitted and when to use them.

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A Gaussian mixture model (GMM) assumes data comes from a mix of several normal distributions, each representing a cluster.

Soft Assignments

Unlike k-means, which assigns each point to one cluster, a GMM gives the probability that each point belongs to each cluster. Useful when groups overlap.

Fitting

GMMs are usually fitted with the expectation-maximisation (EM) algorithm:

  1. E-step: estimate cluster membership probabilities.
  2. M-step: update each component's mean, covariance and weight.
  3. Repeat until stable.

Covariance Types

Components can be spherical, diagonal or fully flexible ellipses. More flexible shapes fit better but need more data.

Choosing the Number of Components

Use criteria such as BIC or AIC, which balance fit against complexity.

Uses

  • Clustering with overlapping groups.
  • Density estimation.
  • Anomaly detection: points with low likelihood are unusual.

Limitations

Sensitive to initialisation; can converge to poor solutions. Run several initialisations and assume roughly elliptical clusters.

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