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 C. Relationships between occurrence
and contingency tables.
If Tc is a centred-by-columns
occurrence table and X contains dummy variables corresponding to the
modalities of a single qualitative variable, then between-analysis is the GSVD
of (TcX, Dp, Dn),
which can be expressed by introducing the projection operator onto X as:
= TctDnX(XtDnX)-1 XtDnX(XtDnX)-1< b>XtDnTcp
= XtDnTc = [fkj
- fkfj] is the (m × p)
centred table of the relative frequencies fkj
of the occurrences that belong to modality k and species j;
fk is the marginal frequency of modality k and
fj the marginal frequency of species j;
Dm-1 = (XtDnX)-
1 is (m × m) diagonal matrix containing modality
weights equal to 1/fk.
The GSVD of (P0,
is a Correspondence Analysis (CA; Hill 1974), while the
GSVD of (P0, Ip, Dm-1)
is a Non-Symmetric Correspondence Analysis (NSCA) on row profiles (Lauro
and D'Ambra 1984).
Hill, M. O. 1974. Correspondence
analysis: a neglegted multivariate method. Journal of the Royal Statistical
Lauro, N., and L. D'Ambra 1984.
L'analyse non symétrique des correspondances. Pages 433–446 in
E. Diday, editor. Data analysis and informatics III, Elsevier, The Netherlands.
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