Cluster Evaluation: Silhouette Score

 Technical Concept:

The Silhouette Score evaluates the cluster quality based on the notion of cohesion and separation both.

That means it evaluate the quality of clustering by measuring how well data points fit within their assigned clusters and how good separation is among clusters.  

It provides insight into the separation and compactness of clusters, combining information about intra-cluster similarity and inter-cluster dissimilarity.

Formula

For a given data point $i$:

• $a(i)$ ->  the average distance between point $i$ and all other points in the same cluster.
• $b(i)$  -> Calculate the data point's average distance to all the objects in any cluster (not containing i data point) . Find the minimum such value with respect to all clusters. 

The Silhouette Score $s(i)$ is computed as:

$$s<span\; face="Arial,\; sans-serif">(</span>\mathrm{<span\; face="Arial,\; sans-serif">i</span>}<span\; face="Arial,\; sans-serif">)</span><span\; face="Arial,\; sans-serif">=</span>\frac{}{}$$b(i)−a(i)max⁡(a(i),b(i))s(i) = \frac{b(i) - a(i)}{\max(a(i), b(i))}

• Silhouette coefficient can vary between -1 and 1.
• Negative value is undesirable. Because this is corresponding to the case where a(i), which is average distance of a point within the cluster is greater than b(i), the minimum average distance to points in another clusters.
• value 0 indicates that a(i) = b(i), that means point is on the boundary of cluster.
• Good value is positive value towards 1. That means a(i) should be lower, indicates high cohesion in cluster and b(i) should be higher, indicates good cluster separation.

• $s(i) \approx 1$: Point is well clustered.
• $s(i) \approx 0$: Point lies on or very close to the decision boundary between clusters.
• $s(i) < 0$: Point is assigned to the wrong cluster.

Example

Let’s consider a small dataset and calculate the Silhouette Score manually:

For point $A$, we compute:

$$a(A) = 2.0, \quad b(A) = 3.0$$
$$s(A) = \frac{3.0 - 2.0}{\max(2.0, 3.0)} = \frac{1.0}{3.0} \approx 0.33$$

Similarly, you can compute the Silhouette Score for all points. The average of these scores gives the overall Silhouette Score for the entire clustering solution.

Limitations

1. Non-convex clusters: It may perform poorly on data with complex, elongated clusters.
2. Imbalanced clusters: If cluster sizes vary significantly, the score can be biased toward larger clusters.
3. High-dimensional data: Silhouette can be less informative as distance measures in high-dimensional spaces can become less meaningful.

Pros and Cons of Silhouette Score

Pros

1. Intuitive: Provides a clear, interpretable score for cluster quality.
2. Balances cohesion and separation: It evaluates how compact the clusters are (intra-cluster distance) and how far apart clusters are (inter-cluster distance).

Cons

1. Assumes convex clusters: It works best for spherical, well-separated clusters and can give misleading results for non-convex clusters.
2. Distance-based: If the chosen distance metric doesn’t align with the underlying structure of the data, the results may not be meaningful.
3. High computational cost: It computes pairwise distances for every point, which can be expensive for large datasets.
4. Sensitive to outliers: Outliers can heavily impact the score by distorting cluster separation.

Comparison: Silhouette Score vs. Other Techniques:

Interview Questions:

Q. What is the Silhouette Score in clustering?

Answer:
The Silhouette Score is a metric used to measure the quality of clusters in a dataset. It evaluates how well data points are clustered by considering both intra-cluster cohesion (how similar points are within the same cluster) and inter-cluster separation (how distinct a point is from points in the nearest cluster). The score ranges from −1-1−1 to 111, where:

• 1 means the point is well clustered,

• 0 means the point is on or near the boundary between clusters,

Negative values mean the point is likely in the wrong cluster.

Q. What does a negative Silhouette Score indicate?

Answer: 

A negative Silhouette Score indicates that a data point is likely assigned to the wrong cluster. In this case, the point is closer (in terms of distance) to points in a different cluster than to points in its own cluster, suggesting poor clustering quality for that particular point.

Q. What does it mean if the Silhouette Score is close to zero?

Answer:

If the Silhouette Score is close to zero, it indicates that the point lies very close to the boundary between two clusters. This means the data point could fit into either of the clusters, which suggests that the clusters may not be well-separated. 

Q. Can the Silhouette Score handle non-convex clusters?

Answer:

No, the Silhouette Score tends to perform poorly with non-convex clusters. It works best for convex or spherical clusters because it relies on distance-based measurements (like Euclidean distance), which may not adequately capture the structure of more complex, elongated, or irregular clusters. 

Q. How does the Silhouette Score handle outliers?

Answer:

The Silhouette Score is sensitive to outliers. Outliers can negatively affect the score since they tend to increase intra-cluster distances (making clusters appear less cohesive) and may also distort the calculation of distances to other clusters. This can lower the overall Silhouette Score and suggest poor clustering, even if the main clusters are well-formed.