This report develops the theoretical foundation for analytical description and quantification of habitat structure. The analytical description of environmental gradients is shown to be an eigenanalysis problem, mathematically equivalent to the largest eigenvector (or first principal component) of a principal components analysis. The analytical representation of an environmental gradient, itself a single variable, is empirically demonstrated to have similar ecological information as the combination of all the original 58 habitat variables describing five Mojave Desert study sites. Two vastly different data bases were analyzed to explore the effects of sample sizes and variable selection on the ordination of study sites in both principal components and canonical variate space. Merits and shortcomings of principal components analysis, canonical analysis of discriminance, and cluster analysis for the ordination and classification of samples are reviewed in detail. Canonical analysis of discriminance is a very effective mechanism for classifying samples into a priori established groups, or for identifying variables that contribute significantly to group discrimination.