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.
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