#dbscan — Unsupervised learning

Group by density: epsilon, MinPts, core, border and noise points — the algorithm that finds arbitrary shapes and ignores intruders.

What you'll play with

Welcome to #dbscan. On the plane, 240 gray points: two interlocked crescents and about twenty intruders scattered at random. A 2-center k-means would slice this cloud into halves with a line and give a cluster to each intruder, whatever the cost. DBSCAN reasons differently: it looks at density. A point that has at least MinPts neighbors (itself included) within radius eps is a core; neighboring cores merge into one cluster, whatever its shape; whatever stays isolated is noise. Current settings, in the right panel: eps = 0.25, MinPts = 5.
Run the algorithm: /run. Watch the order of appearance: DBSCAN starts from a point, finds its neighbors within radius eps (the yellow ring), then the neighbors of its neighbors… the cluster spreads outward like an expanding stain.
Shrink the radius: /eps 0.1. With such a small ring, few points still count 5 neighbors.
The opposite now: /eps 0.5. A radius that big links everything within 0.5, including what shouldn't be linked.
Return to the good setting: /eps 0.25. How to pick it without trial and error? The right panel shows the median 4-distance: the typical distance from a point to its fourth neighbor (≈ 0.13 with this dataset). An eps slightly above yields cores everywhere in the moons without crossing the gap between them.
Second knob: /minpts 12. Being a core now requires 12 neighbors within the same radius.
The same dataset seen by k-means: /compare. k-means looks for as many centers as DBSCAN found clusters (2 here) and cuts the plane with the perpendicular bisector of the two centers (dashed line): each moon is sliced in two, and intruders get a cluster like everyone else.
To finish, replay the algorithm in slow motion: /step. Everything turns gray again, then a single point is revealed, with its eps ring and segments to its neighbors.
Your turn: /minpts 5 to return to the loose setting, then /dataset rings (two concentric circles: k-means cannot, DBSCAN can), /dataset noise (only intruders: DBSCAN finds nothing, or almost), /noise 0.3 (30% intruders), /neighbors 0 0 (place the ring on the point nearest the center), /reset to start over. Limitation to know: DBSCAN has a single eps, so clusters of very different densities escape it — this is the problem HDBSCAN solves. Next steps: #anomaly-detection (DBSCAN's noise as an intruder detector) and #hierarchical-clustering.

Channel commands

/eps <radius=0.05..1> — Epsilon neighborhood radius (reruns DBSCAN if already started).
/minpts <2..20> — Minimum number of neighbors (itself included) required to be a core point.
/run — Compute DBSCAN and replay the cluster expansion.
/step — Reveal the next point in visit order (step-by-step mode).
/neighbors <x=-2..2> <y=-2..2> — Place the eps ring on the point closest to (x, y) and link its neighbors.
/compare — Toggle coloring to the k-means result (and back).
/dataset <moons|blobs|rings|noise> — Change the point set (moons, blobs, rings or pure noise).
/noise <proportion=0..0.3> — Proportion of uniform intruders added to the dataset (0 to 0.3).
/seed <1..99> — New random draw of the points (deterministic).
/reset — Return to moons, 10% noise, eps 0.25, MinPts 5, nothing run.

Glossary

DBSCAN: Density-based clustering algorithm (Ester, Kriegel, Sander, Xu, 1996): two parameters, eps and MinPts, no number of clusters to fix. It finds clusters of arbitrary shape and explicitly labels isolated points as noise.
epsilon (neighborhood radius): Radius eps of the disk drawn around each point: any point at distance ≤ eps is a neighbor. Too small, everything becomes noise; too large, everything merges.
MinPts: Minimum number of neighbors (the point itself included) a point must have within its eps radius to be a core. Usual value: 2 × dimension, so 4 or 5 in 2D; raising it makes the algorithm more demanding and more robust to noise.
core point: A point that has at least MinPts neighbors within its eps radius: it belongs to the dense interior of a cluster and is allowed to expand it to its neighbors.
border point: A point that doesn't have enough neighbors to be a core, but is within the eps radius of some core: it joins that core's cluster without being able to expand it. This is the cluster's edge.
noise / outlier: A point that is neither a core nor a border: no core reaches it. DBSCAN gives it label −1 instead of forcing it into a cluster, which also makes it a rudimentary anomaly detector.
density: Number of points per unit area. DBSCAN measures it locally, counting neighbors within a disk of radius eps: a cluster is a region where this density exceeds the MinPts threshold, separated from others by sparse areas.
cluster expansion: The central mechanism: starting from a core, we add its neighbors to a queue; each neighbor that is itself a core adds its own, and so on until exhaustion. The cluster is the set of points density-reachable from the seed, hence arbitrary shapes.
k-distance plot: For each point, distance to its k-th neighbor (k = MinPts − 1, often 4), sorted. The curve rises gently for cluster points then takes off for intruders: the elbow indicates a good eps.
varying densities (DBSCAN limitation): With a single eps, DBSCAN cannot distinguish a compact cluster from a diffuse one: one gets fragmented or the other merged. HDBSCAN lifts this limit by exploring all eps values at once and keeping the most stable clusters.

Other channels in Unsupervised learning

#k-means — Grouping without labels: centroids that move, inertia that drops, the choice of k — and the shapes where k-means fails.
#pca — Principal component analysis: find the axes where data varies most, project, compress — and measure what is lost.
#hierarchical-clustering — Merge the points two at a time until only one is left: the dendrogram, the linkage criteria, and the cut height that decides the number of clusters.
#dbscan — Group by density: epsilon, MinPts, core, border and noise points — the algorithm that finds arbitrary shapes and ignores intruders.
#anomaly-detection — Spot what fits nothing: z-score / Mahalanobis, Isolation Forest, LOF — three ways to say 'this point is odd'.
#t-sne-umap — Map the high dimensions: t-SNE and UMAP unfold 10-dimensional data into a readable 2D map — perplexity, neighbors, and reading pitfalls.