Two outstanding ordination problems

Abstract

Ordination techniques seem hitherto not to have been applied to two problems of considerable botanical interest: (i) the ordination of a set of groups after a numerical classification, and (ii) the ordination of an asymmetric system. The botanical contexts in which these problems arise are discussed. It is shown that Gower's "principal coordinate analysis" can be simply extended to cover these two cases; computer programming requirements are minimal, and standard programmes are available for the Control Data 3600 computer.

Patterns of Resource Allocation in Fleshy Fruits of Nine European Tall-Shrub Species

“…Each attribute was standardised to zero mean and unit variance, and the correlation matrix used for the PCA. Forming the correlations for a pair of variates over those species with missing values for those two variates gives a correlation matrix which is not semi-definite (Williams et al 1971) and therefore has negative eigenvalues. Moreover, the positions of the species with missing values in the PCA space cannot be calculated.…”

Numerical Analysis of Volatile Oil Data in Systematic Studies of Australian Rainforest Trees

“…For each species, a series of 18 different analyses was performed, comparing all combinations of two different weighting schemes (unweighted and F-weighted), three different standardisation schemes (unstandardised, standardised by range, standardised by standard deviation), and three different distance measures (Squared Euclidean Distance, Euclidean Distance, Manhattan Metric distance). The distance matrix from each was submitted to a principal coordinates analysis (Gower, 1966) using a modified version of the GOWaRD computer program (Williams et al, 1971). In this analysis technique there is a Gower standardisation by row and by column of the distance matrix, and the eigenvectors are standardised so that the length of each is equal to its corresponding root.…”

Phenotypic relationships among Australian and Mexican wheat cultivars

“…The cultivars were ordinated by principal co-ordinate analysis (GOWER, 1966(GOWER, , 1967) by using the matrix of inter-individual squared Euclidean distances output from MULCLAS in the program GOWER (WILLIAMS et al, 1971). This ordination procedure is the dual of principal component analysis.…”