Vitali covering lemma: Difference between revisions

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In [[statistics]], and especially in [[biostatistics]], '''cophenetic correlation'''<ref>Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods.  Taxon, 11:33-40</ref> (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.<ref>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. [[United States Department of Energy|Department of Energy]])</ref> This coefficient has also been proposed for use as a test for nested clusters.<ref>Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets.  Systematic Zool., 17:407-412</ref>
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==Calculating the cophenetic correlation coefficient==
 
Suppose that the original data {''X<sub>i</sub>''} have been modeled using a cluster method to produce a dendrogram {''T<sub>i</sub>''}; 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'') = |&thinsp;''X<sub>i</sub>'' &minus; ''X<sub>j</sub>''&thinsp;|, the ordinary Euclidean distance between the ''i''th and ''j''th observations.
*''t''(''i'', ''j'') = the dendrogrammatic distance between the model points ''T<sub>i</sub>'' and ''T<sub>j</sub>''. 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<ref>[http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/cophenet.html Mathworks statistics toolbox]</ref>
 
:<math>
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]}}.
</math>
 
==See also==
*[[Cophenetic]]
 
==References==
{{Reflist}}
 
==External links==
* [http://people.revoledu.com/kardi/tutorial/Clustering/index.html Numerical example of cophenetic correlation]
* [http://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]
 
 
{{DEFAULTSORT:Cophenetic Correlation}}
[[Category:Covariance and correlation]]

Latest revision as of 07:20, 25 July 2014

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