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.
Keywords
MathematicsPrincipal component analysisMultivariate statisticsEuclidean distanceInterpretation (philosophy)Euclidean spaceSingular valueCombinatoricsSample (material)Euclidean distance matrixPoint (geometry)Set (abstract data type)Matrix (chemical analysis)Representation (politics)StatisticsGeometryEigenvalues and eigenvectors
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Publication Info
- Year
- 1966
- Type
- article
- Volume
- 53
- Issue
- 3-4
- Pages
- 325-338
- Citations
- 3981
- Access
- Closed
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Cite This
J. C. Gower
(1966).
Some distance properties of latent root and vector methods used in multivariate analysis.
Biometrika
, 53
(3-4)
, 325-338.
https://doi.org/10.1093/biomet/53.3-4.325
Identifiers
- DOI
- 10.1093/biomet/53.3-4.325