SymScal: symbolic multidimensional scaling of interval dissimilarities

Multidimensional scaling aims at reconstructing dissimilaritiesbetween pairs of objects by distances in a low dimensional space.However, in some cases the dissimilarity itself is unknown, but therange of the dissimilarity is given. Such fuzzy data fall in thewider class of symbolic data (Bock and Diday, 2000).Denoeux and Masson (2000) have proposed to model an intervaldissimilarity by a range of the distance defined as the minimum andmaximum distance between two rectangles representing the objects. Inthis paper, we provide a new algorithm called SymScal that is basedon iterative majorization. The advantage is that each iteration isguaranteed to improve the solution until no improvement is possible.In a simulation study, we investigate the quality of thisalgorithm. We discuss the use of SymScal on empirical dissimilarityintervals of sounds.iterative majorization;multidimensional scaling;symbolic data analysis;distance smoothing

3WaySym-Scal: three-way symbolic multidimensional scaling

Multidimensional scaling aims at reconstructing dissimilarities between pairs of objects by distances in a low dimensional space.However, in some cases the dissimilarity itself is not known, but the range, or a histogram of the dissimilarities is given. This type of data fall in the wider class of symbolic data (see Bock and Diday (2000)). We model three-way two-mode data consisting of an interval of dissimilarities for each object pair from each of K sources by a set of intervals of the distances defined as the minimum and maximum distance between two sets of embedded rectangles representing the objects. In this paper, we provide a new algorithm called 3WaySym-Scal using iterative majorization, that is based on an algorithm, I-Scal developed for the two-way case where the dissimilarities are given by a range of values ie an interval (see Groenen et al. (2006)).The advantage of iterative majorization is that each iteration is guaranteed to improve the solution until no improvement is possible. We present the results on an empirical data set on synthetic musical tones.2WaySym-Scal;interval data;multidimensional scaling;symbolic data analysis;three-way data

A Distance-Based Test of Association Between Paired Heterogeneous Genomic Data

Due to rapid technological advances, a wide range of different measurements can be obtained from a given biological sample including single nucleotide polymorphisms, copy number variation, gene expression levels, DNA methylation and proteomic profiles. Each of these distinct measurements provides the means to characterize a certain aspect of biological diversity, and a fundamental problem of broad interest concerns the discovery of shared patterns of variation across different data types. Such data types are heterogeneous in the sense that they represent measurements taken at very different scales or described by very different data structures. We propose a distance-based statistical test, the generalized RV (GRV) test, to assess whether there is a common and non-random pattern of variability between paired biological measurements obtained from the same random sample. The measurements enter the test through distance measures which can be chosen to capture particular aspects of the data. An approximate null distribution is proposed to compute p-values in closed-form and without the need to perform costly Monte Carlo permutation procedures. Compared to the classical Mantel test for association between distance matrices, the GRV test has been found to be more powerful in a number of simulation settings. We also report on an application of the GRV test to detect biological pathways in which genetic variability is associated to variation in gene expression levels in ovarian cancer samples, and present results obtained from two independent cohorts

Symbolic Multidimensional Scaling

__Abstract__ Multidimensional scaling (MDS) is a technique that visualizes dissimilarities between pairs of objects as distances between points in a low dimensional space. In symbolic MDS, a dissimilarity is not just a value but can represent an interval or even a histogram. Here, we present an overview of developments for symbolic MDS. We discuss how interval dissimilarities they can be represented by (concentric) circles or rectangles, how replications can be represented by a three-way MDS version, and show how nested intervals of distances can be obtained for representing histogram dissimilarities. The various models are illustrated by empirical examples

THE MEASUREMENT OF THE ECONOMIC DISTANCE ON THE BASIS OF SYMBOLIC DATA

The economic distance defines a dissimilarity level between objects functioning in the economic space. It is one of the most important issues of spatial econometrics. However its measurement is difficult due to the definition, description and estimation problems. The objective of the paper is to indicate the role of symbolic data in describing the economic distance and also the way of its measurement using symbolic data analysis methods. A significance of the economic distance, measurement problems, symbolic data concept and dissimilarity measures, and also an empirical example were presented in the paper

SymScal: symbolic multidimensional scaling of interval dissimilarities

Multidimensional scaling aims at reconstructing dissimilarities between pairs of objects by distances in a low dimensional space. However, in some cases the dissimilarity itself is unknown, but the range of the dissimilarity is given. Such fuzzy data fall in the wider class of symbolic data (Bock and Diday, 2000). Denoeux and Masson (2000) have proposed to model an interval dissimilarity by a range of the distance defined as the minimum and maximum distance between two rectangles representing the objects. In this paper, we provide a new algorithm called SymScal that is based on iterative majorization. The advantage is that each iteration is guaranteed to improve the solution until no improvement is possible. In a simulation study, we investigate the quality of this algorithm. We discuss the use of SymScal on empirical dissimilarity intervals of sounds

3rd Workshop in Symbolic Data Analysis: book of abstracts

This workshop is the third regular meeting of researchers interested in Symbolic Data Analysis. The main aim of the event is to favor the meeting of people and the exchange of ideas from different fields - Mathematics, Statistics, Computer Science, Engineering, Economics, among others - that contribute to Symbolic Data Analysis

3WaySym-Scal: three-way symbolic multidimensional scaling

Multidimensional scaling aims at reconstructing dissimilarities between pairs of objects by distances in a low dimensional space. However, in some cases the dissimilarity itself is not known, but the range, or a histogram of the dissimilarities is given. This type of data fall in the wider class of symbolic data (see Bock and Diday (2000)). We model three-way two-mode data consisting of an interval of dissimilarities for each object pair from each of K sources by a set of intervals of the distances defined as the minimum and maximum distance between two sets of embedded rectangles representing the objects. In this paper, we provide a new algorithm called 3WaySym-Scal using iterative majorization, that is based on an algorithm, I-Scal developed for the two-way case where the dissimilarities are given by a range of values ie an interval (see Groenen et al. (2006)). The advantage of iterative majorization is that each iteration is guaranteed to improve the solution until no improvement is possible. We present the results on an empirical data set on synthetic musical tones