DBSCAN, point by point

The one question. DBSCAN never computes a center. It only ever asks, of one point at a time: are there at least minPts points within ε of me? If yes, that point is core and the cluster grows through it — every point inside its circle is swallowed, and each of those gets asked the same question. If no, the point is parked as noise, though a later cluster may still reach it and claim it as a border point. The pale blobs are the union of the ε-circles of every core point, which is exactly what a DBSCAN cluster is.

Three things worth trying.

1. Two rings, ε = 30. Press Run and watch one cluster crawl the whole way around a ring and stop. This is the case k-means cannot do at any k: its boundaries are straight lines, and no straight line separates two circles that share a center. DBSCAN never draws a boundary at all — it just follows contact.

2. Smiley, ε = 30. Three clusters and nobody supplied k: two eyes and a mouth, with the long curve kept in one piece — the exact answer k-means cannot reach at any k. And it is not a lucky setting: you get the same three clusters anywhere from ε = 20 to 60. Push ε down to 15 instead and it fragments into seven pieces — too short a reach and the algorithm loses the thread along the curve.

3. Elongated — two long thin clusters, the identical points used on the k-means and GMM pages. At the default ε = 30 you get 3 clusters, not 2: the cigars are sparse along their length, so one of them snaps in the middle (82 points and a stray group of 5). Nudge ε up to 35 and it settles on the right answer — 86 and 88 points, 6 noise. Shape was never the problem: DBSCAN does not care that a cluster is stretched, only whether the points are close enough to chain along it. Compare with k-means on these same points, which puts about 40% of them in the wrong group and cannot be fixed by restarting.

4. Uniform, and sweep ε slowly. There is no structure in this data, so watch what the algorithm invents: at ε = 15 almost everything is noise (152 of 160) and it reports 2 clusters; by ε = 30 it claims 11; at ε = 40, 15; and by ε = 60 the whole canvas is a single cluster. Every one of those answers is a fiction. Now compare Gaussian mixture, which sits on 3 clusters from ε = 30 all the way to 70. Stability across parameters is the evidence that structure is real — an answer that moves when you breathe on the slider is not a finding.