Formal Concept Analysis (FCA) transforms a context bigraph, having vertices of type object and attribute, into a lattice digraph, whose vertices and arcs represent formal concepts and their covering relation. The computational complexity of most FCA algorithms is a polynomial function of the numbers of vertices in both the context bigraph and lattice digraph. While the latter quantity is fixed, the former can be decreased by context standardisation, a process which we show is facilitated by the efficient partition refinement algorithm of Spinrad. The Carve algorithm recursively partitions the context bigraph and corresponding lattice digraph by removing universal objects and attributes and their orphans and partitioning the resultant sub-context into its connected components. The associated software prototype uses the resultant tree structure to support coordinated browsing of both. This paper describes an additional, coordinated representation of the context bigraph which makes explicit in its bi-adjacency matrix the pattern of nested sub-contexts discovered by the Carve algorithm. We show that permuting this matrix into doubly-lexical order with the aid of Spinrad’s algorithm groups together into nested rectangles the bigraph edges belonging to these sub-contexts, and facilitates the two key processing steps of the Carve algorithm.
We use a formal concept theoretic approach to computationally validate the classification of all 192 centralising monoids on three-element sets by Machida and Rosenberg. We determine a manageable finite (and row-reduced) context the intents of which are exactly all centralising monoids on {0,1,2} . As an advantage of our method, we are able to compute a list with a witness for each of the 192 monoids, which has not been available to date and allows for an immediate verification of the figure 192 as a lower bound for the number of centralising monoids. We also confirm that the 192 centralising monoids split into 48 conjugacy classes and we provide a list of all conjugacy types including witnesses.
In previous works we proved that fuzzy closure structures such as closure systems, fuzzy closure systems, fuzzy closure operators and fuzzy closure relations are formal concepts of five Galois connections (three antitone and two isotone), together with the commutativity of the fuzzy part of the diagram. In this work we study the commutativity of the crisp diagram formed by three of these Galois connections.
Implications in Formal Concept Analysis (FCA), Horn clauses in Logic, and Functional Dependencies (FDs) in the Relational Database Model, are very important dependency types in their respective fields. Moreover, they have been proved to be equivalent from a syntactical point of view. Then notions and algorithms related to one dependency type in a field can be reused and applied to another dependency type in the other field. One of these notions is that of cover, also known as a basis, i.e., a compact representation of a complete set of implications, FDs, or Horn clauses. Although the notion of cover exists in the three fields, the characterization and the related uses of a cover are different. In this paper, we study and compare, from an FCA perspective, the principles on which rely the most important covers in each field. Finally, we discuss some open questions that are of interest in the three fields, and especially to the FCA community.
The Hasse diagrams of Formal Concept Analysis (FCA) concept lattices have the disadvantages that users need to be trained in reading the diagrams and diagrams of larger lattices tend to be too cluttered to be comprehensible. This paper therefore discusses how to reduce lattices and then represent them with a specific type of Euler diagram instead of Hasse diagrams. A semi-automated process of reducing concept lattices is described and supported by algorithms.
We give a complete list of lattices associated to closure systems generated by four maximal cliques and generalize a known characterization to uniform hypergraphs.
Many applications of Formal Concept Analysis (FCA) and its diverse extensions have been carried out in recent years. Among these extensions, Relational Concept Analysis (RCA) is one approach for addressing knowledge discovery in multi-relational datasets. Applying RCA requires stating a question of interest and encoding the dataset into the input RCA data model, i.e. an Entity-Relationship model with only Boolean attributes in the entity description and unidirectional binary relationships. From the various concrete RCA applications, recurring encoding patterns can be observed, that we aim to capitalize taking software engineering design patterns as a source of inspiration. This capitalization work intends to rationalize and facilitate encoding in future RCA applications. In this paper, we describe an approach for defining such design patterns, and we present two design patterns: “Separate/Gather Views” and “Level Relations”.
A number of extensions have been proposed for Formal Concept Analysis (FCA). Among them, Pattern Structures (PS) bring complex descriptions on objects, as an extension to sets of binary attributes; while Graph-FCA brings n-ary relationships between objects, as well as n-ary concepts. We here introduce a novel extension named Graph-PS that combines the benefits of PS and Graph-FCA. In conceptual terms, Graph-PS can be seen as the meet of PS and Graph-FCA, seen as sub-concepts of FCA. We demonstrate how it can be applied to RDFS graphs, handling hierarchies of classes and properties, and patterns on literals such as numbers and dates.
Fuzzy relation equations (FRE) is a formal theory broadly studied in the literature and applied to decision making, optimization problems, image processing, etc. It is usual that the initial data contains uncertain, imperfect or incomplete information, which can imply, for instance, the existence of inconsistencies. As a consequence, the FRE that arises from the data may be unsolvable. Taking advantage of the relationship between FRE and concept lattices, this paper is focused on three mechanisms for approximating unsolvable FRE. Several properties have been introduced and different distances for determining the best approximation are considered and applied to an example.
