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

Appendix A. A worked example of spatial covariance in plant communities.

Example data

The example data set consists of two artificial variables x1 and x2 that describe the presence and absence of two species in three quadrats A, B, and C along a transect y. The probability of occurrence, pi, is 2/3 for either species in any quadrat, and species richness S ranges from 1 to 2:

Basic elements of spatial covariance

The basic elements of spatial covariance are calculated using Eq. 8. For species variables x1 and x2 and the three unique pairs of quadrats a and b:

The spatial covariance ij(a,b) for each combination of species and quadrats is tabulated in the following matrices:

The variance of species richness

This paragraph contrasts the calculations for the empirical variance (left) with the calculations for the unbiased variance estimator (right) and shows that the variogram definition is equivalent to the latter.

Global variance of species richness S.---Using the standard textbook formulae, the global variance of species richness S in the N = 3 quadrats of the example data set is either 2/9 (empirical variance, left) or 1/3 (unbiased variance estimator, right):

Matrix of distances h.---Whether the variogram provides the empirical variance or the unbiased variance estimator depends on whether all pairs of quadrats are considered. Common geostatistical analysis omits distances of h = 0. This means that a quadrat is not compared to itself, as doing so would not provide any information. Analysis is often restricted to unique pairs in order to reduce calculations. The different approaches are illustrated in the following distance matrices, which list the distance h between pairs of quadrats as measured in transect coordinates y:

 All possible pairs: Unique pairs, a b:

Variograms and cross-variograms.---An empirical variogram i(h) is calculated using Eq. 4, as is illustrated here for species variable x1 and distance class h = 1, based on unique pairs:

Eq. 6 is used for calculating an empirical cross-variogram ij(h), as is illustrated here for species variables x1 and x2 and distance class h = 2, based on unique pairs:

The variograms i(h) and cross-variograms ij(h) for all species and distance classes are listed below. A weighted average of the semi-variance per distance class, weighted by nh, the number of pairs of quadrats, provides the global variance for each species or covariance for each pair of species.

Variance of species richness as the sum of (cross-)variograms.---The variance of species richness is obtained by summing the variograms i(h) and crossvariograms ij(h) of the individual species (Eq. 7). Depending on the inclusion of distance class h = 0, this results in the empirical variance (left) or the unbiased estimator (right) of the global variance of species richness S as calculated above.

In detail, the calculations for the unbiased estimator are:

The spatial variance test of species richness.---For the spatial variance test of species richness S, the variogram of species richness,

,

is divided, separately for each distance class, by the variogram of complementarity,

For distance h = 0, the ratio would result in a division by 0. Therefore only the unbiased version is shown:

The average of the ratios for each distance class h, weighted by nh, reconstructs the ratio of the global variance test.

Multi-scale ordination

Global variance-covariance matrix C.---Principal component analysis (PCA) uses the global empirical variance-covariance matrix C:

Variogram matrix.---The variogram matrix provides a partitioning of the global empirical variance-covariance matrix C by distance class h. In the variogram matrix, the variograms i(h) and cross-variograms ij(h) are summarized in a set of distance-dependent variance-covariance matrices C(h). In order to reconstruct the matrix C, a matrix C(0) for distance class h = 0 is needed:

The matrix C with its elements cij is the sum of the distance-dependent variance-covariance matrices C(h), weighted by nh, the number of pairs of quadrats. As is illustrated here for the covariance c12 between species variables x1 and x2, the weighted sum is equal to the global empirical covariance of the two species.

Matrix U of eigenvectors.---PCA replaces the original s variables by s artificial, uncorrelated variables, the PCA axes. The eigenvector uf defines the translation of the original variables into PCA axes. For the example data set, the matrix U of eigenvectors uf is (calculations not shown):

The global eigenvalue f of PCA axis f, as listed above for the example data set, results from matrix multiplication of U and C:

Partitioning of eigenvalues f by distance class.---The variogram f(h) of eigenvalue f is obtained by matrix multiplication of the eigenvector uf with each of the matrices C(h) (Eq. 18):

The following table summarizes the partitioning of the eigenvalues f for the example data set:

The sum of the variogram f(h) of PCA axis f, weighted by nh, the number of pairs of quadrats per distance class h, provides the global eigenvalue f. The calculation is illustrated here for PCA axis 1 (Eq. 19):

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