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An Introduction to Symbolic Data Analysis and the Sodas Software
 Journal of Symbolic Data Analysis
, 2003
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Symbolic Objects in Formal Concept Analysis
 Proceedings of the Second International Symposium on Knowledge, Retrieval, Use and Storage for Efficiency, G. Mineau & A. Fall (eds
, 1997
"... . 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 langu ..."
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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...
Knowledge Discovery From Symbolic Data And The Sodas Software
 Conf. on Principles and Practice of Knowledge Discovery in Databases, PPKDD2000
, 2000
"... 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 under ..."
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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
Generalization of the Principal Components Analysis to Histogram Data
 In Principles and Practice of knowledge discovery in databases
, 2000
"... 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 ..."
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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 histogramindividual by a succession of k intervalindividuals (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 vectorsuccession 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 vectorsuccession of intervals associated is: x # ij = # # # # # [a, b] [a...
Multidimensional IntervalData: Metrics and Factorial Analysis
"... Abstract. Statistical units described by intervalvalued 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 intervalvalued data that are based on the definition of the Hausdor ..."
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Abstract. Statistical units described by intervalvalued 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 intervalvalued 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 intervalvalued 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.
Pyramidal Clustering Algorithms in ISO3D Project
, 2000
"... Pyramidal clustering method generalizes hierarchies by allowing nondisjoint classes at a given level instead of a partition. Moreover, the clusters of the pyramid are intervals of a total order on the set being clustered. [Diday 1984], [Bertrand, Diday 1990] and [Mfoumoune 1998] proposed algorithms ..."
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Pyramidal clustering method generalizes hierarchies by allowing nondisjoint classes at a given level instead of a partition. Moreover, the clusters of the pyramid are intervals of a total order on the set being clustered. [Diday 1984], [Bertrand, Diday 1990] and [Mfoumoune 1998] proposed algorithms to build a pyramid starting with an arbitrary order of the individual. In this paper we present two new algorithms name CAPS and CAPSO. CAPSO builds a pyramid starting with an order given on the set of the individuals (or symbolic objects) while CAPS finds this order. These two algorithms allows moreover to cluster more complex data than the tabular model allows to process, by considering variation on the values taken by the variables, in this way, our method produces a symbolic pyramid. Each cluster thus formed is defined not only by the set of its elements (i.e. its extent) but also by a symbolic object, which describes its properties (i.e. its intent). These two algorithms were implemented in C++ and Java to the ISO3D project. 1 Definitions Diday in [5, Diday (1984)] proposes the algorithm CAP to build numeric pyramids. Algorithms are also presented with this purpose in [2, Bertrand y Diday (1990)], [10, Gil (1998)] and [11, Mfoumoune (1998)]. Paula Brito in [3, Brito (1991)] proposes a macroalgorithm that generalizes the algorithm to build numeric pyramids proposed by Bertrand to the symbolic case. In this article we propose two algorithm designed to build symbolic pyramids (CAPS and CAPSO), that is to say, a pyramid in which each node is again a symbolic object. These algorithms also calculate the extension of each one of these symbolic objects and verifie its completeness. Notation:  # the set of individuals.  O j the description space for the variable j....
Formal Concept Analysis for General Objects
, 2000
"... 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. ..."
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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.
Principal Component Analysis of Interval Data:
 Computational Statistics
, 2000
"... 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 ..."
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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