χristoφe Демко

In this article, we present a new data type agnostic algorithm calculating a concept lattice from heterogeneous and complex data. Our NᴇxᴛPʀɪᴏʀɪᴛʏCᴏɴᴄᴇᴘᴛ algorithm is first introduced and proved in the binary case as an extension of Bordat’s algorithm with the notion of strategies to select only some predecessors of each concept, avoiding the generation of unreasonably large lattices. The algorithm is then extended to any type of data in a generic way. It is inspired by the pattern structure theory, where data are locally described by predicates independent of their types, allowing the management of heterogeneous data.

Concept lattices and closed set lattices are graphs with the lattice property. They have been increasingly used this last decade in various domains of computer science, such as data mining, knowledge representation, databases or information retrieval. A fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table. A consequence is the existence of a bijective link between lattices, contexts (via the table) and a set of implicational rules (via the canonical (direct) basis). The possible transformations between these objects give rise to relevant tools for data analysis. In this paper, we present a survey of lattice theory, from the algebraic definition of a lattice, to that of a concept lattice, through closure systems and implicational rules; including the exploration of fundamental bijective links between lattices, reduced contexts and bases of implicational rules; and concluding with the presentation of the main generation algorithms of these objects.

In this paper we deal with the clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes. The method that we propose is formulated as a constrained minimization problem, whose solution depends on a fuzzy objective function in which reject options are introduced. Two types of rejection have been included: the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes. To compute these rejections, we propose an extension of the fuzzy c-means (FcM) algorithm of Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. This algorithm is called the fuzzy c+2-means (Fc+2M). These measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes. The advantages of our method are (1) the degree of membership to the reject classes for a pattern xk are learned in the iterative clustering problem; (2) it is not necessary to compute other characteristics to determine the reject and ambiguity degrees; (3) the partial ambiguity rejections introduce a discounting process between the classical FcM membership functions in order to avoid the memberships to be spread across the classes; (4) the membership functions are more immune to noise and correspond more closely to the notion of compatibility. Preliminary computational experiences on the developed algorithm are encouraging and compared favorably with results from other methods as FcM, FPcM and F(c+1)M (fuzzy c+1-means: clustering with solely distance rejection) algorithms on the same data sets. The differences in the performance can be attributed to the fact that ambiguous patterns are less accounted in for the computing of the centers.