Abstract not found

SUMMARY Regression problems with a number of related response variables are typically analyzed by separate multiple regressions. This paper shows how these regressions can be visualized jointly in a biplot based on reduced-rank regression. Reduced-rank regression combines multiple regression and principal components analysis and can therefore be carried out with standard statistical packages. The proposed biplot highlights the major aspects of the regressions by displaying the least-squares approximation of fitted values, regression coefficients and associated t-ratio's. The utility and interpretation of the reduced-rank regression biplot is demonstrated with an example using public health data that were previously analyzed by separate multiple regressions.

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis, The plot of structure correlations is shown to be optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biptots may be interpreted. Key words: biplot, canonical correlation analysis, canonical weight, interbattery factor analy-sis, partial analysis, redundancy analysis, regression coefficient, reduced rank regression, struc-ture correlations.

canonical correlations, principal component analysis. We propose a new procedure to perform Reduced Rank Regression (RRR) in non-Gaussian contexts, based on Multivariate Dispersion Models. Reduced-Rank Multivariate Dispersion Models (RR-MDM) generalise RRR to a very large class of distributions, which include continuous distributions like the normal, Gamma, Inverse Gaussian, and discrete distributions like the Poisson and the binomial. A multivariate distribution is created with the help of the Gaussian copula and estimation is performed using maximum likelihood. We show how this method can be amended to deal with the case of discrete data. We perform Monte Carlo simulations and show that our estimator is more efficient than the traditional Gaussian RRR. In the framework of MDM’s we introduce a procedure analogous to canonical correlations, which takes into account the distribution of the data. ∗ The authors would like to thank Luc Bauwens and Léopold Simar for helpful discussions and suggestions. The usual disclaimers apply. 1 1

Over the last century, Component Analysis (CA) methods such as Principal Component Analysis

... (SC) have been extensively used as a feature extraction step for modeling, clustering, classification, and visualization. CA techniques are appealing because many can be formulated as eigen-problems, offering great potential for learning linear and non-linear representations of data in closed-form. However, the eigen-formulation often conceals important analytic and computational drawbacks of CA techniques, such as solving generalized eigen-problems with rank deficient matrices (e.g., small sample size problem), lacking intuitive interpretation of normalization factors, and understanding commonalities and differences between CA methods. This paper proposes a unified least-squares framework to formulate many CA methods. We show how PCA, LDA, CCA, LE, SC, and their kernel and regularized extensions, correspond to a particular instance of least-squares weighted kernel reduced rank regression (LS-WKRRR). The LS-WKRRR formulation of CA methods has several benefits: (1) provides a clean connection between many CA techniques and an intuitive framework to understand normalization factors; (2) yields efficient numerical schemes to solve CA techniques; (3) overcomes the small sample size problem; (4) provides a framework to easily extend CA methods. We derive new weighted generalizations of PCA, LDA, CCA and SC, and several novel CA techniques.