Triadic concept approximation

Abstract

Formal Concept Analysis is a mathematical theory for knowledge representation as well as data analysis and visualization. It provides a mechanism for understanding and mining data through concept lattice construction and exploration. This theory assists in data processing by providing a framework for applying different analysis techniques. One of its potentials lies in its mathematical foundation that enables the generation, ordering, and visualization of knowledge in the form of formal concepts described in a hierarchy known as a concept lattice. However, as the input dataset grows, the search and especially the visualization and exploration of concepts becomes prohibitive. With the extension of the classical FCA approach to Triadic Concept Analysis, this problem becomes even more evident due to the complexity of the inherent structures in triadic concepts and relationships. In this work, we propose an approach to find triadic concepts when a query in the form of a triple is given, allowing the visualization and exploration of a Hasse diagram of triadic concepts.

Introduction

Due to the large volume of data currently being generated by different applications, knowledge exploration and extraction from data have become increasingly important paths for understanding human behaviour and assisting in decision making. In this context, Formal Concept Analysis (FCA) provides a mathematical framework for representation and extraction of knowledge from data. Originally proposed by Wille and Ganter in the 80’s [2], FCA is based on the binary relationship between two sets of elements, called objects and attributes. The great interest in the use of FCA in different fields of applications can be attributed to its formalism and the possibility of identifying formal concepts, which can be represented through concept lattices [19]. The lattices are used to understand the data through a hierarchical visualization of ordered formal concepts, and to extract implications and association rules.

Although successful in many applications in the literature, some situations require an extension to the classical approach by adding a third dimension, in order to have a better characterization and representation of data. FCA was indeed extended by Rudolf Wille and Fritz Lehmann in [11] to describe a ternary relation between object, attribute, and condition sets. This new approach is known as Triadic Concept Analysis (TCA or 3FCA). An example of such scenario is the social resource sharing system represented by a folksonomy [9] which expresses a ternary relation between users, resources, and keywords (tags) used to annotate such resources by users. In [11] the authors proposed a graphical representation of a trilattice as a three-dimensional diagram. Despite coming from FCA, whose graphical representation is intuitive, the three-dimensional complexity of the data in TCA makes it difficult to use these structures. The graphical representation is not intuitive even for small datasets due to the complexity of the triadic settings, resulting in a loss of TCA power of interpretation and visualization. For example, Fig. 1 shows a trilattice from [7], represented as a 3-net that displays all the triconcepts and their respective equivalence classes. As can be seen, the diagram becomes very complex and difficult to perceive even for a small context.

Some studies try to reduce the complexity of visualization and navigation of the triadic lattice [15], [17] by proposing navigation strategies and graphical representations for triadic settings, based on the classical Hasse diagram. In [15], the authors defined T-iPred procedure as an adaptation of the iPred algorithm [1] (used to compute concept lattices) to represent the Hasse diagram of triadic concepts.

The graphical representation proposed in [15] can be used to explore and search for knowledge hidden in triadic contexts. For example, it would be possible to identify concepts in the diagram that are exact or approximate answers to a given query. In case of an approximation, the closest concepts to the query are sought in the Hasse diagram.

In this work we propose a method to uncover patterns from triadic contexts using the T-iPred as a graphical basis to display and retrieve the answer to a user query, using the clarity and expressiveness of this diagram. Patterns are represented by formal concepts that match with or approximate the query expressed by a triple of object, attribute, and condition sets. Our proposed lattice search strategy uses the diagram generated by T-iPred to display the result of the query as the exact answer to the query or the upper and lower covers for concepts that approximate the triple (query). Our research work was mostly implemented in Java and its software architecture can be seen in Fig. 2. Our contributions are in the third module, where we propose three types of queries that can be addressed to the output of the second module using all the triadic dimensions or any combination of them:

• 1. The Data-Peeler algorithm [5] computes the triadic concepts

• 2. The T-iPred algorithm builds the precedence links and displays the Hasse diagram of triadic concepts

• 3. The concept approximation is performed using three types of queries.

This article is organized as follows: Section 2 presents the theoretical basis on Triadic Concept Analysis while Section 3 gives a brief overview about concept approximation and lattice visualization in FCA. Section 4 presents the concept approximation and the three types of queries that can be based on objects, attributes, and condition sets, and any combination of these. Section 5 shows the empirical study and, finally, in Section 6, we present our conclusions and possible future work.