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. Symbolic objects are the basic elements for knowledge representation in symbolic data analysis. This paper aims to integrate symbolic objects into formal concept analysis in order to compare and tie together both approaches. 1 Introduction Symbolic objects are the basic elements of a formal language which has been developed since 1987 in symbolic data analysis. The general aim was to extend the field of application, methods and algorithms of classic data analysis to more complex data. Meanwhile, the formalism of symbolic objects is not only used in a broad field of data analysis, but also in knowledge representation and knowledge processing. From the point of view of formal concept analysis, the most interesting parts of symbolic data analysis are those which are concerned with knowledge processing and conceptual classifications. These parts of symbolic data analysis and formal concept analysis both emphasize the intensional view. Hence, there are various points of common interest w...

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The data descriptions of the units are called "symbolic" when they are more complex than the standard ones due to the fact that they contain internal variation and are structured. Symbolic data happen from many sources, for instance in order to summarise huge Relational Data Bases by their underlying concepts. "Extracting knowledge" means getting explanatory results, that why, "symbolic objects" are introduced and studied in this paper. They model concepts and constitute an explanatory output for data analysis. Moreover they can be used in order to define queries of a Relational Data Base and propagate concepts between Data Bases. We define "Symbolic Data Analysis" (SDA) as the extension of standard Data Analysis to symbolic data tables as input in order to find symbolic objects as output. In this paper we give an overview on recent development on SDA. We present some tools and methods of SDA and introduce the SODAS software prototype (issued from the work of 17 teams of nine countries involved in an European project of EUROSTAT). 1

In this article we propose an algorithm for Principal Components Analysis when the variables are histogram type. This algorithm also works if the data table has variables of interval type and histogram type mixed. If all the variables are interval type it produces the same output as the one produced by the algorithm of the Centers Method propose in [5, Cazes, Chouakria, Diday and Schektman (1997)]. 1 The algorithm In this algorithm we use the idea proposed in [9, Diday (1998)]. We represent each histogram--individual by a succession of k interval--individuals (the first one included in the second one, the second one included in the third one and so on) where k is the maximum number of modalities taken by some variable in the input symbolic data table. Instead of representing the histograms in the factorial plane, we are going to represent the Empirical Distribution Function F Y defined, in [3, Bock and Diday (2000)] associated with each histogram. In other words if we have an histogram variable Y on a set E = {a 1 , a 2 , . . .} of objects with domain Y represented by the mapping Y (a) = (U(a), # a ), for a # E, where # a is frequency distribution, then in the algorithm we will use the function F (x) = # i / # i #x # i instead of the histogram. Definition 1. Let X = (x ij ) i=1,2,...,m, j=1,2,...,n be a symbolic data table with variables type continuous, interval and histogram, and let be k = max{s, where s is the number of modalities of Y j , j = 1, 2, . . . , n} where Y j is a variable of histogram type 1 . We define the vector--succession of intervals associated with each cell of X as: 1 If all the variables are interval type then k = 1. 1. if x ij = [a, b] then the vector--succession of intervals associated is: x # ij = # # # # # [a, b] [a...

Abstract. Statistical units described by interval-valued variables represent a special case of Symbolic Objects, where all descriptors are quantitative variables. In this context, the paper presents two different metrics in R p for interval-valued data that are based on the definition of the Hausdorff distance in R. Hausdorff distance in R p (for any p≥1) is a L ∞ norm between pairs of closed sets. However, when p> 1 the problem complexity leads towards the definition of L2 norms approximating as well as possible the Hausdorff distance. Given a set of n units described by p interval-valued variables, we compute and represent the distances over factorial planes that are defined by factorial analyses that are consistent with the two distance measure definitions.

General objects are classes of individual objects that are considered to be extents of concepts of a formal context. In this paper, different contexts with general objects are defined and their conceptual structure and relation to other contexts is analyzed with methods of Formal Concept Analysis.

this paper we do not consider the interval central value, but we only point the attention on the minimum and maximum values. These are evaluated as two di#erent and related aspects of the same phenomenon