Canonical analysis of distance biplots

Canonical Analysis of Distance (AoD) extends Canonical Variate Analysis (CVA) to cope with a wide class of distance functions. In this presentation attention  is restricted to Euclidean embeddable distances that are also  additive e.g. Pythagorean distance, Clark’s distance, Chi-squared distance and the extended matching coefficient (EMC). It will be shown how canonical AoD deals with both continuous variables and categorical variables.  Extending the procedures for dealing with mixed variables is straight forward.

A key feature of canonical AoD is the analysis of the grouping structure of the data in a space of dimension not larger than K–1 where K denotes the number of groups. Biplot visualisations (predictive and interpolative) can be made showing: points representing the group centroids; points and/or regions representing within group variation; linear or nonlinear trajectories for quantitative variables and Category Level Points (CLPs) for categorical variables as well as prediction regions for different categories.

Generalised biplots give nonlinear biplot trajectories for continuous variables and CLPs for categorical variables.  These CLPs have interesting properties, analogous to those associated with Multiple Correspondence Analysis (MCA), which is a special case. Canonical AoD extends these properties to the analysis of grouped data leading to a technique that may be termed categorical CVA.