WARREN S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....procedures for numerical optimization typically nd local minimizers that may not be global minimizers, the choice of an initial con guration from which to begin searching for an optimal con guration is crucial. A popular choice of initial con guration is the classical solution of Torgerson [21]. We exploit results from the theory of distance matrices to derive two alternatives, each guaranteed to be at least as good as the classical solution, and present empirical evidence that they are usually substantially better. Key words: Distance matrices, distance geometry, spectral ....

....s e = 1, the linear transformation s de ned by s (D) 1 I es D I es is an inverse of . Johnson and Tarazaga [11] demonstrated that the image of s is a face of the polyhedral cone n (n) The inverse transformation 1 obtained by setting s = e=n was introduced by Torgerson [21]. See Critchley [1] for a detailed study of the properties of and 1 . Implicit in Torgerson s [21] and Gower s [5] pioneering studies of metric MDS was an optimization problem. Given a dissimilarity matrix = ij ] let r denote the matrix with entries ( ij ) Let k k F denote the ....

[Article contains additional citation context not shown here]

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401-419, 1952.

....Section 4 we derive the algorithm and the Slater constraint qualification result. We conclude with several remarks and numerical tests in Section 5. In addition, we include Section 5.1 with some technical details on the SDP algorithm. 2. DISTANCE GEOMETRY It is well known, e.g. 34] 15] 16] [38], that a pre distance matrix D is a EDM if and only if D is negative semidefinite on the orthogonal complement of e, where e is the vector of all ones. Thus the set of all EDMs is a convex cone, which we denote by . We exploit this result to translate the cone to the cone of semidefinite matrices ....

WARREN S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika 17:401-419 1952.

....with the original FastMap heuristic. Keywords: Data Mining, Distributed Databases, Information Systems, Parallel and Distributed Algorithms 1 Introduction A set S of points in a d dimensional space often belong to an embedded manifold of dimension d d. Classic dimension reduction techniques [3, 8, 5] compute an optimal k dimensional representation of S for a speci ed k d and a given optimality criterion. Techniques related to principal components [3] begin with coordinates of the points, whereas those related to multidimensional scaling [8, 5] begin with a complete set of pairwise ....

.... d. Classic dimension reduction techniques [3, 8, 5] compute an optimal k dimensional representation of S for a speci ed k d and a given optimality criterion. Techniques related to principal components [3] begin with coordinates of the points, whereas those related to multidimensional scaling [8, 5] begin with a complete set of pairwise distances. All of these require at least quadratic running time, making them reasonable reduction candidates only as long as S is not too large. The focus of this paper, however, is on the case in which S is of some immense size N , with its elements ....

[Article contains additional citation context not shown here]

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrica., 17:401-419, 1952.

....techniques address this issue and allow the user to better analyze or visualize complex data sets. These techniques may be distinguished into two classes. In the first one are linear methods like the Principal Component Analysis (PCA, 8] or the original metric multidimensional scaling (MDS, [14]) In the second class are nonlinear algorithms like Kohonen s Self Organizing Map (SOM, 9, 10] or nonlinear variants of the MDS. Contrarily to the linear PCA, the last ones do not use a criterion based on variance preservation. Instead, they try to reproduce in the projection space the pairwise ....

W. S. Torgerson. Multidimensional Scaling, I: Theory and Method. Psychometrika, 17:401--419, 1952.

....steps, and the parameters schedule during this sub convergence is empiric up to now. This continuous mapping is invertible: backward ,tapping can be obtained by simply swapping input and output weights and spaces and using the same algorithm. V. COMPARISON WITH OTHER ALGORITHMS Classical MDS [21] and PCA [7] both find the axes of maximum variance of input data and represent them by a linear projection onto a subspace of re duced dimension [15] In the case of Euclidean dis tances, the cost function to be minimized is E : Zi Zji(Xj Yi) under constraint that Yij Xij 2 . 2(10 NLM: ....

W.S. Torgerson. Multidimensional scaling, i: theory and method. Psychometrika, 17:401 419, 1952.

....the visualizations. The methods di er in what properties of the data set they try to preserve. The simplest methods, such as the principal component analysis (PCA) 3] are based on linear projection. A more complex set of traditional methods, that are based on multidimensional scaling (MDS) [10], try to preserve the pairwise distances of the data samples as well as possible. That is, the pairwise distances after the projection approximate the original distances. In a variant of nonlinear MDS, nonmetric MDS [8] only the rank order of the distances is to be preserved. Another variant, ....

