Ecological Archives E084-006-A1

Raphaël Pélissier, Pierre Couteron, Stéphane Dray, and Daniel Sabatier. 2003. Consistency between ordination techniques and diversity measurements: two strategies for species occurrence data. Ecology 84:242–251.

Appendix A. Triplet notation.

Let Tc be a (n × p) matrix derived from any data table, Dn be a (n × n) diagonal matrix containing the associated row weights, and p be a (p × p) diagonal matrix containing the associated column weights. The resulting statistical triplet (sensu Escoufier 1987) is (Tc, p, Dn). The generalized singular value decomposition (GSVD; Greenacre 1984) of (Tc, p, Dn) consists in finding right-hand eigenvectors of TctDnTc p and Tc pTctDn:

svd(TctDnTc p) = UpDVpt

svd(TcpTc tDn) = UnDVnt

where, Dis the diagonal matrix of eigenvalues, such that Iner(Tc, p, Dn) = tr(D).

Rows (y) and columns (x) scores are then computed as:

x = Tc pVp and  y = Tc tDnVn

Literature cited

Escoufier, Y. 1987. The duality diagram: a means of better practicle applications. Pages 139–156 in P. Legendre and L. Legendre, editors. Development in numerical ecology, Springer-Verlag, Berlin, Germany.

Greenacre, M. J. 1984. Theory and apllications of correspondence analysis. Academic Press, London, UK.



[Back to E084-006]