Gaussian Mixture Model: Expectation-Maximization Clustering

Mixture Components

Number of clusters K 3

Steps per second 2.0

EM Progress

Iteration

Log-likelihood

—

Δ log-L: —

How it works

A Gaussian Mixture Model explains the data as a weighted sum of K Gaussian components. Expectation-Maximization fits it by alternating two steps until the likelihood stops improving:

E-step (soft assignment):
  r_ik = π_k N(x_i | μ_k, Σ_k) / Σ_j π_j N(x_i | μ_j, Σ_j)

M-step (refit each component):
  N_k = Σ_i r_ik
  μ_k = (1/N_k) Σ_i r_ik x_i
  Σ_k = (1/N_k) Σ_i r_ik (x_i-μ_k)(x_i-μ_k)^T
  π_k = N_k / N

r_ik is the "responsibility" of component k for point i — a probability, not a hard label. A point's color here is the blend of all its responsibilities, so purple-ish dots straddle two clusters.
Ellipses are the 2σ contour of each component's covariance Σ_k — its orientation and stretch come straight from the eigenvectors/eigenvalues of that 2×2 matrix, so a component can tilt and elongate to fit anisotropic clusters, unlike k-means' circular boundaries.
Log-likelihood Σ_i log(Σ_k π_k N(x_i|μ_k,Σ_k)) is guaranteed to never decrease across an E+M step; the run is "converged" once its change drops below 1e-4.
A small diagonal regularizer (1e-3·I) is added to every Σ_k so a component that collapses onto a single point never produces a singular, non-invertible covariance.

Unlike k-means (hard, spherical clusters) or DBSCAN (density contours, no explicit shape), a GMM gives every cluster its own ellipsoidal shape and a genuine probability of membership — the same soft-clustering idea underlies speaker diarization, background subtraction in video, and the E-step used inside many other latent-variable models.

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