W. S. Torgerson. Multidimensional scaling I|theory and methods. Psychometrica, 17:401-419, 1952.

....There are two main branches to MDS: the original metric method, and the more commonly used non metric method. In metric MDS, the distances in the configuration are intended to directly correspond to the given dissimilarities. In the original, and largely dominant, metric model, classical MDS [28], the configuration is obtained by an analytic method (via a spectral decomposition of the centred inner product matrix) and, if the dissimilarities correspond to a matrix of Euclidean distances between a set of points in some space, it can be shown to be equivalent to a PCA of those original ....

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401--419, 1952.

....of getting a single hidden state from a MFA, we can use the following procedure. 1) Estimate the similarity between analyzor centres using average separation in time between data points for which they are active. 2) Use standard embedding techniques such as multidimensional scaling (MDS) [42] to place the MFA centres in a Euclidean space of dimension k. 3) Time independent state inference for each observation now consists of the responsibility weighted low dimensional MFA centres, where the responsibilities are the posterior probabilities of each analyzor given the observation under ....

W. S. Torgerson, \Multidimensional scaling I. Theory and method," Psychometrika, vol. 17, pp. 401-419, 1952.

....only if f(D ) is zero. Furthermore, 1) can be posed as a semidefinite programming problem be exploiting the relation between the cone E and P , the cone of positive semidefinite matrices. This relation is established in the following subsection. 2. 1 Distance Geometry It is well known, e.g. [21, 7, 9, 22], that a symmetric matrix D with nonnegative elements and with zero diagonal is a EDM if and only if D is negative semidefinite on M : n x 2 n : x T e = 0 o ; the orthogonal complement of e, the vector of all ones. Let S n denote the space of symmetric matrices of order n. Define the ....

WARREN S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.

....by D(X) It is obvious that a distance matrix is necessarily a dissimilarity matrix. Determining whether or not a specified dissimilarity matrix is a distance matrix is a famous problem in classical distance geometry. We state the standard solution of this problem, implicit in Torgerson s [24] formulation of multidimensional scaling and demonstrated by Gower [9] The standard solution is a trivial modification of the solution independently discovered by Schoenberg [20] and by Young and Householder [31] Its statement requires some additional definitions: Definition 4 Let A = a ij ] ....

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401--419, 1952.

....and data mining: the objects can now be plotted as points in 2 d or 3 d space, revealing potential clusters, correlations among attributes and other regularities that data mining is looking for. We introduce an older method from pattern recognition, namely, Multi Dimensional Scaling (MDS) Tor52] although unsuitable for indexing, we use it as yardstick for our method. Then, we propose a much faster algorithm to solve the problem in hand, while in addition it allows for indexing. Experiments on real and synthetic data indeed show that the proposed algorithm is significantly faster than ....

....positions of the k d points. Intuitively, it treats each pair wise distance as a spring between the two points; then, the algorithm tries to re arrange the positions of the k d points to minimize the stress of the springs. The above version of MDS is called metric multidimensional scaling [Tor52] because the distances are given as numbers. Several generalizations and extensions have been proposed to the above basic algorithm: Kruskal [KW78] proposed a method that automatically determines a good value for k; Shepard [She62] and Kruskal [Kru64] proposed the non metric MDS where the ....

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401-- 419, 1952.

....data. Unfortunately the method does not provide any measures of the quality of this visualization. The multidimensional scaling (MDS) technique is a statistical clusterization method almost unknown outside of the mathematical psychology field. It has foundations in the work of Torgerson [4] and in the Coombs theory of data [5] Computer programs and applications of MDS have been developed, among others, by Kruskal [6] at the Bell Laboratories, by Lingoes, Roskam and Borg [7] in Ann Arbor, and by Shepard [8] in Palo Alto [9] MDS was used by experts in mathematical psychology who ....

W.S. Torgerson, Multidimensional scaling. I. Theory and method. Psychometrika, 17 (1952) 401-419

....fd ij g, match as well as possible the original dissimilarities fffi ij g [CC94] There are two main categories of MDS methods : the metric and the nonmetric ones. The difference between these two families of MDS methods lies in the kind of input dissimilarities they used. For metric MDS methods [Tor52] the dissimilarities fffi ij g are used in a quantitative metric sense, that means the dissimilarities are such that the value of each fd ij g approximates 1 Centre de Recherches de Royallieu UMR CNRS 6599. Universit e de Technologie de Compi egne, B.P. 20529 60205 Compi egne cedex France, ....

W.S. Torgerson. Multidimensional scaling. i: Theory and method. Psychometrika, 17:401--419, 1952.

