A method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis. In this method, the original data are first decomposed into several components according to external information. The components are then subjected to principal component analysis to explore structures within the components. It is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices. The numerical method based on the QR decomposition is described, which simplifies the computation considerably. The proposed method includes a number of interesting special cases, whose relations to existing methods are discussed. Examples are given to demonstrate practical uses of the method.

We propose a two-way Bayesian vector spatial procedure incorporating dimension reparameterization with a variable selection option to determine the dimensionality and simultaneously identify the significant covariates that help interpret the derived dimensions in the joint space map. We discuss how we solve identifiability problems in a Bayesian context that are associated with the two-way vector spatial model, and demonstrate through a simulation study how our proposed model outperforms a popular benchmark model. In addition, an empirical application dealing with consumers’ ratings of large sport utility vehicles is presented to illustrate the proposed methodology. We are able to obtain interpretable and managerially insightful results from our proposed model with variable selection in comparison with the benchmark model.

Finding the intersection of $n$-dimensional spheres in $\mathbb{R}^{n}$ is an interesting problem with applications in trilateration, global positioning systems, multidimensional scaling and distance geometry. In this paper, we generalize some known results on finding the intersection of spheres, based on QR decomposition. Our main result describes the intersection of any number of $n$-dimensional spheres without the assumption that the centres of the spheres are affinely independent. A possible application in the interval distance geometry problem is also briefly discussed.