---
title: Silhouette
showthedate: false
math: true
---

This technique validates the consistency within clusters of data. It provides a succinct graphical representation of how well each object lies in its cluster.

The silhouette ranges from -1 to 1 where a high value indicates that the object is consistent within its own cluster and poorly matched to neighboring clustesr.

A low or negative silhouette value can mean that the current clustering configuration has too many or too few clusters.

## Definition

For each datum $i$, let $a(i)$ be the average distance of $i$ with all other data within the same cluster.

$a(i)$ can be interpreted as how well $i$ is assigned to its cluster. (lower values mean better agreement)

We can then define the average dissimilarity of point $i$ to a cluster $c$ as the average distance from $i$ to all points in $c$.

Let $b(i)$ be the lowest average distance of $i$ to all other points in any other cluster in which i is not already a member.

The cluster with this lowest average dissimilarity is said to be the neighboring cluster of $i$. From here we can define a silhouette:
$$
s(i) = \frac{b(i) - a(i)}{max\{a(i), b(i)\}}
$$
The average $s(i)$ over all data of a cluster is a measure of how tightly grouped all the data in the cluster are. A silhouette plot may be used to visualize the agreement between each of the data and its cluster.

![img](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Silhouette-plot-orange.png/800px-Silhouette-plot-orange.png)

### Properties

Recall that $a(i)$ is a measure of how dissimilar $i$ is to its own cluster, a smaller value means that it's in agreement to its cluster. For $s(i)$ to be close to 1, we require $a(i) << b(i)$ .

If $s(i)$ is close to negative one, then by the same logic we can see that $i$ would be more appropriate if it was clustered in its neighboring cluster.

$s(i)$ near zero means that the datum is on the border of two natural clusters.

## Determining the number of Clusters

This can also be used in helping to determine the number of clusters in a dataset. The ideal number  of cluster is one that produces the highest silhouette value.

Also a good indication that one has too many clusters is if there are clusters with the majority of observations being under the mean silhouette value.

https://kapilddatascience.wordpress.com/2015/11/10/using-silhouette-analysis-for-selecting-the-number-of-cluster-for-k-means-clustering/

![plot_kmeans_silhouette_analysis_004](https://kapilddatascience.files.wordpress.com/2015/11/plot_kmeans_silhouette_analysis_004.png?w=660&h=257)