The Past, Present, and Future of Multidimensional Scaling

Multidimensional scaling (MDS) has established itself as a standard tool for statisticians and applied researchers. Its success is due to its simple and easily interpretable representation of potentially complex structural data. These data are typically embedded into a 2-dimensional map, where the objects of interest (items, attributes, stimuli, respondents, etc.) correspond to points such that those that are near to each other are empirically similar, and those that are far apart are different. In this paper, we pay tribute to several important developers of MDS and give a subjective overview of milestones in MDS developments. We also discuss the present situation of MDS and give a brief outlook on its future.

[1]  J. Ramsay.  Maximum likelihood estimation in multidimensional scaling , 1977 .

[2]  J. Meulman.  A Distance Approach to Nonlinear Multivariate Analysis , 1986 .

[3]  J. Meulman.  The integration of multidimensional scaling and multivariate analysis with optimal transformations , 1992 .

[4]  Willem J. Heiser,et al.  A Permutation-Translation Simulated Annealing Algorithm for L1 and L2 Unidimensional Scaling , 2005, J. Classif..

[5]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[6]  V. Pliner.  Metric unidimensional scaling and global optimization , 1996 .

[7]  Patrick Mair,et al.  Multidimensional Scaling Using Majorization: SMACOF in R , 2008 .

[8]  Thierry Denoeux,et al.  Multidimensional scaling of interval-valued dissimilarity data , 2000, Pattern Recognit. Lett..

[9]  P. Groenen,et al.  The majorization approach to multidimensional scaling for Minkowski distances , 1995 .

[10]  A. Gifi,et al.  NONLINEAR MULTIVARIATE ANALYSIS , 1990 .

[11]  A. Buja,et al.  Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling , 1994 .

[12]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. I. , 1962 .

[13]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[14]  Hans-Hermann Bock,et al.  Analysis of Symbolic Data , 2000 .

[15]  E. Rothkopf.  A measure of stimulus similarity and errors in some paired-associate learning tasks. , 1957, Journal of experimental psychology.

[16]  T. Kloek,et al.  International Comparisons of Prices and Quantities Consumed , 1965 .

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

[18]  P. Groenen,et al.  Modern multidimensional scaling , 1996 .

[19]  Suzanne Winsberg,et al.  Multidimensional Scaling of Histogram Dissimilarities , 2006, Data Science and Classification.

[20]  Reginald G. Golledge,et al.  Matrix reorganization and dynamic programming: Applications to paired comparisons and unidimensional seriation , 1981 .

[21]  Michael J. Brusco.  A Simulated Annealing Heuristic for Unidimensional and Multidimensional (City-Block) Scaling of Symmetric Proximity Matrices , 2001, J. Classif..

[22]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[23]  P. Groenen,et al.  Embedding the Organizational Culture Profile Into Schwartz’s Theory of Universals in Values , 2011 .

[24]  P. GroenenA,et al.  I-Scal: Multidimensional scaling of interval dissimilarities , 2006 .

[25]  J. Leeuw.  Applications of Convex Analysis to Multidimensional Scaling , 2000 .

[26]  Ingwer Borg,et al.  A direct approach to individual differences scaling using increasingly complex transformations , 1978 .

[27]  Forrest W. Young,et al.  Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features , 1977 .

[28]  Andreas Buja,et al.  Visualization Methodology for Multidimensional Scaling , 2002, J. Classif..

[29]  Joseph L. Zinnes,et al.  Theory and Methods of Scaling. , 1958 .

[30]  J. Leeuw.  Fitting Distances by Least Squares , 1993 .

[31]  I. Borg,et al.  Facet Theory: Form and Content , 1995 .

[32]  Ivo A. van der Lans,et al.  Multidimensional Scaling with Regional Restrictions for Facet Theory: An Application to Levi's Political Protest Data , 2006 .

[33]  Ludo Waltman,et al.  Software survey: VOSviewer, a computer program for bibliometric mapping , 2009, Scientometrics.

[34]  P. Groenen,et al.  Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing , 1999 .

[35]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[36]  J. Leeuw.  Multidimensional Scaling Using Majorization : SMACOF in , 2008 .

[37]  J. Leeuw.  Convergence of the majorization method for multidimensional scaling , 1988 .

[38]  Deborah F. Swayne,et al.  Data Visualization With Multidimensional Scaling , 2008 .

[39]  P. Groenen,et al.  The tunneling method for global optimization in multidimensional scaling , 1996 .

[40]  C. Horan.  Multidimensional scaling: Combining observations when individuals have different perceptual structures , 1969 .

[41]  J. Gower.  Some distance properties of latent root and vector methods used in multivariate analysis , 1966 .

[42]  D. Defays.  A short note on a method of seriation , 1978 .

[43]  Willem J. Heiser,et al.  Global Optimization in Any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling , 2007, J. Classif..

[44]  M. Hill,et al.  Nonlinear Multivariate Analysis. , 1990 .