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Journal ArticleDOI

Fuzzy multidimensional scaling

01 Nov 2006-Computational Statistics & Data Analysis (Elsevier Science Publishers B. V.)-Vol. 51, Iss: 1, pp 335-359
TL;DR: Two algorithms are proposed and illustrated that represent both the structure and the vagueness of dissimilarity measurements in a low-dimensional space using the Euclidean and spherical models.
About: This article is published in Computational Statistics & Data Analysis.The article was published on 2006-11-01. It has received 14 citations till now. The article focuses on the topics: Multidimensional scaling & Fuzzy number.
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: 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 "Fuzzy multidimensional scaling"

  • ...The paper by Hébert et al. (2007) is devoted to the construction of multidimensional scaling techniques, when the dissimilarity matrix is imprecisely observed....

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  • ...In the paper by Groenen et al. (2007), one of the models also considered by Hébert et al. (2007) is studied in more detail....

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Journal ArticleDOI

16 citations


Cites background from "Fuzzy multidimensional scaling"

  • ...…Masson (2004b), D’Urso and Giordani (2005), Giordani and Kiers (2006) and Calcagnì et al. (2016) Multidimensional scaling Denoeux and Masson (2000) and Hébert et al. (2006) Self-organizing maps D’Urso et al. (2014) Clusterwise regression analysis D’Urso and Santoro (2006) Correspondence analysis…...

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Journal ArticleDOI
24 Mar 2017
TL;DR: An overview of the developments in the exploratory multivariate analysis of imprecise data is presented and the advantages of the fuzzy approach in providing a deeper and more comprehensive insight into the management of uncertainty in this branch of Statistics are shown.
Abstract: In the last few decades, there has been an increase in the interest of the scientific community for multivariate statistical techniques of data analysis in which the data are affected by uncertainty, imprecision, or vagueness. In this context, following a fuzzy formalization, several contributions and developments have been offered in various fields of the multivariate analysis. In this paper—to show the advantages of the fuzzy approach in providing a deeper and more comprehensive insight into the management of uncertainty in this branch of Statistics—we present an overview of the developments in the exploratory multivariate analysis of imprecise data. In particular, we give an outline of these contributions within an overall framework of the general fuzzy approach to multivariate statistical analysis and review the principal exploratory multivariate methods for imprecise data proposed in the literature, i.e., cluster analysis, self-organizing maps, regression analysis, principal component analysis, multidimensional scaling, and other exploratory statistical approaches. Finally, we point out certain potentially fruitful lines of research that could enrich the future developments in this interesting and promising research area of Statistical Reasoning. All in all, the main purpose of this paper is to involve the Granular Computing scientific community and to stimulate and focus its interest on these statistical fields.

10 citations

Book ChapterDOI
15 Jul 2014
TL;DR: A very efficient computational method is proposed for approximating bounds of the Kolmogorov-Smirnov homogeneity test by using a p-box representation of the samples.
Abstract: In this paper, we are interested in extending the classical Kolmogorov-Smirnov homogeneity test to compare two samples of interval-valued observed measurements. In such a case, the test result is interval-valued, and one major difficulty is to find the bounds of this set. We propose a very efficient computational method for approximating these bounds by using a p-box (pairs of upper and lower cumulative distributions) representation of the samples.

10 citations


Cites background from "Fuzzy multidimensional scaling"

  • ...However, treating this imprecision usually leads to an increased computational costs, as shown by various authors in the past [6,7,3]....

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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

01 Jan 1975

8,942 citations


"Fuzzy multidimensional scaling" refers methods in this paper

  • ...Concepts of fuzzy correlation were introduced in Liu and Kao (2002) and Hébert et al. (2003); Denœux et al. (2005) to measure the degree of association between fuzzy-valued attributes, by applying the extension principle (Zadeh, 1975)....

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  • ...Applying the extension principle (Zadeh, 1975), the fuzzy distance between two fuzzy regions R̃i et R̃j can be defined as: µ d̃ij (w) = sup x,y∈Rp min(µ R̃i (x), µ R̃j (y)), (20) where the supremum is computed under the constraint ‖x−y‖ = w....

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Journal ArticleDOI
Joseph B. Kruskal1
TL;DR: The numerical methods required in the approach to multi-dimensional scaling are described and the rationale of this approach has appeared previously.
Abstract: We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.

4,561 citations


"Fuzzy multidimensional scaling" refers methods in this paper

  • ...In this case, the disparities are computed using a transformation referred to as isotonic regression (Kruskal, 1964) which insures that dij ≤ dkl whenever δij ≤ δkl. This nonmetric or ordinal approach will not be considered further in this paper, although a nonmetric MDS procedure for interval-valued data was proposed in Denoeux and Masson (2000)....

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  • ...In this case, the disparities are computed using a transformation referred to as isotonic regression (Kruskal, 1964) which insures that dij ≤ dkl whenever δij ≤ δkl....

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Journal ArticleDOI
TL;DR: This book introduced many novel mathematical operations based on this concept of level of confidence and have presented many generalizations, and presented several operations and functions of fuzzy numbers, such as integer modulo operations, trigonometric functions, and hyperbolic functions.
Abstract: We were rather pleased to read the review of our book, Introduction to Fuzzy Arithmetic: Theory and Applications. This review was done quite carefully by Caroline M. Eastman of the University of South Carolina, and we are grateful to her for pointing out many interesting, positive aspects as well as some shortcomings of our book. As members of the fuzzy community, we are concerned with studies and developments of concepts and techniques basic to the analysis of uncertainty arising from human perception, thinking, and reasoning processes. In this book we present such concepts and some novel tools for dealing with uncertainties. We start our introduction with the definition for the interval of confidence [al, a2], where al and a2 represent, respectively, the lower and upper bounds of our (subjective) confidence. Next, we introduce some arithmetic operations on these numbers. We then introduce the level of presumption ue [13, 1] and, using it, introduce the uncertain or fuzzy number that is so pervasive in our reasoning process. The reviewer has rightly pointed out that in certain situations, interval arithmetic can be considered a subset of fuzzy arithmetic, the main topic of our book. However, we intentionally did not want to confuse the issue by introducing interval arithmetic and then giving a generalization. We liked our approach, as have many other researchers and students who have used the book. In our approach, we have been guided throughout by a desire to lay a firm foundation for the definition of fuzzy numbers using the basic concept of level of confidence. We have introduced many novel mathematical operations based on this concept and have presented many generalizations. In addition, we have presented several operations and functions of fuzzy numbers, such as integer modulo operations, trigonometric functions, and hyperbolic functions. These studies have been included for students as well as researchers who wish to have an extended view of the theory. We have attempted to give a thorough exposition of fuzzy numbers; this exposition is illustrated by about 115 worked-out examples, 150 diagrams, and 90 tables. We did not include problems or exercises, which would have put this book in the category of a textbook. The subtitle of the book is \"Theory and Applications,\" but as is rightly

2,238 citations


Additional excerpts

  • ...If R̃i and R̃j are multidimensional fuzzy numbers (Kaufmann and Gupta, 1991, page 146), each α-cut of d̃ij is a closed interval αd̃ij = [ αd̃−ij, αd̃+ij], whose bounds are respectively the minimum and maximum distances between the α-cuts of R̃i and R̃j....

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