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Top1. Introduction
Formal Concept Analysis (FCA) is a data mining model that introduces the relation among attributes in a visual form. It was introduced in the early 80s by Wille (1982) to study how objects can be hierarchically grouped together according to their common attributes. It serves in different fields such as biology (Bertaux et al., 2007), medicine (Motameny et al., 2008) and system analysis (Düwel, 2009). This tool is of a great interest for mining association rules in real life data, specially the numerical ones. The basic structure of FCA is the formal context which is a binary-relation between a set of objects and a set of attributes. The formal context is based on the ordinary set, whose elements has one of two values, 0 or 1 (Kaytoue et al., 2009; Cole et al., 1998). A context materializes a set of individuals called objects, a set of properties called attributes, and a binary relation usually represented by a binary table relating objects to attributes.
These mappings are called Galois connections or concepts. Such concepts are ordered in FCA within a lattice structure called concept lattice within the FCA. Concept lattices can be represented by diagrams giving clear visualization of classes of objects in each domain. At the same time, the edges of these diagrams give essential knowledge about objects, by introducing association rules between attributes which describe the objects (Sergei, 2002). Mostly, the real-world data are not available as binary data. Such data could be either numerical or categorical. To represent a numerical or a categorical data in the form of a formal context, such data should be transformed using conceptual scaling. In this approach, the attribute of numerical values are discredited. Then each interval of entry values, have to be considered as binary attributes (Motameny et al., 2008). The transformation of such data, i.e., conceptual scaling, allows one to apply FCA tools. Such procedure may dramatically increase the complexity of computation and representation. Hence, it worsens the visualization of results. This scaling may produce large and dense binary data (Snsel et al. 2008), which are hard to process with the existing FCA algorithms. As it is based on arbitrary choices, the data may be scaled in a lot of different ways that lead to different results. Its interpretations could lead also to classification problems. The work done by Fu and Nguifo (2003) proposed a scalable lattice-based algorithm ScalingNextClosure to decompose the search space for finding formal concepts in large datasets into partitions and then generate independently concepts (or closed item sets) in each partition. This paper replaces the scaling technique by a method that uses of the chiMerge algorithm into binarization of the numerical data attributes and into the validation of the generated formal concept lattice. The binarization technique is applied using the chiMerge algorithm through discretizing the continuous attribute values into only of two values, 0 or 1. Then the resulted binary table is used in generation of the formal concept lattice. Then the chiMerge method here is used to validate this generated lattice. For continuous data sets, the ChiMerge method is used to automatically select proper Chi-square χ2 values to evaluate the worth of each attribute with respect to the corresponding classes (Thabtah et al., 2002; Kumar & Rao, 2009). These χ2 values are used to compare value of each attribute. Such values were calculated according to a novel formula based on the generated formal concept lattice. If both evaluations are matched, then the generated lattice is considered to be representing the actual structure of the data. Hence, the binarization method does not corrupt the generated lattice. Finally, it led to a valid lattice. The conceptual computation and the lattice visualization are performed using a tool for formal concept lattice generation named (OpenFCA, http://code.google.com/p/openfca). The introduced techniques, the binarization, visualization and validation methods, is applied on two data sets in the medical field from the UCI database; the Indian Diabetes data set and the Breast Cancer data set.