Multidimensional scaling by iterative majorization using radial basis functions

Abstract This paper considers the use of radial basis functions for modelling the non-linear transformation of a data set obtained by a multidimensional scaling analysis. This approach has two advantages over conventional nonmetric multidimensional scaling. It reduces the number of parameters to estimate and it provides a transformation that may be used on an unseen test set. A scheme based on iterative majorization is proposed for obtaining the parameters of the network.

[1]  H. Hotelling Analysis of a complex of statistical variables into principal components. , 1933 .

[2]  David J. Hand,et al.  A Handbook of Small Data Sets , 1993 .

[3]  Andrew R. Webb An approach to non-linear principal components analysis using radially symmetric kernel functions , 1996, Stat. Comput..

[4]  D. A. Wolf Recent advances in descriptive multivariate analysis , 1996 .

[5]  I. D. Hill,et al.  An Efficient and Portable Pseudo‐Random Number Generator , 1982 .

[6]  J. Barra,et al.  Recent Developments in Statistics , 1978 .

[7]  W. Heiser A generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative , 1991 .

[8]  W. J. Krzanowski,et al.  Recent Advances in Descriptive Multivariate Analysis. , 1996 .

[9]  Trevor F. Cox,et al.  Discriminant analysis using non-metric multidimensional scaling , 1993, Pattern Recognit..

[10]  John W. Sammon,et al.  A Nonlinear Mapping for Data Structure Analysis , 1969, IEEE Transactions on Computers.

[11]  J. Kruskal Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[12]  Keinosuke Fukunaga,et al.  A Nonlinear Feature Extraction Algorithm Using Distance Transformation , 1972, IEEE Transactions on Computers.

[13]  I. Borg,et al.  Geometric Representations of Relational Data , 1981 .

[14]  D. Lowe Novel 'topographic' nonlinear feature extraction using radial basis functions for concentration coding in the 'artificial nose' , 1993 .