We present a novel approach to compute formal concepts of formal context. In terms of operations with Boolean matrices, the presented algorithm computes all maximal rectangles of the input Boolean matrix which are full of 1s. The algorithm combines b

We address important issues of dimensionality reduction of transactional data sets where the input data consists of lists of transactions, each of them being a finite set of items. The reduction consists in finding a small set of new items, so-called factor-items, which is considerably smaller than the original set of items while comprising full or nearly full information about the original items. Using this type of reduction, the original data set can be represented by a smaller transactional data set using factor-items instead of the original items, thus reducing its dimensionality. The procedure utilized in this paper is based on approximate Boolean matrix decomposition. In this paper, we focus on the role of frequent closed item sets that can be used to determine factor-items. We present the factorization problem, its reduction to Boolean matrix decompositions, experiments with publicly available data sets, and an algorithm for computing decompositions.

We present two input data preprocessing methods for machine learning (ML). The first one consists in extending the set of attributes describing objects in input data table by new attributes and the second one consists in replacing the attributes by new attributes. The methods utilize formal concept analysis (FCA) and boolean factor analysis, recently described by FCA, in that the new attributes are defined by so-called factor concepts computed from input data table. The methods are demonstrated on decision tree induction. The experimental evaluation and comparison of performance of decision trees induced from original and preprocessed input data is performed with standard decision tree induction algorithms ID3 and C4.5 on several benchmark datasets.

We present a fast bottom-up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees, along with the partial order on the set of all fixpoints. Fuzzy closure operators appear in several areas of fuzzy logic and its applications, including formal concept analysis (FCA) that we use as a reference area of application in this paper. Several problems in FCA, such as computing all formal concepts from data with graded attributes or computing non-redundant bases of all attribute dependencies, can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data. The development of a general algorithm that is applicable, in particular, to these problems is the ultimate purpose of this paper. We present the algorithm, its theoretical foundations, and experimental evaluation.

Concept lattices are systems of conceptual clusters, called formal concepts, which are partially ordered by the subconcept/superconcept relationship. Concept lattices are basic structures used in formal concept analysis. In general, a concept lattice may contain overlapping clusters and need not be a tree. On the other hand, tree-like classification schemes are appealing and are produced by several clustering methods. In this paper, we present necessary and sufficient conditions on input data for the output concept lattice to form a tree after one removes its least element. We present these conditions for input data with yes/no attributes as well as for input data with fuzzy attributes. In addition, we show how Lindig's algorithm for computing concept lattices gets simplified when applied to input data for which the associated concept lattice is a tree after removing the least element. The paper also contains illustrative examples.