DBSCAN (Density-Based Spatial Clustering of Applications with Noise) employs density measures to mark points in high-density regions and those in low-density regions – the noise. Because of this natural behaviour of the algorithm, it is particularly useful in signal processing and similar application domains. ^[1].

One of the essential concepts is the point's p neighbourhood, which is the set of points reachable within the user-defined distance eps (epsilon):

, where:

• p – the point of interest;
• N(p) – neighbourhood of the point p;
• q – any other point;
• distance(p,q) – Euclidian distance between points q and p;
• eps – epsilon – user-defined distance constant;
• D – the initial set of points available for the algorithm;

The algorithm treats different points differently depending on density and neighbouring points distribution around the point – its neighbourhood:

1. Core Points:
   • A point is a core point if it has at least MinPts neighbours within a distance eps, where MinPts and eps are user-defined parameters, i.e. N(p) ≥ MinPts.
2. Directly Density-Reachable points:
   • A point is directly density-reachable from a core point if it lies within the distance eps of the core point.
3. Border Points:
   • A border point is not a core point but within the eps distance of a core point. Border points are part of a cluster but do not have enough neighbours to be core points.
4. Noise points:
   • Points that are not core and are not reachable from any core point are considered noise or outliers.

• The main steps in DBSCAN:
  • Selects a point from the data set – it might be the first in the data set or any randomly selected;
  • If it is a core point, form a cluster by grouping it with all directly density-reachable points.
  • Move to the next unvisited point and return to step 1.
  • Border points are added to the nearest cluster, and points that are not reachable from any core point are marked as noise.

• Advantages:
  • Can detect clusters of arbitrary shape.
  • Naturally identifies outliers or noise.
  • Unlike K-means, DBSCAN does not require specifying the number of clusters upfront.

• Disadvantages:
  • Results depend heavily on the choice of eps and MinPts.
  • It struggles with clusters of varying densities since eps is fixed.

DBSCAN is great for discovering clusters in data with noise, especially when clusters are not circular or spherical.