Vitali covering lemma: Difference between revisions
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The | 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> | ||
==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'') = | ''X<sub>i</sub>'' − ''X<sub>j</sub>'' |, 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]] | |||
Revision as of 10:30, 24 January 2014
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]
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) = | Xi − Xj |, 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]
See also
References
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External links
- ↑ Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40
- ↑ 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)
- ↑ Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412
- ↑ Mathworks statistics toolbox