Cluster Analysis: Gaussian Mixture Model

Gaussian Mixture Model

Components (K) 3

EM speed 4/s

Iteration

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Converged

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Log-likelihood

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How it works

A Gaussian Mixture Model assumes the data was generated by K weighted Gaussian "blobs". Expectation-Maximization fits it by alternating two steps until the fit stops improving. The E-step computes, for every point, a soft responsibility — the probability it belongs to each component given the current parameters. The M-step then re-estimates each component's mean, variance and weight as a responsibility-weighted average over all points. Unlike k-means' hard 0/1 assignment, every point here contributes fractionally to every cluster — that's why the dots blend colour instead of snapping to one.

E:  γ_ik = π_k·N(x_i|μ_k,Σ_k) / Σ_j π_j·N(x_i|μ_j,Σ_j)
M:  μ_k = Σ_i γ_ik·x_i / Σ_i γ_ik
    σ_k² = Σ_i γ_ik·(x_i−μ_k)² / Σ_i γ_ik
    π_k = (1/N)·Σ_i γ_ik
LL = Σ_i log( Σ_k π_k·N(x_i|μ_k,Σ_k) )

Components (K) — how many Gaussians the model fits. Set it below the true number of underlying blobs and the ellipses merge; above it and they split a single blob in two.
EM speed — how many E/M iterations run per second; slow it down to watch a single responsibility update and ellipse reshape.
New data — resamples a fresh point cloud from 2–4 random overlapping blobs.
Reset fit — reinitialises μ/Σ/π and restarts EM on the same points, so you can compare runs that land on different local optima.

The log-likelihood is EM's proof of correctness in action: each E/M pass is guaranteed to never decrease it, so the number you're watching should only ever climb — a flattening curve is exactly what "converged" means.

Component 1 Component 2 Component 3

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