Some distance properties of latent root and vector methods used in multivariate analysis

TLDR

The paper examines representing a multivariate sample as points in Euclidean space and notes that Q and R techniques are dual when they produce identical inter‑point distances. The study investigates how to interpret distances between sample points in common multivariate analyses such as Q and R techniques. The authors derive a method that, given all pairwise distances, computes coordinates relative to principal axes and discuss interpreting these distances in Q and R analyses. They establish necessary and sufficient conditions for a real Euclidean solution, derive pairs of dual techniques, and show that in factor analysis distances between estimated factor‑score points correspond to D² with a singular dispersion matrix.

Abstract

This paper is concerned with the representation of a multivariate sample of size n as points P1, P2, …, Pn in a Euclidean space. The interpretation of the distance Δ(Pi, Pj) between the ith and jth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean sapce is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinrates are the estimated factor scores can be interpreted as D2 with a singular dispersion matrix.