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2011 | OriginalPaper | Buchkapitel

3. Association Measures and Matrices

verfasst von : Daniel Borcard, François Gillet, Pierre Legendre

Erschienen in: Numerical Ecology with R

Verlag: Springer New York

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Abstract

Most methods of multivariate analysis, in particular ordination and clustering techniques, are explicitly or implicitly based on the comparison of all possible pairs of objects or descriptors. The comparisons take the form of association measures (often called coefficients or indices), which are assembled in a square and symmetrical association matrix, of dimensions n × n when objects are compared, or p × p when variables are compared.

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Vorheriges Kapitel Exploratory Data Analysis

Nächstes Kapitel Cluster Analysis

The association measure among objects may be implicit. It is the Euclidean distance in principal component analysis (PCA, Chap. 5) and k-means partitioning (Chap. 4), for example, and the chi-square distance in correspondence analysis (CA, Chap. 5).

Although the term “coefficient” is sometimes narrowly defined as a multiplicative factor of a variable in a mathematical expression, it has been applied for decades to the broader sense used in this book.

The use of the words symmetrical /asymmetrical for this distinction, as opposed to symmetric/asymmetric (same value of the coefficient between n ₁ and n ₂ as between n ₂ and n ₁), follows Legendre and Legendre (1998).

In this book, symbols and numbering of similarity and distance measures are taken from Legendre and Legendre (1998).

This way of presenting several distance measures (as pre-transformations followed by computation of a Euclidean distance) was described by Legendre and Gallagher (2001). More about this topic in Sect. 3.5 at the end of this chapter.

Gower, J. C. and Legendre, P.: Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification 3, 5–48 (1986)MATHCrossRefMathSciNet

Jaccard, P.: Étude comparative de la distribution florale dans une portion des Alpes et du Jura. Bulletin de la Société Vaudoise des Sciences Naturelles 37, 547–579 (1901)

Legendre, P.: Species Associations: The Kendall coefficient of concordance revisited. Journal of Agricultural, Biological, and Environmental Statistics 10, 226–245 (2005)CrossRef

Legendre, P. and Gallagher, E. D.: Ecologically meaningful transformations for ordination of species data. Oecologia 129, 271–280 (2001)CrossRef

Legendre, P. and Legendre, L.: Numerical ecology. 2nd English edition. Elsevier, Amsterdam (1998)

Titel: Association Measures and Matrices
verfasst von: Daniel Borcard
François Gillet
Pierre Legendre
Verlag: Springer New York
Buch: Numerical Ecology with R
Print ISBN: 978-1-4419-7975-9

Electronic ISBN: 978-1-4419-7976-6

Copyright-Jahr: 2011
DOI: https://doi.org/10.1007/978-1-4419-7976-6_3