Optimal triangular decompositions of matrices with entries from residuated lattices

AbstractWe describe optimal decompositions of an n×m matrix I into a triangular product I=A◁B of an n×k matrix A and a k×m matrix B. We assume that the matrix entries are elements of a residuated lattice, which leaves binary matrices or matrices which contain numbers from the unit interval [0,1] as special cases. The entries of I, A, and B represent grades to which objects have attributes, factors apply to objects, and attributes are particular manifestations of factors, respectively. This way, the decomposition provides a model for factor analysis of graded data. We prove that fixpoints of particular operators associated with I, which are studied in formal concept analysis, are optimal factors for decomposition of I in that they provide us with decompositions I=A◁B with the smallest number k of factors possible. Moreover, we describe transformations between the m-dimensional space of original attributes and the k-dimensional space of factors. We provide illustrative examples and remarks on the problem of computing the optimal decompositions. Even though we present the results for matrices, i.e. for relations between finite sets in terms of relations, the arguments behind are valid for relations between infinite sets as well

Adding Background Knowledge to Formal Concept Analysis via Attribute Dependency Formulas

ABSTRACT We present a way to add user's background knowledge to formal concept analysis. The type of background knowledge we deal with relates to relative importance of attributes in the input data. We introduce AD-formulas which represent this type of background knowledge. The background knowledge serves as a constraint. The main aim is to make extraction of clusters from the input data more focused by taking into account the background knowledge. Particularly, only clusters which are compatible with the background knowledge are extracted from data. As a result, the number of extracted clusters becomes smaller, leaving out non-interesting clusters. We present illustrative examples and results on entailment of background knowledge such as efficient testing of entailment and a complete systems of deduction rules

Triadic fuzzy Galois connections as ordinary connections

Abstract-The paper presents results on representation of the basic structures related to ternary fuzzy relations by the structures related to ordinary ternary relations, such as Galois connections, closure operators, and trilattices (structures of maximal Cartesian subrelations). These structures appear as the fundamental structures in relational data analysis such as formal concept analysis or association rules. We prove several representation theorems that allow us to automatically transfer some of the known results from the ordinary case to fuzzy case. The transfer is demonstrated by examples. I. INTRODUCTION Relations play a fundamental role in mathematics, computer science, and their applications. Many results about ordinary relations have been generalized to the setting of fuzzy relations in the past. There has always been a fundamental question of how the various fuzzifications are related to the ordinary notions and results. Needless to say, this question is important both from a practical and theoretical point of view and is treated to some extent in textbooks, see e.g. In this paper we deal with basic structures associated to ternary relations that appear as fundamental ones in the methods of relational data analysis, namely the closure-like structures such as Galois connections, closure operators, structures of their fixpoints and the like. Such structures appear e.g. in formal concept analysis The most common way of looking at the relationship between ordinary notions and their fuzzy counterparts is in terms of a-cuts of fuzzy relations (see e.g. [15]) but there are additional possible views at the question as well. One of them, utilized in this paper, is provided in [3, Section 3.1.2]. Our paper is organized as follows. We first provide preliminaries in Section II. In Section III, we introduce the Galoi

Grouping fuzzy sets by similarity

a b s t r a c t The paper presents results on factorization of systems of fuzzy sets. The factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a. An obstacle to such factorization, well known in fuzzy set theory, is the fact that ''being similar at least to degree a" is not an equivalence relation because, in general, it is not transitive. As a result, ordinary factorization using equivalence classes cannot be used. This obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a. We show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice. This lattice plays the role of a factor structure for the original system of fuzzy sets. Particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices

Bivalent and other solutions of fuzzy relational equations via linguistic hedges

Abstract We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints. © 2011 Elsevier B.V. All rights reserved. Motivation Fuzzy relational equations play an important role in fuzzy set theory and its applications, see and every fuzzy relation U satisfying the first or the second equality is called a solution of the respective fuzzy relational equation. The nature of the unknown relationship represented by U may impose additional constraints on U. For example, one may require that U be a bivalent (crisp) relation (see Section 3 for an illustrative example). More generally

Omecamtiv mecarbil in chronic heart failure with reduced ejection fraction, GALACTIC‐HF: baseline characteristics and comparison with contemporary clinical trials

Aims: The safety and efficacy of the novel selective cardiac myosin activator, omecamtiv mecarbil, in patients with heart failure with reduced ejection fraction (HFrEF) is tested in the Global Approach to Lowering Adverse Cardiac outcomes Through Improving Contractility in Heart Failure (GALACTIC‐HF) trial. Here we describe the baseline characteristics of participants in GALACTIC‐HF and how these compare with other contemporary trials. Methods and Results: Adults with established HFrEF, New York Heart Association functional class (NYHA) ≥ II, EF ≤35%, elevated natriuretic peptides and either current hospitalization for HF or history of hospitalization/ emergency department visit for HF within a year were randomized to either placebo or omecamtiv mecarbil (pharmacokinetic‐guided dosing: 25, 37.5 or 50 mg bid). 8256 patients [male (79%), non‐white (22%), mean age 65 years] were enrolled with a mean EF 27%, ischemic etiology in 54%, NYHA II 53% and III/IV 47%, and median NT‐proBNP 1971 pg/mL. HF therapies at baseline were among the most effectively employed in contemporary HF trials. GALACTIC‐HF randomized patients representative of recent HF registries and trials with substantial numbers of patients also having characteristics understudied in previous trials including more from North America (n = 1386), enrolled as inpatients (n = 2084), systolic blood pressure < 100 mmHg (n = 1127), estimated glomerular filtration rate < 30 mL/min/1.73 m2 (n = 528), and treated with sacubitril‐valsartan at baseline (n = 1594). Conclusions: GALACTIC‐HF enrolled a well‐treated, high‐risk population from both inpatient and outpatient settings, which will provide a definitive evaluation of the efficacy and safety of this novel therapy, as well as informing its potential future implementation