Pierre Couteron and Sébastien Ollier. 2005. A generalized, variogram-based framework for multi-scale ordination. Ecology 86:828–834.

Appendix A. Matrix-algebraic presentation of the concepts and computations. A pdf file of this appendix is also available.

General denotation

Let X be a table expressing a measure of the abundance x_ai of S species (columns) within Q quadrats (rows). x_i and x_j are two columns of table X, relating to species i and j, respectively. Let D be a matrix containing quadrat weights (_a , ) on its main diagonal and zeros for all off-diagonal values, and let W be a S by S matrix containing the square root of species weights ( ) on its main diagonal and zeros outside.  (In the main paper, Table 1 gives some options for abundance re-scaling and for quadrat and species weighting.)

Contiguity relationships

Let L_h be a Q by Q matrix expressing a contiguity relationship (sensu Lebart 1969) between the quadrats. For our variogram-based approach, we consider quadrats a and b as neighbors if the distance between the two is within the bounds of the distance class centered around h:

Lh(a,b) = 1  if ha,bh  and Lh(a,b) = 0 otherwise.

(A.1)

To introduce quadrat weights into the analysis, we define the matrix M_h and the vector E_h such that:

and

(A.2)

where 1_Qis the vector containing Q values equal to 1.

M_hcontains, for each pair (a,b) of neighboring quadrats at "scale" h, theproduct _ab of their weights. E_h features, for each quadrat a, the sum of the weights of its neighbors multiplied by _a.Let N_h be the Q by Q matrix with E_h on its main diagonal and zeros elsewhere.

We shall assume that distance classes include all pairs of quadrats while being mutually exclusive. In such a case, the two following matrices:

and

(A.2b)

are such that M_T is a Q by Q matrix that containing zeros on the diagonal while all values off the diagonal are equal to _ab; N_T is a Q by Q matrix thatcontaining (1 – _a)_a values on the diagonal and zeros elsewhere. With M_T and N_T it is as if each quadrat has all other quadrats as neighbors. Denoting I_Q the Q by Q diagonal identity matrix, we can also write

and

(A.3)

(where the exponent '^t' is the matrix transpose). Thus:

.

(A.4)

Equivalent expressions of the generalized variance–covariance matrix

Let G_T be the generalized variance–covariance matrix, irrespective of distance classes, that can be directly computed from table X using weighting options for rows and columns defined by matrices D and W, respectively. G_T contains, for each species couple (i,j), the generalized covariances, g_ij as defined by Eq. 1 and Eq. 2 in the main paper:

.

(A.5)

Usual algebraic manipulations allow us to re-write Eq. 1 and Eq. A.5 as:

(A.6)

where and are the D-weighted means of x_i and x_j, respectively.

( or, equivalently, ).

The matrix expression of g_ij is thus

(A.7)

 which generalizes into

	(A.8)
where  and where  .	(A.9)

Note that we may also write

.

(A.10)

On the other hand, it is important to note that G_T can be directly computed as

.

(A.11)

Proof of Eq. A.11:
(using Eq. A.4)
(using Eq. A.9)
Noting that , allows us to write:
.	(A.12)

Partition of the generalized variance–covariance matrix among distance classes

The very definition of matrices N_T and M_T (Eq. A.2b), along with Eq. A.11, enables partition of  G_T into strictly additive components, G_h, that relate each to a distance class

.

(A.13)

G_his the generalized variance–covariance matrix defined for the distance class h by the neighboring relationship expressed by the matrices  and . G_htranslates easily into generalization of Wagner's variogram matrix (2003) by a division of all its values by

or  .

(A.14)

Equations A.2, A.13, and A.14 are used for easy programming of the method as well as efficient  computations via any matrix-oriented programming environment, as we did with Matlab and R (Ihaka and Gentleman 1996):  see the freely available library "msov" on< http://pbil.univ-lyon1.fr/CRAN/>.)

For a particular species couple i and j we obtain

.

(A.15)

Dividing by the scaling factor K(h) gives the value at "scale" h of the generalized version of either cross-variogram (ij) or variogram (i = j)

.

(A.16)

Multiscale ordination

All the ordination methods mentioned in Table 1 of the main paper are based on the singular values decomposition (SVD) of the appropriate version of G_T to compute eigenvectors, u_f, and associated eigenvalues, _f. Let U_f be the matrix having all the eigenvectors u_f as columns and let be the diagonal matrix having the eigenvalues _f on its diagonal. Both eigenvectors and eigenvalues of  G_T can be partitioned by distance classes

and  .

(A.17)

F_h is the variance–covariance matrix of the eigenvectors at scale h. Scale-dependent variogram/cross-variogram matrices of the eigenvectors are deduced by the appropriate scaling (Eq. A.16). Note also that

.

(A.18)

Taking environmental heterogeneity into account

Let us now suppose that a table, Z, containing assessments of P environmental variables for the Q quadrats, is available in addition to table of species composition. It is well established that the centered by columns table, , may be partitioned into an approximated table,

(A.19)

and a residual table,  (Sabatier et al. 1989).

In the same manner, it may also have a direct decomposition of the initial table X:

and   .

(A.20)

After factoring out the environmental variables, residual spatial patterns may be studied by the multiscale analysis of spatial covariances derived from table R or . The total residual variance–covariance matrix, G_RT, is computed as

(A.21)

and is broken down with respect to distance classes

.

(A.22)

The additive partitioning of  G_RT with respect to distance classesthus enables an investigation of the residual spatial patterns by a multiscale ordination scheme analogous to that defined by Eq. A.17 and Eq. A.18.

LITERATURE CITED

Ihaka, R., and R. Gentleman. 1996. R: a language for data analysis and graphics. Journal of Computational and Graphical Statistics 5:299–314.

Lebart, L. 1969. Analyse statistique de la contiguïté. Publications de l'Institut de Statistiques de l'Université de Paris 28:81–112.

Sabatier, R., J.-D. Lebreton, and D. Chessel. 1989. Principal component analysis with instrumental variables as a tool for modelling composition data. Pages 341–352 in R. Coppi and S. Bolasco, Editors. Multiway data analysis, Elsevier Science Publishers, Amsterdam, The Netherlands.

Wagner, H. H. 2003. Spatial covariance in plant communities: integrating ordination, geostatistics, and variance testing. Ecology 84:1045–1057.