....and data mining: the objects can now be plotted as points in 2 d or 3 d space, revealing potential clusters, correlations among attributes and other regularities that data mining is looking for. We introduce an older method from pattern recognition, namely, Multi Dimensional Scaling (MDS) [51]; although unsuitable for indexing, we use it as yardstick for our method. Then, we propose a much faster algorithm to solve the problem in hand, while in addition it allows for indexing. Experiments on real and synthetic data indeed show that the proposed algorithm is significantly faster than ....

....the positions of the k d points. Intuitively, it treats each pair wise distance as a spring between the two points; then, the algorithm tries to rearrange the positions of the k d points to minimize the stress of the springs. The above version of MDS is called metric multidimensional scaling [51], because the distances are given as numbers. Several generalizations and extensions have been proposed to the above basic algorithm: Kruskal [29] proposed a method that automatically determines a good value for k; Shepard [48] and Kruskal [28] proposed the non metric MDS where the distance ....

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401--419, 1952.

....1 Delta 0 2 D n (p) if and only if B 0 = Delta 0 Delta 0 ) 2 Omega n (p) Furthermore, if B 0 2 Omega n (p) then the factorization B 0 = XX 0 produces an n Theta p configuration matrix X for which D(X) Delta 0 . Theorem 1 is a slight modification, due to Torgerson [37], of a result that was independently discovered and proved by Schoenberg [34] and by Young and Householder [47] It provides an affirmative answer to Question 1: if we compute the spectral decomposition ( Delta 0 Delta 0 ) QQ 0 and let X denote the first p = 3 columns of Q 1=2 , then ....

.... Gamma 1) 2 interatomic distances, stored in the n Theta n dissimilarity matrix Delta 0 . Can we determine a configuration matrix X such that D(X) Delta 0 Question 2 virtually defines metric MDS, for which a variety of approaches are available. The classical approach of Torgerson [37] and Gower [13] finds an approximate solution to the embedding problem of Section 3 with fallible data. This approach can be formulated as the following optimization problem: minimize kB Gamma ( Delta Delta)k 2 F subject to B 2 Omega n (p) 5) The objective function is sometimes ....

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401--419, 1952.

....1 Delta 0 2 Dn (p) if and only if B 0 = Delta 0 Delta 0 ) 2 Omega n (p) Furthermore, if B 0 2 Omega n (p) then the factorization B 0 = XX 0 produces an n Theta p configuration matrix X for which D(X) Delta 0 . Theorem 1 is a slight modification, due to Torgerson [17], of a result that was independently discovered and proved by Schoenberg [14] and by Young and Householder [22] It provides an affirmative answer to Question 1: if we compute the spectral decomposition ( Delta 0 Delta 0 ) QQ 0 and let X denote the first p = 3 columns of Q 1=2 , ....

.... Gamma 1) 2 interatomic distances, stored in the n Theta n dissimilarity matrix Delta 0 . Can we determine a configuration matrix X such that D(X) Delta 0 Question 2 virtually defines metric MDS, for which a variety of approaches are available. The classical approach of Torgerson [17] and Gower [6] finds an approximate solution to the embedding problem of Section 3.1 with fallible data. This approach can be formulated as the following optimization problem: minimize kB Gamma ( Delta Delta)k 2 F subject to B 2 Omega n (p) 6) Mardia [12] appears to have been the ....

[Article contains additional citation context not shown here]

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401--419, 1952.

....and MDS is scaling in the case that the target space is Euclidean. Kruskal and Wish [27] provided an elementary introduction to basic MDS methodology, as well as many enlightening examples. The present paper addresses two very specific, but very important problems in MDS. As in classical MDS [39, 40, 17], two assumptions are made about the nature of the information provided about the interpoint distances. Formally, a symmetric n Theta n matrix Delta = ffi ij ) is called a dissimilarity matrix if ffi ij 0 (nonnegative elements) and ffi ii = 0 (zero diagonal elements) From a given ....

....ffi ij have been termed metric. In contrast, nonmetric techniques minimize some measure of discrepancy between the interpoint distances and a set of dissimilarity matrices whose elements have the same rank ordering as the given ffi ij . Early MDS techniques, e.g. the methods of Torgerson [39], were exclusively metric. However, since the pioneering work of Shepard [34, 35] and Kruskal [25, 26] the psychometric and statistical communities have tended to emphasize nonmetric MDS. Nevertheless, metric MDS has remained critically important, because solving nonmetric MDS problems typically ....

[Article contains additional citation context not shown here]

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401--419, 1952.

