Here’s how the silhouette coefficient is calculated for each data point:
- a(i): The average distance from the ith data point to the other data points in the same cluster. It measures the cohesion within the cluster.
- b(i): The average distance from the ith data point to the data points in the nearest cluster (i.e., the cluster that the data point is not a part of). It measures the separation from other clusters.
The silhouette coefficient \(s(i)\) for a data point is given by the formula:
\(s(i) = \frac{b(i) – a(i)}{\max{a(i), b(i)}}\)
The overall silhouette coefficient for the entire clustering is the average of the silhouette coefficients for all data points. Mathematically, for a set of (n) data points:
\(\text{Silhouette Score} = \frac{1}{n} \sum_{i=1}^{n} s(i)\)
Interpretation of silhouette coefficient values:
- (\(s(i) \approx 1\)): The data point is well matched to the assigned cluster.
- (\(s(i) \approx 0\)): The data point is on or very close to the boundary between two adjacent clusters.
- (\(s(i) \approx -1\)): The data point may be assigned to the wrong cluster.
The silhouette coefficient is a useful metric for assessing the quality of clustering results, and it is often used to find the optimal number of clusters in techniques like k-means clustering. Higher silhouette coefficients indicate better-defined clusters with appropriate cohesion and separation.
