*Ecological Archives*
E084-006-A3

**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 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 (**Tc**_{X}, **D**_{p}, **D**_{n}),
which can be expressed by introducing the projection operator onto X as:

** Tc**_{X}^{t}**D**_{n}**Tc**_{X}_{p}
= **P**_{X}(**Tc**)^{t}**D**_{n}**P**_{X}(**Tc**)_{p}

= (**X**(**X**^{t}**D**_{n}**X**)^{-1}**X**^{t}**D**_{n}
**Tc**)^{t}**D**_{n}**X**(**X**^{t}**D**_{n}**X**)^{-1}**X**^{t}**D**_{n}**Tc**_{p}

= **Tc**^{t}**D**_{n}**X**(**X**^{t}**D**_{n}**X**)^{-1} **X**^{t}**D**_{n}**X**(**X**^{t}**D**_{n}**X**)^{-1}< b>X^{t}**D**_{n}**Tc**_{p}

= **Tc**^{t}**D**_{n}**X**(**X**^{t}**D**_{n}**X**)^{-1}
**X**^{t}**D**_{n}**Tc**_{p}

= **P**_{0}^{t}**D**_{m}^{-1}**P**_{0}_{p}

where: **P**_{0}
= **X**^{t}**D**_{n}**Tc** = [*f*_{kj}
- *f*_{k}f_{j}] is the (*m* × *p*)
centred table of the relative frequencies *f*_{kj}
of the occurrences that belong to modality *k* and species *j*;
*f*_{k} is the marginal frequency of modality *k* and
*f*_{j} the marginal frequency of species *j*;
**D**_{m}^{-1} = (**X**^{t}**D**_{n}**X**)^{-
1} is (*m* × *m*) diagonal matrix containing modality
weights equal to 1/*f*_{k}.

The GSVD of (**P**_{0},
**D**_{p}^{-1}, **D**_{m}^{-1})
is a Correspondence Analysis (CA; Hill 1974), while the
GSVD of (**P**_{0}, **I**_{p}, **D**_{m}^{-1})
is a Non-Symmetric Correspondence Analysis (NSCA) on row profiles (Lauro
and D'Ambra 1984).

**Literature cited**

Hill, M. O. 1974. Correspondence
analysis: a neglegted multivariate method. Journal of the Royal Statistical
Society **23**:340–354.

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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