# skfolio.utils.stats.compute_optimal_n_clusters ### skfolio.utils.stats.compute_optimal_n_clusters(distance, linkage_matrix) Compute the optimal number of clusters based on Two-Order Difference to Gap Statistic [[1]](#re0e718a4c413-1). The Two-Order Difference to Gap Statistic has been developed to improve the performance and stability of the Tibshiranis Gap statistic. It applies the two-order difference of the within-cluster dispersion to replace the reference null distribution in the Gap statistic. The number of cluster $k$ is determined by: $$ \begin{cases} \begin{aligned} &\max_{k} & & W_{k+2} + W_{k} - 2 W_{k+1} \\ &\text{s.t.} & & 1 \ge c \ge max\bigl(8, \sqrt{n}\bigr) \\ \end{aligned} \end{cases} $$ with $n$ the sample size and $W_{k}$ the within-cluster dispersions defined as: $$ W_{k} = \sum_{i=1}^{k} \frac{D_{i}}{2|C_{i}|} $$ where $|C_{i}|$ is the cardinality of cluster $i$ and $D_{i}$ its density defined as: $$ D_{i} = \sum_{u \in C_{i}} \sum_{v \in C_{i}} d(u,v) $$ with $d(u,v)$ the distance between u and v. * **Parameters:** **distance** *ndarray of shape (n, n)* : Distance matrix. **linkage_matrix** *ndarray of shape (n - 1, 4)* : Linkage matrix. * **Returns:** **value** *int* : Optimal number of clusters. ### References * **[1]** “Application of two-order difference to gap statistic”. Yue, Wang & Wei (2009)