Cophenetic correlation


Cophenetic correlation

In statistics, and especially in biostatistics, cophenetic correlation[1] (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the field of biostatistics (typically to assess cluster-based models of DNA sequences, or other taxonomic models), it can also be used in other fields of inquiry where raw data tend to occur in clumps, or clusters.[2] This coefficient has also been proposed for use as a test for nested clusters.[3]

Contents

Calculating the cophenetic correlation coefficient

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

  • x(i, j) = | XiXj |, the ordinary Euclidean distance between the ith and jth observations.
  • t(i, j) = the dendrogrammatic distance between the model points Ti and Tj. This distance is the height of the node at which these two points are first joined together.

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


c = \frac {\sum_{i<j} (x(i,j) - x)(t(i,j) - t)}{\sqrt{[\sum_{i<j}(x(i,j)-x)^2] [\sum_{i<j}(t(i,j)-t)^2]}}.

See also

References

  1. ^ Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40
  2. ^ Dorthe B. Carr, Chris J. Young, Richard C. Aster, and Xioabing Zhang, Cluster Analysis for CTBT Seismic Event Monitoring (a study prepared for the U.S. Department of Energy)
  3. ^ Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412
  4. ^ Mathworks statistics toolbox

External links


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