Gaussian Mixture Models (GMM)
Gaussian Mixture Models (GMM) are a sophisticated type of clustering that assumes all data points are generated from a mixture of a finite number of Gaussian (Normal) Distributions with unknown parameters.
Think of GMM as a "generalized" version of K-Means. While K-Means creates circular clusters, GMM can handle elliptical shapes and provides the probability of a point belonging to a cluster.
1. Hard vs. Soft Clustering
Most clustering algorithms provide "Hard" assignments. GMM provides "Soft" assignments.
- Hard Clustering (K-Means): "This point belongs to Cluster A. Period."
- Soft Clustering (GMM): "There is a 70% chance this point is in Cluster A, and a 30% chance it is in Cluster B."
2. How it Works: Expectation-Maximization (EM)
GMM uses a clever two-step iterative process to find the best-fitting Gaussians:
- Expectation (E-step): For each data point, calculate the probability that it belongs to each cluster based on current Gaussian parameters (mean, variance).
- Maximization (M-step): Update the Gaussian parameters (moving the center and stretching the shape) to better fit the points assigned to them.
3. The Power of Covariance Shapes
The "shape" of a Gaussian distribution is determined by its Covariance. In Scikit-Learn, you can control the flexibility of these shapes:
- Spherical: Clusters must be circular (like K-Means).
- Diag: Clusters can be ellipses, but only aligned with the axes.
- Tied: All clusters must share the same shape.
- Full: Each cluster can be any oriented ellipse. (Most Flexible)
4. Implementation with Scikit-Learn
from sklearn.mixture import GaussianMixture
# 1. Initialize the model
# n_components is the number of clusters
gmm = GaussianMixture(n_components=3, covariance_type='full', random_state=42)
# 2. Fit the model
gmm.fit(X)
# 3. Predict 'Soft' probabilities
# Returns an array of shape (n_samples, n_clusters)
probs = gmm.predict_proba(X)
# 4. Predict 'Hard' labels (picks the highest probability)
labels = gmm.predict(X)
5. Choosing the number of clusters: BIC and AIC
Since GMM is a probabilistic model, we don't use the "Elbow Method." Instead, we use information criteria:
- BIC (Bayesian Information Criterion)
- AIC (Akaike Information Criterion)
We look for the number of clusters that minimizes these scores. They reward a good fit but penalize the model for becoming too complex (having too many clusters).
6. GMM vs. K-Means
| Feature | K-Means | GMM |
|---|---|---|
| Cluster Shape | Strictly Circular (Spherical) | Flexible Ellipses |
| Assignment | Hard (0 or 1) | Soft (Probabilities) |
| Math | Distance-based | Density-based (Statistical) |
| Flexibility | Low | High |
References for More Details
- Scikit-Learn GMM Guide: Understanding advanced variants like Bayesian Gaussian Mixture Models.
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