On Some Functional Characterizations of (Fuzzy) Set-Valued Random Elements
One of the most common spaces to model imprecise data through (fuzzy) sets is that of convex and compact (fuzzy) subsets in \(\mathbb {R}^p\) . The properties of compactness and convexity allow the identification of such elements by means of the so-called
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Abstract One of the most common spaces to model imprecise data through (fuzzy) sets is that of convex and compact (fuzzy) subsets in R p . The properties of compactness and convexity allow the identification of such elements by means of the so-called support function, through an embedding into a functional space. This embedding satisfies certain valuable properties, however it is not always intuitive. Recently, an alternative functional representation has been considered for the analysis of imprecise data based on the star-shaped sets theory. The alternative representation admits an easier interpretation in terms of ‘location’ and ‘imprecision’, as a generalized idea of the concepts of mid-point and spread of an interval. A comparative study of both functional representations is made, with an emphasis on the structures required for a meaningful statistical analysis from the ontic perspective.
1 Introduction The statistical analysis of (fuzzy) set-valued data from the so-called ‘ontic’ perspective has frequently been developed as a generalization of the statistics for interval data (see, e.g., [1]). From this ‘ontic’ perspective, (fuzzy) set-valued data are considered as whole entities, in contrast to the epistemic approach, which considers (fuzzy) set-valued data as imprecise measurements of precise data (see, e.g., [2]). Both the arithmetic and metric structure to handle this ‘ontic’ data is often based on an extension of the Minkowski arithmetic and the distance between either infima and suprema or mid-points and spreads for intervals. In this way, key concepts such as the expected value or the variability, are naturally defined as an extension of the classical notions within the context of (semi-)linear metric spaces. The generalization of the concept of interval to R p keeps the compactness and convexity properties, and this allows the identification of the contour of the convex A. Colubi (B) · G. Gonzalez-Rodriguez Indurot and Department of Statistics, University of Oviedo, 33007 Oviedo, Spain e-mail: [email protected] G. Gonzalez-Rodriguez e-mail: [email protected] © Springer International Publishing Switzerland 2017 M.B. Ferraro et al. (eds.), Soft Methods for Data Science, Advances in Intelligent Systems and Computing 456, DOI 10.1007/978-3-319-42972-4_17
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and compact sets in R p by means of the support function (see, e.g., [6]). The support function is coherent with the Minkowski arithmetic, but sometimes this is not easy to interpret. In [4] the so-called kernel-radial characterization is investigated as an alternative to the support function based on a representation on polar coordinates. This polar representation is established in the context of the star-shaped sets, and is connected with the developments in [3]. It is coherent with alternative arithmetics and distances generalizing the concepts of location and imprecision in an intuitive way, which are of paramount importance in the considered context. The aim is to show a comparative study of the support fun
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