Cophenetic correlation
In statistics, and especially in biostatistics, cophenetic correlationSokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40 (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.Dorthe B. Carr, Chris J. Young, Richard C. Aster, and Xioabing Zhang, [http://www.osti.gov/bridge/servlets/purl/9576-lcvvCD/webviewable/9576.pdf Cluster Analysis for CTBT Seismic Event Monitoring] (a study prepared for the U.S. Department of Energy) This coefficient has also been proposed for use as a test for nested clusters.Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412 ([http://life.bio.sunysb.edu/ee/rohlf/reprints/RohlfFisher_1968.pdf link])
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.
- , the Euclidean distance between the ith and jth observations.
- , the dendrogrammatic distance between the model points and . This distance is the height of the node at which these two points are first joined together.
Then, letting be the average of the x(i, j), and letting be the average of the t(i, j), the cophenetic correlation coefficient c is given by[http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/cophenet.html Mathworks statistics toolbox]
:
c = \frac {\sum_{i
Software implementation
It is possible to calculate the cophenetic correlation in R using the dendextend R package.{{Cite web|url=https://cran.r-project.org/web/packages/dendextend/vignettes/dendextend.html|title=Introduction to dendextend}}
In Python, the SciPy package also has an implementation.{{Cite web|url=https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.cluster.hierarchy.cophenet.html|title=scipy.cluster.hierarchy.cophenet — SciPy v0.14.0 Reference Guide|website=docs.scipy.org|access-date=2019-07-11}}
In MATLAB, the Statistic and Machine Learning toolbox contains an implementation.{{Cite web|url=https://www.mathworks.com/help/stats/cophenet.html|title=Cophenetic correlation coefficient - MATLAB cophenet}}
See also
References
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External links
- [http://people.revoledu.com/kardi/tutorial/Clustering/index.html Numerical example of cophenetic correlation]
- [https://stackoverflow.com/questions/5639794/in-r-how-can-i-plot-a-similarity-matrix-like-a-block-graph-after-clustering-d Computing and displaying Cophenetic distances]
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