Silhouette Coefficient

The silhouette coefficient is defined as:

$s(i)=\{\begin{array}{ll}1-a(i)/b(i)& \text{ if }a(i)<b(i)\\ 0& \text{ if }a(i)=b(i)\\ b(i)/a(i)-1& \text{ if }a(i)>b(i)\end{array}$

Where:

• $a(i)=\frac{1}{|C_i|-1}{\sum }_{j\in C_i,j\ne i}\delta (i,j)$ is the means distance between $i$ and all other data points in the same cluster $C_i$;
• $b(i)={\min }_{k\ne i}\frac{1}{|C_k|}{\sum }_{j\in C_k}\delta (i,j)$ is the smallest distance of $i$ to all points in any other cluster $C_k\ne C_i$.

The higher the coefficient, the better.

---

clustering