Some distance properties of latent root and vector methods used in multivariate analysis
1966/01/01 by J. C. Gower, J. C. GOWER · 4,106 citations
Agricultural and Biological Sciences · Chemistry · Mathematics · #Advanced Statistical Methods and Models #Combinatorics #Eigenvalues and eigenvectors #Euclidean distance #Euclidean distance matrix #Euclidean space #Geometry #Interpretation (philosophy) #Mathematics #Matrix (chemical analysis) #Multivariate statistics #Point (geometry) #Principal component analysis #Representation (politics) #Sample (material) #Sensory Analysis and Statistical Methods #Set (abstract data type) #Singular value #Spectroscopy and Chemometric Analyses #Statistics
paper · doi:10.1093/biomet/53.3-4.325
published in Biometrika 53(3-4), 325-338 (Oxford University Press)
openalex publication_date 1966/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.
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