Solving the puzzle: Simulation of multivariate data with control over the structure of columns and rows using the two-sided orthogonal procrustes problem

Abstract

Researchers interested in developing new multivariate statistical methods often need to be able to generate multivariate datasets with specific characteristics to test the effectiveness of their data analysis algorithms under specific conditions.

In this paper, we present a family of methods for generating multivariate centred datasets by simultaneously controlling features of the cross-product matrices $X^{\top }X$ and $X{X}^{\top }$. This provides an interesting trade-off to control for the variance structure in the data, important for the family of algorithms that operate on the data matrix, like, e.g., Principal Component Analysis, and control for the distances among objects, important for algorithms that operate on the distance matrix, like Multidimensional Scaling. The proposed methods form a general framework that can be understood as a jigsaw puzzle, joining pieces obtained from the spectral decomposition of a target covariance matrix and the singular value decomposition of a target data matrix. These methods have in common that they are derived from a two-sided orthogonal Procrustes problem.