Abstract
In the multidimensional scaling of samples described by categorical variables, each l-level categorical variable is represented by a set of l points forming the vertices of a simplex. Any sample that has level i of a categorical variable will be nearer the corresponding vertex C i than to any other vertex of the simplex, so defining convex neighbour-regions for each level. In r-dimensional approximations, predictions of levels are obtained by examining the intersections of the neighbour-regions with the r-dimensional space. The case r = 2 is of special practical importance and is the main concern of this paper; the methodology easily generalises. Examples are given of the forms taken by the neighbour-regions in planar sections of the (lā1)-dimensional simplex and an algorithm is proposed for their construction.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Anderson T. W. (1958), An Introduction to Multivariate Statistical Analysis, New York, John Wiley.
Devijver P.A. and Diekesei M. (1985), Computing multidimensional Delauney tessellations, Pattern Recognition Letters, 1, 311ā6.
Eckart C. and Young G. (1936), The approximation of one matrix by another of lower rank, Psychometrika, 1, 211ā8.
Gabriel K. R. (1971), The biplot-graphic display of matrices with applications to principal components analysis, Biometrika, 58, 453ā67.
Gifi A. (1990), Non-linear Multivariate Analysis, New York, J. Wiley and Son.
Gower J. C. (1966), Some distance properties of latent root and vector methods used in multivariate analysis, Biometrika, 53, 325ā38.
Gower J. C. (1968), Adding a point to vector diagrams in multivariate analysis, Biometrika, 55, 582ā5.
Gower J.C. (1982), Euclidean distance geometry, Tie Mathematical Scientist, 7, 1ā14.
Gower J.C. (1985), Properties of Euclidean and non-Euclidean distance matrices, Linear Algebra and its Applications, 67, 81ā97.
Gower J. C. (1991), Generalised biplots, Research Report RR-91-02, Leiden, Department of Data Theory.
Gower J.C. (1992), Biplot Geometry.
Gower J. C. and Harding S. (1988), Non-linear biplots. Biometrika, 73, 445ā55.
Greenacre M.J. (1984), Theory and Applications of Correspondence Analysis. London, Academic Press
Sibson R. (1980), The Dirichlet tessellation as an aid to data analysis, Scandinavian Journal of Statistics, 7, 14ā20.
Torgerson W. S. (1955) Theory and Methods of Scaling. New York, John Wiley.
Watson D. F. (1981) Computing the n-dimensional Delauney tessellation with applications to Voronoi polytopes, The Computer Journal, 24, 167ā72.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 1993 Springer-Verlag Berlin Ā· Heidelberg
About this paper
Cite this paper
Gower, J.C. (1993). The Construction of Neighbour-Regions in Two Dimensions for Prediction with Multi-Level Categorical Variables. In: Opitz, O., Lausen, B., Klar, R. (eds) Information and Classification. Studies in Classification, Data Analysis and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50974-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-50974-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56736-3
Online ISBN: 978-3-642-50974-2
eBook Packages: Springer Book Archive