integrating and extending the pioneering work of Kruskal (1964); Guttman (1968); ... This general loss function is called Stress as a tribute to Kruskal (1964).
PROXSCAL: A Multidimensional Scaling Program for Individual Differences Scaling with Constraints Frank M.T.A. Busing Jacques J.F. Commandeur Willem J. Heiser Department of Data Theory Leiden University P.O. Box 9555 2300 RB Leiden The Netherlands
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Summary The program PROXSCAL performs multidimensional scaling of proximity ((dis)similarity, distance-like) data to find a least squares representation of the objects in a low-dimensional Euclidean space. It is an integration and extension of the pioneering work by Kruskal, Guttman and Carroll. Various options are provided for metric as well as nonmetric data. Weights can be supplied for every proximity and the program can handle missing data. Either one or several symmetric data matrices (sources) may be analyzed, under a variety of models and constraints. An extended example is treated to show some of the input and output features of PROXSCAL. 1
Introduction
PROXSCAL is a computer program for multidimensional scaling (MDS) and individual differences scaling (IDS) of proximities, or PROXimity SCALing for short. It is a consolidated version of the older SMACOF series, except that the SMACOF-III unfolding program is not included. These programs were all designed from a single theoretical and computational point of view, integrating and extending the pioneering work of Kruskal (1964); Guttman (1968); Carroll (1972). The major tools used in this approach are alternating least squares (ALS) and iterative majorization (IM). For the theory of PROXSCAL, we refer to de Leeuw and Heiser (1980); Meulman and Heiser (1984); de Leeuw (1988); Heiser (1995) as basic papers, and to Heiser (1988, 1991); Meulman (1992); Meulman and Verboon (1993); Groenen, Mathar, and Heiser (1995) for various extensions in programs that are currently under development. For details of algorithm construction and computation in PROXSCAL, we refer to Heiser (1985); Heiser and Stoop (1986); Commandeur and Heiser (1993). 2
Individual Differences Scaling with Kruskal’s Stress
In MDS we try to find a configuration of points in which the distances between these points match as well as possible the proximities between the objects. The term proximity is the generic name used for either similarity or dissimilarity data. Large similarities should correspond to small distances, whereas small similarities should correspond to large distances. Given the (dis)similarities δijk between n objects (i, j = 1, . . . , n) for m sources (k = 1, . . . , m), PROXSCAL determines m configurations Xk of order (n×p), such that the Euclidean distances dij (Xk ) between the rows of the Xk ’s (conceived of as n points in p dimensions) approximate the given (dis)similarities δijk as well as possible for all i, j = 1, . . . , n and k = 1, . . . , m. The formal problem PROXSCAL solves in the most general case is the minimization of the least squares loss function n
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1 XX wijk [φ(δijk ) − dij (Xk )]2 , f (X1 , . . . , Xm ; φ) ≡ m k=1 i
