Principal Components Biplots and Alpha and Beta Diversity

Attention is drawn to some useful but not generally known properties of principal components analysis (PCA). Noncentered PCA of proportion data gives site ordinations that display approximate alpha diversities of sites and beta diversities of groups of sites, as measured by the Simpson index and mean squared Euclidean distance, respectively. Species centering allows a better approximation to beta diversities. Alpha diversities can still be visualized after centering if the true origin is projected into the plane of the ordination. The approximate species composition of each site can also be visualized if the site ordination is combined with a species ordination. The resulting plot of site scores and species loadings is called a PCA biplot. Finally, in a PCA biplot that displays both species composition and diversity, diversity values can be explained in terms of the main species contributing to diversity. In such a biplot the sum of squares of the species loadings must be scaled to unity, while the site scores must be scaled to a sum of squares equal to the corresponding eigenvalue. This type of biplot is termed a "distance biplot." For a simple illustration noncentered and species—centered distance biplots were produced for some diatom samples taken from Dutch moorland pools in the 1920s and 1978. The distance biplot is concluded to be among the most powerful analytical tools for species—composition data and derives some of its power from properties not possessed by, for example, reciprocal averaging. One problem is that it attaches little weight to rare species, but this problem can be solved by various possible data transformations based on the theory of diversity indices.