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

Abstract

This paper is concerned with the representation of a multivariate sample of size n as points P 1 , P 2 , …, P n in a Euclidean space. The interpretation of the distance Δ( P i , P j ) between the i th and j th 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 D 2 with a singular dispersion matrix.