Studies in the robustness of multidimensional scaling: euclidean models and simulation studies

This series of papers is devoted to the investigation ot the extent to which the accuracy oi operation of multidimensional scaling can be put onto a quantitative footing. In thls third and final paper, four probabilistic models for the generation of euclidean-distance-like dissimilarity function are proposed; these models reflect some of the ways in which dissimilarities actually arise. and allow such effects as dependence between dissimilarities to be studied. Using these models. simulation experiments are carried out to assess the response of both classical and ordinal (non-metric) scaling to errors, with procrustes statistics being used to measure accuracy of recovery. A further scaling method. leait bquarer scaling. is discussed briefly and shown to display empirically a useful combination of properties. as is a technique used to preprocess the dissimilarity matrix.

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

[2]  M. S. Bartlett,et al.  The spectral analysis of two-dimensional point processes , 1964 .

[3]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

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

[5]  R. Shepard Metric structures in ordinal data , 1966 .

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

[7]  Herbert H. Stenson,et al.  GOODNESS OF FIT FOR RANDOM RANKINGS IN KRUSKAL'S NONMETRIC SCALING PROCEDURE * , 1969 .

[8]  H J Spaeth,et al.  THE USE AND UTILITY OF THE MONOTONE CRITERION IN MULTIDIMENSIONAL SCALING. , 1969, Multivariate behavioral research.

[9]  R. E. Miles On the homogeneous planar Poisson point process , 1970 .

[10]  Forrest W. Young Nonmetric multidimensional scaling: Recovery of metric information , 1970 .

[11]  DAVID G. KENDALL,et al.  Construction of Maps from “Odd Bits of Information” , 1971, Nature.

[12]  P. Padmos,et al.  QUANTITATIVE INTERPRETATION OF STRESS IN KRUSKAL'S MULTIDIMENSIONAL SCALING TECHNIQUE , 1971 .

[13]  J. Gower A General Coefficient of Similarity and Some of Its Properties , 1971 .

[14]  A. J. B. Anderson,et al.  Ordination Methods in Ecology , 1971 .

[15]  C. R. Sherman,et al.  Nonmetric multidimensional scaling: A monte carlo study of the basic parameters , 1972 .

[16]  R. Sibson Order Invariant Methods for Data Analysis , 1972 .

[17]  Ian Spence,et al.  A monte carlo evaluation of three nonmetric multidimensional scaling algorithms , 1972 .

[18]  I Spence,et al.  A TABLE OF EXPECTED STRESS VALUES FOR RANDOM RANKINGS IN NONMETRIC MULTIDIMENSIONAL SCALING. , 1973, Multivariate behavioral research.

[19]  L. Tucker,et al.  A reliability coefficient for maximum likelihood factor analysis , 1973 .

[20]  Richard C. T. Lee,et al.  A Heuristic Relaxation Method for Nonlinear Mapping in Cluster Analysis , 1973, IEEE Trans. Syst. Man Cybern..

[21]  R. Shepard Representation of structure in similarity data: Problems and prospects , 1974 .

[22]  Paul D. Isaac,et al.  On the determination of appropriate dimensionality in data with error , 1974 .

[23]  Lawrence E. Jones,et al.  The effects of random error and subsampling of dimensions on recovery of configurations by non-metric multidimensional scaling , 1974 .

[24]  J. Graef,et al.  THE DETERMINATION OF THE UNDERLYING DIMENSIONALITY OF AN EMPIRICALLY OBTAINED MATRIX OF PROXIMITIES. , 1974, Multivariate behavioral research.

[25]  David G. Kendall,et al.  Review Lecture, The recovery of structure from fragmentary information , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[26]  Dissociated random variables , 1975 .

[27]  B. Silverman,et al.  Limit theorems for dissociated random variables , 1976, Advances in Applied Probability.

[28]  W. G. McGinley Some Optimisation Problems in Data Analysis , 1977 .

[29]  Bruce Bloxom,et al.  Constrained multidimensional scaling inN spaces , 1978 .

[30]  R. Sibson Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics , 1978 .

[31]  Robin Sibson,et al.  Computing Dirichlet Tessellations in the Plane , 1978, Comput. J..

[32]  R. Sibson Studies in the Robustness of Multidimensional Scaling: Perturbational Analysis of Classical Scaling , 1979 .

[33]  Robin Sibson,et al.  The Dirichiet Tessellation as an Aid in Data Analysis , 1980 .