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Journal ArticleDOI

Multidimensional scaling of fuzzy dissimilarity data

16 Jun 2002-Fuzzy Sets and Systems (North-Holland)-Vol. 128, Iss: 3, pp 339-352
TL;DR: This paper extendsMultidimensional scaling to the case where dissimilarities are expressed as intervals or fuzzy numbers, and each object is no longer represented by a point but by a crisp or a fuzzy region.
About: This article is published in Fuzzy Sets and Systems.The article was published on 2002-06-16. It has received 30 citations till now. The article focuses on the topics: Fuzzy number & Multidimensional scaling.
Citations
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Journal ArticleDOI
TL;DR: This work proposes a variant of the EM algorithm that iteratively maximizes the maximization of a generalized likelihood criterion, which can be interpreted as a degree of agreement between the statistical model and the uncertain observations.
Abstract: We consider the problem of parameter estimation in statistical models in the case where data are uncertain and represented as belief functions. The proposed method is based on the maximization of a generalized likelihood criterion, which can be interpreted as a degree of agreement between the statistical model and the uncertain observations. We propose a variant of the EM algorithm that iteratively maximizes this criterion. As an illustration, the method is applied to uncertain data clustering using finite mixture models, in the cases of categorical and continuous attributes.

249 citations

Journal ArticleDOI
TL;DR: The main fuzzy approaches for defining spatial relationships including topological (set relationships, adjacency) and metrical relations (distances, directional relative position) are reviewed.

232 citations


Cites background from "Multidimensional scaling of fuzzy d..."

  • ...[57], and the corresponding distance density is expressed as:...

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Journal ArticleDOI
TL;DR: In spite of a growing literature concerning the development and application of fuzzy techniques in statistical analysis, the need is felt for a more systematic insight into the potentialities of cross fertilization between Statistics and Fuzzy Logic.

129 citations


Cites background or methods from "Multidimensional scaling of fuzzy d..."

  • ...…of the fuzzy-possibilistic approach to regression analysis as well as to other statistical analyses, such as principal components, multidimensional scaling, etc. have followed the seminal paper by Tanaka et al. (1982) (see, for instance, D’Urso and Giordani, 2005; Masson and DenWux, 2002)....

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  • ...This consists of minimizing the overall fuzziness of the estimated responses ỹi , while imposing the constraint of inclusion of the observed responses yi (i = 1, . . . , n) within the support of the respective fuzzy theoretical values ỹi (i = 1, . . . , n)....

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Journal ArticleDOI
TL;DR: This paper describes an extension of principal component analysis allowing the extraction of a limited number of relevant features from high-dimensional fuzzy data, and the concept of correlation coefficient is extended to fuzzy numbers, allowing the interpretation of the new features in terms of the original variables.
Abstract: This paper describes an extension of principal component analysis (PCA) allowing the extraction of a limited number of relevant features from high-dimensional fuzzy data. Our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as PCA, without explicit matrix diagonalization. Fuzzy input values are propagated through the network using fuzzy arithmetics, and the weights are adjusted to minimize a suitable error criterion, the inputs being taken as target outputs. The concept of correlation coefficient is extended to fuzzy numbers, allowing the interpretation of the new features in terms of the original variables. Experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data.

56 citations


Cites methods from "Multidimensional scaling of fuzzy d..."

  • ...Multidimensional scaling, a technique to map objects to a multidimensional feature space based on observed dissimilarities between objects, has recently been extended to interval-valued and fuzzy dissimilarities [12], [30]....

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Journal ArticleDOI
TL;DR: The proposed method assigns to each object a basic belief assignment defined on the set of clusters, in such a way that the belief and the plausibility that any two objects belong to the same cluster reflect, respectively, the observed lower and upper dissimilarity values.

49 citations


Cites methods from "Multidimensional scaling of fuzzy d..."

  • ...For representing interval-valued dissimilarities [9] or fuzzy dissimilarities [17] in a low dimensional space, we have proposed several extensions of classical multidimensional scaling methods (MDS)....

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References
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Journal ArticleDOI
TL;DR: Much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.

12,530 citations

Journal ArticleDOI
Joseph B. Kruskal1
TL;DR: The fundamental hypothesis is that dissimilarities and distances are monotonically related, and a quantitative, intuitively satisfying measure of goodness of fit is defined to this hypothesis.
Abstract: Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.

6,875 citations


"Multidimensional scaling of fuzzy d..." refers methods in this paper

  • ...To assess the quality of the approximation of ∆ by D(X), the following loss function, known in the literature as the Stress function [11], is used: σ(X) = ∑ i<j (dij(X)− δij)(2) ....

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Journal ArticleDOI
TL;DR: In this paper, an individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common "psychological space" and a corresponding method of analyzing similarities data is proposed, involving a generalization of Eckart-Young analysis to decomposition of three-way (or higher-way) tables.
Abstract: An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.

4,520 citations


"Multidimensional scaling of fuzzy d..." refers methods in this paper

  • ...Another popular method is the INDSCAL model (INdividual DIfferences SCALing) [2]....

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Journal ArticleDOI
TL;DR: An algorithm for the analysis of multivariate data is presented along with some experimental results that is based upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data "structure" is approximately preserved.
Abstract: An algorithm for the analysis of multivariate data is presented along with some experimental results. The algorithm is based upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data "structure" is approximately preserved.

3,460 citations


"Multidimensional scaling of fuzzy d..." refers background in this paper

  • ...An early use of MDS has been dimension reduction [13]....

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