A Simulated Annealing Heuristic for Unidimensional and Multidimensional (City-Block) Scaling of Symmetric Proximity Matrices

This paper presents a heuristic for uni- and multidimensional (city-block) scaling of symmetric proximity matrices. The heuristic begins with the partition of each coordinate axis into equally spaced discrete points. A simulated annealing algorithm is used to search the lattice defined by these points with the objective of minimizing either a least-squares or least absolute deviation loss function. The solution obtained by the simulated annealing algorithm is subsequently refined by passing the object permutations for each dimension to a postprocessor that finds a locally optimal set of coordinates. Experimental testing was conducted for uni- and two-dimensional (city-block) scaling using a well-studied proximity matrix. Although the simulated annealing based heuristic is effective in its own right, the results herein suggest that perhaps its greatest utility is the provision of very good starting solutions for combinatorial algorithms designed for uni- and multidimensional scaling.

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

[2]  R N SHEPARD,et al.  Analysis of Proximities as a Technique for the Study of Information Processing in Man1 , 1963, Human factors.

[3]  L. Hubert,et al.  Multidimensional scaling in the city-block metric: A combinatorial approach , 1992 .

[4]  Michael J. Brusco,et al.  Using Quadratic Assignment Methods to Generate Initial Permutations for Least-Squares Unidimensional Scaling of Symmetric Proximity Matrices , 2000, J. Classif..

[5]  I. Barrodale,et al.  An Improved Algorithm for Discrete $l_1 $ Linear Approximation , 1973 .

[6]  Phipps Arabie,et al.  Was euclid an unnecessarily sophisticated psychologist? , 1991 .

[7]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

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

[9]  P. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 1999 .

[10]  Willem J. Heiser,et al.  City-Block Scaling: Smoothing Strategies for Avoiding Local Minima , 1998 .

[11]  L. Hubert,et al.  Quadratic assignment as a general data analysis strategy. , 1976 .

[12]  Joseph Heller.  Catch-22: A Novel , 1961 .

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

[14]  Lawrence Hubert,et al.  The analysis of proximity matrices through sums of matrices having (anti‐)Robinson forms , 1994 .

[15]  L. Hubert,et al.  Normative location theory: Placement in continuous space , 1976 .

[16]  Patrick J. F. Groenen,et al.  The majorization approach to multidimensional scaling : some problems and extensions , 1993 .

[17]  I. Barrodale,et al.  An Efficient Algorithm for Discrete $l_1$ Linear Approximation with Linear Constraints , 1978 .

[18]  M. Brusco.  Morph-Based Local-Search Heuristics for Large-Scale Combinatorial Data Analysis , 1999 .

[19]  Phipps Arabie,et al.  Concerning Monte Carlo evaluations of nonmetric multidimensional scaling algorithms , 1973 .

[20]  Kin-Nam Lau,et al.  A Nonlinear Programming Approach to Metric Unidimensional Scaling , 1998 .

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

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

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

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

[25]  H. Späth.  Mathematical algorithms for linear regression , 1991 .

[26]  Phipps Arabie,et al.  Combinatorial Data Analysis: Optimization by Dynamic Programming , 1987 .

[27]  Ian Barrodale,et al.  Algorithm 552: Solution of the Constrained I1 Linear Approximation Problem [F4] , 1980, TOMS.

[28]  Evangelia Simantiraki,et al.  Unidimensional scaling: A linear programming approach minimizing absolute deviations , 1996 .

[29]  Lawrence Hubert,et al.  Linear and circular unidimensional scaling for symmetric proximity matrices , 1997 .

[30]  Keith T. Poole.  Least squares metric, unidimensional scaling of multivariate linear models , 1990 .

[31]  Lawrence J Hubert,et al.  Assignment methods incombinatorial data analysis , 1986 .

[32]  W. S. Robinson.  A Method for Chronologically Ordering Archaeological Deposits , 1951, American Antiquity.

[33]  Edward L. Frome,et al.  A revised simplex algorithm for the absolute deviation curve fitting problem , 1979 .

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