The Limitations Of Elbow Curve And What You Should Replace It With

A better alternative to the Elbow curve.

We commonly use the Elbow curve to determine the number of clusters (k) for KMeans.

However, the Elbow curve:

• has a subjective interpretation

• involves ambiguity in determining the Elbow point accurately

• only considers a within-cluster distance, and more.

Silhouette score is an alternative measure used to evaluate clustering quality.

It is computed as follows:

• For every data point (i), find:

  • a(i): average distance to every other data point within the cluster

    

    Calculate a(i)

  • b(i): average distance to every data point in the nearest cluster.

  

  Calculate b(i)

• Silhouette score for a specific data point (i) is:

\(s_{i} = \frac{b_{i}-a_{i}}{max(a_{i},b_{i})}\)

• Silhouette score for the whole clustering is:

\(s = \frac{1}{n} \sum_{i}^{n}{s_{i}}\)

Some properties of the Silhouette score are:

• it ranges from [-1,1]

• a higher score indicates better clustering

• it can be used as an evaluation metric for clustering in the absence of ground truth labels

In contrast to the Elbow curve, the Silhouette score:

• provides a quantitative (and objective) measure

• involves no ambiguity

• considers BOTH within-cluster and between-cluster distance.

The visual below compares the Elbow curve and the Silhouette plot.

Elbow vs. Silhouette curve

It’s clear that the Elbow curve is highly misleading and inaccurate.

In a dataset with 25 clusters:

• The Elbow curve depicts 4 as the number of optimal clusters.

• The Silhouette curve depicts 25 as the number of optimal clusters.

Get started with Silhouette score here: Sklearn Docs.

👉 Over to you: What are some other measures to evaluate clustering quality?

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Find the code for my tips here: GitHub.

I like to explore, experiment and write about data science concepts and tools. You can read my articles on Medium. Also, you can connect with me on LinkedIn and Twitter.