Silhouette Score

Score a clustering without any ground truth labels.

For each point i:

a(i) = mean distance from i to the other points in its own cluster
b(i) = min over other clusters C of the mean distance from i to the points in C
s(i) = (b(i) - a(i)) / max(a(i), b(i))

Return the mean of s(i) over all points.

The score runs from -1 to 1. Near 1 means points sit much closer to their own cluster than to any other; near 0 means they sit on a boundary; negative means they would be better off elsewhere.

Two cases need handling:

• A point alone in its cluster has no other members to average over, so a(i) is undefined. Define s(i) = 0 for singletons, which is the standard convention.
• If there is only one cluster in total, b(i) does not exist. Return 0.0.

Note that a(i) excludes the point itself. Including its own zero distance would shrink a towards zero and inflate every score.

Use Euclidean distance.

Function signature

def silhouette_score(X, labels):

Example

Input

X = np.array([[0.0], [0.1], [10.0], [10.1]])
labels = np.array([0, 0, 1, 1])
silhouette_score(X, labels)

Output

0.9899997499937498

Reasoning

For each point, a is 0.1, the distance to its one cluster mate. b is about 10, the mean distance to the other cluster. So every s(i) is close to 1 - 0.1 / 10 = 0.99.

Constraints and notes

• a(i) leaves out the point itself.
• A point alone in its cluster scores 0.
• With only one cluster, return 0.0.