I-Scal: Multidimensional scaling of interval dissimilarities
TL;DR: A new algorithm called I-Scal, based on iterative majorization, that has the advantage that each iteration is guaranteed to improve the solution until no improvement is possible is developed.
About: This article is published in Computational Statistics & Data Analysis.The article was published on 2006-11-01. It has received 46 citations till now. The article focuses on the topics: Multidimensional scaling & Interval (mathematics).
Citations
More filters
TL;DR: This new method shows the importance of range information in prediction performance as well as the use of inequality constraints to ensure mathematical coherence between the predicted values of the lower and upper boundaries of the interval value of the dependent variable.
188 citations
Cites methods from "I-Scal: Multidimensional scaling of..."
...Groenen et al. (2006) introduced a multidimensional scaling method for managing interval dissimilarities....
[...]
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 methods from "I-Scal: Multidimensional scaling of..."
...In the paper by Groenen et al. (2007), one of the models also considered by Hébert et al. (2007) is studied in more detail....
[...]
...The uncertainty related to model specification and estimation in the traditional approaches, is managed here by means of a flexible system of fuzzy rules embodying the vagueness of the economical relationships and the imprecision of the econometric measurements....
[...]
TL;DR: This paper introduces three approaches to forecasting interval-valued time series based on multilayer perceptron (MLP) neural networks and Holt’s exponential smoothing methods, respectively.
117 citations
Proceedings Article•
21 Jun 2014
TL;DR: It is proved that in the large sample limit it is enough to know "local ordinal information" in order to perfectly reconstruct a given point configuration, which leads to the Local Ordinal Embedding algorithm, which can also be used for graph drawing.
Abstract: We study the problem of ordinal embedding: given a set of ordinal constraints of the form distance(i, j) < distance(k, l) for some quadruples (i, j, k, l) of indices, the goal is to construct a point configuration x1,..., xn in Rp that preserves these constraints as well as possible. Our first contribution is to suggest a simple new algorithm for this problem, Soft Ordinal Embedding. The key feature of the algorithm is that it recovers not only the ordinal constraints, but even the density structure of the underlying data set. As our second contribution we prove that in the large sample limit it is enough to know "local ordinal information" in order to perfectly reconstruct a given point configuration. This leads to our Local Ordinal Embedding algorithm, which can also be used for graph drawing.
69 citations
Cites methods from "I-Scal: Multidimensional scaling of..."
...To construct a majorizing function for our objective function Errsoft, we take inspiration from Groenen et al. (2006)....
[...]
TL;DR: This article forms interval-valued variables as bivariate random vectors and introduces the bivariate symbolic regression model based on the generalized linear models theory which provides much-needed exibility in practice.
Abstract: Interval-valued variables have become very common in data analysis. Up until now, symbolic regression mostly approaches this type of data from an optimization point of view, considering neither the probabilistic aspects of the models nor the nonlinear relationships between the interval response and the interval predictors. In this article, we formulate interval-valued variables as bivariate random vectors and introduce the bivariate symbolic regression model based on the generalized linear models theory which provides much-needed exibility in practice. Important inferential aspects are investigated. Applications to synthetic and real data illustrate the usefulness of the proposed approach.
61 citations
References
More filters
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
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
TL;DR: In this paper, the authors derived necessary and sufficient conditions for a solution to exist in real Euclidean space for a multivariate multivariate sample of size n as points P1, P2,..., PI in a Euclidian space and discussed the interpretation of the distance A(Pi, Pj) between the ith and jth members of the sample.
Abstract: SUMMARY This paper is concerned with the representation of a multivariate sample of size n as points P1, P2, ..., PI in a Euclidean space. The interpretation of the distance A(Pi, Pj) between the ith andjth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean space is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinates are the estimated factor scores can be interpreted as D2 with a singular dispersion matrix.
3,746 citations
TL;DR: The four Purposes of Multidimensional Scaling, Special Solutions, Degeneracies, and Local Minima, and Avoiding Trivial Solutions in Unfolding are explained.
Abstract: Fundamentals of MDS.- The Four Purposes of Multidimensional Scaling.- Constructing MDS Representations.- MDS Models and Measures of Fit.- Three Applications of MDS.- MDS and Facet Theory.- How to Obtain Proximities.- MDS Models and Solving MDS Problems.- Matrix Algebra for MDS.- A Majorization Algorithm for Solving MDS.- Metric and Nonmetric MDS.- Confirmatory MDS.- MDS Fit Measures, Their Relations, and Some Algorithms.- Classical Scaling.- Special Solutions, Degeneracies, and Local Minima.- Unfolding.- Unfolding.- Avoiding Trivial Solutions in Unfolding.- Special Unfolding Models.- MDS Geometry as a Substantive Model.- MDS as a Psychological Model.- Scalar Products and Euclidean Distances.- Euclidean Embeddings.- MDS and Related Methods.- Procrustes Procedures.- Three-Way Procrustean Models.- Three-Way MDS Models.- Modeling Asymmetric Data.- Methods Related to MDS.
3,096 citations