Boolean concept lattices are fundamental structures in formal concept analysis, both from a theoretical and an applied point of view. There are multiple ways to generalise them in the triadic concept analysis framework and one of them, the so-called powerset trilattice, has already been proposed by Biedermann in 1998. However, it lacks some interesting properties such as extremality in the number of triconcepts for tricontexts of a given size. In this paper, we discuss another generalisation of Boolean concept lattices that exhibit such properties. We argue that those structures form equivalence classes and should be studied as such, and investigate the minimum number of objects required to produce them.
In this paper we introduce and study description quivers as compact representations of concept lattices and respective ensembles of decision trees. Formally, description quivers are directed multigraphs where vertices represent concept intents and (multiple) edges represent generators of intents. We study some properties of description quivers and shed light on their use for describing state-of-the-art symbolic machine learning models based on decision trees. We also argue that a concept lattice can be considered as a cornerstone in constructing an efficient machine learning model. We show that the proposed description quivers allow us to fuse decision trees just as we can sum linear regressions, while proposing a way to select the most important rules in decision models, just as we can select the most important coefficients in regressions.
There are well-known problems in extremal set theory that can be formulated as enumeration of the maximal independent sets or counting their total number in certain graphs. Here we provide an FCA-based solution on the number of maximal independent sets of the covering graph of a hypercube. In addition, we consider the related maximal independence polynomials for n up to 6, and prove several properties of the polynomials' coefficients and the corresponding concept lattices.
Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic definitions are in flux. Our contribution was triggered by the study of concept lattices for tree classifiers and the scaling methods used there. We extend some basic notions, give precise mathematical definitions for them and introduce the concept of scaling dimension. In addition to a detailed discussion of its properties, including an example, we show theoretical bounds related to the order dimension of concept lattices. We also study special subclasses, such as the ordinal and the interordinal scaling dimensions, and show for them first results and examples.
Decision trees and their ensembles are very popular models of supervised machine learning. In this paper we merge the ideas underlying decision trees, their ensembles and FCA by proposing a new supervised machine learning model which can be constructed in polynomial time and is applicable for both classification and regression problems. Specifically, we first propose a polynomial-time algorithm for constructing a part of the concept lattice that is based on a decision tree. Second, we describe a prediction scheme based on a concept lattice for solving both classification and regression tasks with prediction quality comparable to that of state-of-the-art models.
We equip our algorithm LinCbO with a pruning technique similar to that of LCM. Our experimental evaluation shows that it significantly improves the performance of the algorithm.
Conceptual exploration as provided by Formal Concept Analysis is potentially suited as a tool for developing learning materials for teaching mathematics. But even just a few mathematical notions can lead to complex conceptual structures which may be difficult to be learned and comprehended by students. This paper discusses how the complexity of diagrammatic representations of conceptual structures can potentially be reduced with Semiotic Conceptual Analysis. The notions of “simultaneous polysemy” and “observational advantage” are defined to describe the special kind of relationship between representations and their meanings which frequently occurs with diagrams.
We propose a method of visualizing statistical information in concept lattice diagrams. To this end, we examine the characteristics of support, confidence, and lift, which are parameters used in association analysis. Based on our findings, we develop the notion of cascading line diagrams , a visualization method that combines the properties of additive line diagrams with association analysis. In such diagrams, one can read the size of a concept’s extent from the height of the corresponding node in the diagram and, at the same time, the geometry of the formed quadrangles illustrates whether two attributes are statistically independent or dependent and whether they are negatively or positively correlated. In order to demonstrate this visualization method, we have developed a program generating such diagrams.
. This paper is a short feedback on a collaborative research work by computer scentists and hydroecologists. We have applied Relational Concept Analysis on complex data about running water characteristics (physical, biological and chemical parameters), to answer various questions. Two approaches are presented and discussed: the first one extracts patterns from temporal data, the second one extracts rules from a multi-relational dataset
Formal Concept Analysis (FCA) comes with a range of relevant techniques for knowledge analysis, such as conceptual structures or implications. The Duquenne-Guigues basis of implications provides a cardinality minimal set of non-redundant implications. The concern of a domain expert is to discover new knowledge within this implication set. The objective of this prospective paper is to collect and discuss the different patterns of implications extracted from a dataset on plants used in medical care or consumed as food. We identify 16 patterns combining 3 types of knowledge elements (KE). The patterns highlight redundant KEs, or KEs of little interest, in particular, those corresponding to plant taxonomy, as it is familiar knowledge for the experts. Removing these KEs from the implications would make them tacit. We suggest a postprocess for cleaning up the implications before reporting them to the experts. In addition, we discuss the different patterns and how an implication classification based on patterns could help the experts.
. In this application note we revisit previous work on the exploratory analysis of Multilabel Classification (MLC) tasks in Machine Learning. We combine Information Theory and Formal Concept Analysis (FCA) to formalize the intuition that inference of classifiers in MLC tasks can only proceed when the training and testing data concerning the labels define the same Concept Lattice. We instantiate our procedure on the emotions dataset, but the procedure is independent of the dataset being explored. An R language interactive notebook carrying out the procedure is available upon request.
