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

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

On central tendency and dispersion measures for intervals and hypercubes

The uncertainty or the variability of the data may be treated by considering, rather than a single value for each data, the interval of values in which it may fall. This paper studies the derivation of basic description statistics for interval-valued datasets. We propose a geometrical approach in the determination of summary statistics (central tendency and dispersion measures) for interval-valued variables

Linear regression for numeric symbolic variables: an ordinary least squares approach based on Wasserstein Distance

In this paper we present a linear regression model for modal symbolic data. The observed variables are histogram variables according to the definition given in the framework of Symbolic Data Analysis and the parameters of the model are estimated using the classic Least Squares method. An appropriate metric is introduced in order to measure the error between the observed and the predicted distributions. In particular, the Wasserstein distance is proposed. Some properties of such metric are exploited to predict the response variable as direct linear combination of other independent histogram variables. Measures of goodness of fit are discussed. An application on real data corroborates the proposed method

Representing complex data using localized principal components with application to astronomical data

Often the relation between the variables constituting a multivariate data space might be characterized by one or more of the terms: ``nonlinear'', ``branched'', ``disconnected'', ``bended'', ``curved'', ``heterogeneous'', or, more general, ``complex''. In these cases, simple principal component analysis (PCA) as a tool for dimension reduction can fail badly. Of the many alternative approaches proposed so far, local approximations of PCA are among the most promising. This paper will give a short review of localized versions of PCA, focusing on local principal curves and local partitioning algorithms. Furthermore we discuss projections other than the local principal components. When performing local dimension reduction for regression or classification problems it is important to focus not only on the manifold structure of the covariates, but also on the response variable(s). Local principal components only achieve the former, whereas localized regression approaches concentrate on the latter. Local projection directions derived from the partial least squares (PLS) algorithm offer an interesting trade-off between these two objectives. We apply these methods to several real data sets. In particular, we consider simulated astrophysical data from the future Galactic survey mission Gaia.Comment: 25 pages. In "Principal Manifolds for Data Visualization and Dimension Reduction", A. Gorban, B. Kegl, D. Wunsch, and A. Zinovyev (eds), Lecture Notes in Computational Science and Engineering, Springer, 2007, pp. 180--204, http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-173750210-

On the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum

Multifunctional proteins revealed by overlapping clustering in protein interaction network

Motivation: Multifunctional proteins perform several functions. They are expected to interact specifically with distinct sets of partners, simultaneously or not, depending on the function performed. Current graph clustering methods usually allow a protein to belong to only one cluster, therefore impeding a realistic assignment of multifunctional proteins to clusters