Euclidean clusters and mixture components
Consider K-means with K=3 in two dimensions.
Describe the cluster boundaries and draw an example showing sample observations, the three fitted centres and the assignment regions.
Explain two differences between this method and a Gaussian finite mixture with component-specific covariance matrices, including the consequences for interpreting a fitted clustering.
Compare both the assignment rule and the object being fitted. Hard clusters and mixture components differ even when a picture makes their groups look similar.
- Nearest-centre assignment gives convex Voronoi cells. Pairwise boundaries are active segments or rays of perpendicular bisectors, with equation
2(\mathbf{m}_k-\mathbf{m}_j)^\top\mathbf{x}=\|\mathbf{m}_k\|^2-\|\mathbf{m}_j\|^2.
The sketch must include three centres, data and valid nearest-centre regions. The three-centre construction in What a K-means diagram must show supplies one exact picture: centres (0,0), (4,0) and (0,4), with active boundaries x_1=2 for x_2\le 2, x_2=2 for x_1\le 2, and x_1=x_2 for x_1\ge 2. Collinear centres, producing parallel boundaries, are also valid.
- Two differences: hard assignments against posterior membership probabilities, and one Euclidean distance against component densities that may have different covariances and weights. The consequences include a representation of uncertainty, ellipsoidal or differently oriented components, and curved boundaries for the maximum-a-posteriori assignment. Both are fitted summaries. Neither automatically discovers biological classes.