Cophenetic correlation

Suppose that the original data {X_i} have been modeled using a cluster method to produce a dendrogram {T_i}; that is, a simplified model in which data that are "close" have been grouped into a hierarchical tree. Define the following distance measures.

• ${\displaystyle x(i,j)=|X_{i}-X_{j}|}$, the Euclidean distance between the ith and jth observations.
• ${\displaystyle t(i,j)}$, the dendrogrammatic distance between the model points ${\displaystyle T_{i}}$ and ${\displaystyle T_{j}}$. This distance is the height of the node at which these two points are first joined together.

Then, letting ${\displaystyle {\bar {x}}}$ be the average of the x(i, j), and letting ${\displaystyle {\bar {t}}}$ be the average of the t(i, j), the cophenetic correlation coefficient c is given by[4]

${\displaystyle c={\frac {\sum _{i<j}[x(i,j)-{\bar {x}}][t(i,j)-{\bar {t}}]}{\sqrt {\sum _{i<j}[x(i,j)-{\bar {x}}]^{2}\sum _{i<j}[t(i,j)-{\bar {t}}]^{2}}}}.}$

It is possible to calculate the cophenetic correlation in R using the dendextend R package.[5]

In Python, the SciPy package also has an implementation.[6]

In MATLAB, the Statistic and Machine Learning toolbox contains an implementation. [7]

• Cophenetic

• Numerical example of cophenetic correlation
• Computing and displaying Cophenetic distances