....the importance of one property over the other. We here take the point of view that an independent evaluation of the different properties is to be preferred; the investigator can draw his or her own conclusions from the resp. numbers. E. 3 Multi dimensional scaling Multi dimensional scaling (MDS) [27] aims at preserving the mutual distances in a bijective map of data vectors onto a lower dimensional representation space. Consider a dissimilarity matrix with entries dV (v i ; v j ) of N data vectors v i and a representation of the data points in an output space with respective dissimilarities ....

....measure a renormalization scheme has not been provided as yet, so the examples investigated here go beyond what the measure was designed for. B. Comparison to PCA and MDS Topographic vector quantization has be achieved by techniques [13] 30] 38] 39] related to multi dimensional scaling [27], 29] The topography measures reviewed in this paper are also applicable to the projections obtained by these methods. This parallel development has led to comparisons between these techniques and topographic maps obtained by the SOM algorithm [13] 39] In these investigations, an optimization ....

W. S. Torgerson, Multidimensional Scaling. I: Theory and Method. Psychometrica 17, 401-419, 1952.

....Section 4 we derive the algorithm and the Slater constraint qualification result. We conclude with several remarks and numerical tests in Section 5. In addition, we include Section 5.1 with some technical details on the SDP algorithm. 2. DISTANCE GEOMETRY It is well known, e.g. 34] 15] 16] [38], that a pre distance matrix D is a EDM if and only if D is negative semidefinite on M : Phi x 2 n : x t e = 0 Psi ; the orthogonal complement of e, where e is the vector of all ones. Thus the set of all EDMs is a convex cone, which we denote by E : We exploit this result to ....

WARREN S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.

....we seek a configuration of n points in p . Then the (unweighted) SSTRESS problem for metric MDS is the following unconstrained optimization problem: minimize X i;j p X k=1 (x ik Gamma x jk ) 2 Gamma ffi 2 ij # 2 : 1) Unlike the STRAIN criterion used in classical metric MDS [19, 4], no explicit formula for a minimizer of the SSTRESS criterion is known. Hence, solutions to Problem (1) must be computed by an iterative algorithm for numerical optimization. A survey of some of the more efficient algorithms for solving Problem (1) was made by Kearsley, Tapia, and Trosset [10] ....

....linear function s defined by s (D) Gamma 1 2 (I Gamma es 0 )D(I Gamma es 0 ) is an inverse of . Johnson and Tarazaga [9] showed that the image of s is a face of the nonpolyhedral cone Omega n (n) The inverse function 1 obtained by setting s = e=n was introduced by Torgerson [19]. Implicit in his pioneering paper was the formulation of (metric) MDS as the optimization problem minimize kB Gamma 1 ( Delta Delta)k 2 F subject to B 2 Omega n (p) 3) This formulation of classical MDS has been explicitly discussed by Mardia [12] by de Leeuw and Heiser [2] and by ....

W. S. Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17:401--419, 1952.

.... = d(x i ; x j ) for all x i ; x j 2 R k , all x i ; x j 2 R p , i; j = 1; n Gamma 1 and d ( d) being a measure of distance in R k (R p ) Techniques for finding such transformations Phi are, among others, various forms of multidimensional scaling 2 (MDS) like metric MDS [Torgerson 52] nonmetric MDS [Shepard 62] or Sammon mapping [Sammon 69] but also principal component analysis (PCA) see e.g. Jolliffe 86] or SOM. Sammon mapping is doing MDS by minimizing the following via steepest descent: 1 P n Gamma1 i=0 P j i d(x i ; x j ) n Gamma1 X i=0 X j i (d(x i ; x j ) ....

Torgerson W.S.: Multidimensional Scaling, I: theory and method, Psychometrika, 17, 401-419, 1952.

No context found.

WARREN S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional scaling. 1. Theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional Scaling, I: Theory and Method. Psychometrika, 17:401--419, 1952.

No context found.

Warren Torgerson. Multidimensional scaling: I. Theory and method. Psychometrika, 17, 401-419. 1952.

No context found.

W.S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrica., 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W. Torgerson. Multidimensional scaling:i. theory and method. Psychometrika, 17:401--419, 1952.

No context found.

W.S. TORGERSON. Multidimensional scaling. I. Theory and method. Psychometrika, 17:401--419, 1952.

No context found.

Torgerson, W.S. (1952), \Multidimensional Scaling: I. Theory and Method," Psychometrika, 17, 401-419.

No context found.

W. Torgerson. Multidimensional scaling:i. theory and method. Psychometrika, 17:401--419, 1952.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC