Petko Valtchev

Formal Concept Analysis for Knowledge Discovery and Data Mining: The New Challenges

Data mining (DM) is the extraction of regularities from raw data, which are further transformed within the wider process of knowledge discovery in databases (KDD) into non-trivial facts intended to support decision making. Formal concept analysis (FCA) offers an appropriate framework for KDD, whereby our focus here is on its potential for DM support. A variety of mining methods powered by FCA have been published and the figures grow steadily, especially in the association rule mining (ARM) field. However, an analysis of current ARM practices suggests the impact of FCA has not reached its limits, i.e., appropriate FCA-based techniques could successfully apply in a larger set of situations. As a first step in the projected FCA expansion, we discuss the existing ARM methods, provide a set of guidelines for the design of novel ones, and list some open algorithmic issues on the FCA side. As an illustration, we propose two on-line methods computing the minimal generators of a closure system.

Towards Rare Itemset Mining

We describe here a general approach for rare itemset mining. While mining literature has been almost exclusively focused on frequent itemsets, in many practical situations rare ones are of higher interest (e.g., in medical databases, rare combinations of symptoms might provide useful insights for the physicians). Based on an examination of the relevant substructures of the mining space, our approach splits the rare itemset mining task into two steps, i.e., frequent itemset part traversal and rare itemset listing. We propose two algorithms for step one, a naive and an optimized one, respectively, and another algorithm for step two. We also provide some empirical evidence about the performance gains due to the optimized traversal.

Relational concept discovery in structured datasets

Relational datasets, i.e., datasets in which individuals are described both by their own features and by their relations to other individuals, arise from various sources such as databases, both relational and object-oriented, knowledge bases, or software models, e.g., UML class diagrams. When processing such complex datasets, it is of prime importance for an analysis tool to hold as much as possible to the initial format so that the semantics is preserved and the interpretation of the final results eased. Therefore, several attempts have been made to introduce relations into the formal concept analysis field which otherwise generated a large number of knowledge discovery methods and tools. However, the proposed approaches invariably look at relations as an intra-concept construct, typically relating two parts of the concept description, and therefore can only lead to the discovery of coarse-grained patterns. As an approach towards the discovery of finer-grain relational concepts, we propose to enhance the classical (object × attribute) data representations with a new dimension that is made out of inter-object links (e.g., spouse, friend, manager-of, etc.). Consequently, the discovered concepts are linked by relations which, like associations in conceptual data models such as the entity-relation diagrams, abstract from existing links between concept instances. The borders for the application of the relational mining task are provided by what we call a relational context family, a set of binary data tables representing individuals of various sorts (e.g., human beings, companies, vehicles, etc.) related by additional binary relations. As we impose no restrictions on the relations in the dataset, a major challenge is the processing of relational loops among data items. We present a method for constructing concepts on top of circular descriptions which is based on an iterative approximation of the final solution. The underlying construction methods are illustrated through their application to the restructuring of class hierarchies in object-oriented software engineering, which are described in UML.

A partition-based approach towards constructing Galois (concept) lattices

Galois lattices and formal concept analysis of binary relations have proved useful in the resolution of many problems of theoretical or practical interest. Recent studies of practical applications in data mining and software engineering have put the emphasis on the need for both efficient and flexible algorithms to construct the lattice. Our paper presents a novel approach for lattice construction based on the apposition of binary relation fragments. We extend the existing theory to a complete characterization of the global Galois (concept) lattice as a substructure of the direct product of the lattices related to fragments. The structural properties underlie a procedure for extracting the global lattice from the direct product, which is the basis for a full-scale lattice construction algorithm implementing a divide-and-conquer strategy. The paper provides a complexity analysis of the algorithm together with some results about its practical performance and describes a class of binary relations for which the algorithm outperforms the most efficient lattice-constructing methods.

Generating frequent itemsets incrementally: two novel approaches based on Galois lattice theory

Galois (concept) lattice theory has been successfully applied in data mining for the resolution of the association rule problem. In particular, structural results about lattices have been used in the design of efficient procedures for mining the frequent patterns (itemsets) in transaction databases. Since such databases are often dynamic, we propose a detailed study of the incremental aspects in lattice construction to support effective procedures for incremental mining of frequent closed itemsets (FCIs). Based on a set of descriptive results about lattice substructures involved in incremental updates, the paper presents a novel algorithm for lattice construction that explores only limited parts of a lattice for updating. Two new methods for incremental FCI mining are studied: the first inherits its extensive search strategy from a classical lattice method, whereas the second applies the new lattice construction strategy to the itemset mining context. Unlike batch techniques based on FCIs, both methods avoid rebuilding the FCI family from scratch whenever new transactions are added to the database and/or when the minimal support is changed.

Relational concept analysis: mining concept lattices from multi-relational data

The processing of complex data is admittedly among the major concerns of knowledge discovery from data (kdd). Indeed, a major part of the data worth analyzing is stored in relational databases and, since recently, on the Web of Data. This clearly underscores the need for Entity-Relationship and rdf compliant data mining (dm) tools. We are studying an approach to the underlying multi-relational data mining (mrdm) problem, which relies on formal concept analysis (fca) as a framework for clustering and classification. Our relational concept analysis (rca) extends fca to the processing of multi-relational datasets, i.e., with multiple sorts of individuals, each provided with its own set of attributes, and relationships among those. Given such a dataset, rca constructs a set of concept lattices, one per object sort, through an iterative analysis process that is bound towards a fixed-point. In doing that, it abstracts the links between objects into attributes akin to role restrictions from description logics (dls). We address here key aspects of the iterative calculation such as evolution in data description along the iterations and process termination. We describe implementations of rca and list applications to problems from software and knowledge engineering.

A proposal for combining formal concept analysis and description logics for mining relational data

Recent advances in data and knowledge engineering have emphasized the need for formal concept analysis (FCA) tools taking into account structured data. There are a few adaptations of the classical FCA methodology for handling contexts holding on complex data formats, e.g. graph-based or relational data. In this paper, relational concept analysis (RCA) is proposed, as an adaptation of FCA for analyzing objects described both by binary and relational attributes. The RCA process takes as input a collection of contexts and of inter-context relations, and yields a set of lattices, one per context, whose concepts are linked by relations. Moreover, a way of representing the concepts and relations extracted with RCA is proposed in the framework of a description logic. The RCA process has been implemented within the Galicia platform, offering new and efficient tools for knowledge and software engineering.

Building Concept (Galois) Lattices from Parts: Generalizing the Incremental Methods

Formal concept analysis is increasingly used as a data mining technique, whence the need of efficient algorithms for handling large sets of volatile data. Recently, we designed a general framework for constructing concept (Galois) lattices from fragmented and/or evolving data based on a lattice assembly operation. In this paper, the framework is adapted to the maintenance of concept lattices upon the insertion of a set of objects into the context, a problem which generalizes the insertion of individual objects considered by the existing incremental methods. The paper provides a set of structural results for the case of single object insertions which underlie a new incremental algorithm. Our method is shown to improve a key flaw of the major incremental technique.

Refactorings of design defects using relational concept analysis

Software engineers often need to identify and correct design defects, i.e., recurring design problems that hinder development and maintenance by making programs harder to comprehend and/or evolve. While detection of design defects is an actively researched area, their correction- mainly a manual and time-consuming activity- is yet to be extensively investigated for automation. In this paper, we propose an automated approach for suggesting defect-correcting refactorings using relational concept analysis (RCA). The added value of rca consists in exploiting the links between formal objects which abound in a software re-engineering context. We validated our approach on instances of the Blob design defect taken from four different open-source programs.

Formal concept analysis-based class hierarchy design in object-oriented software development

The class hierarchy is an important aspect of object-oriented software development. Design and maintenance of such a hierarchy is a difficult task that is often accomplished without any clear guidance or tool support. Formal concept analysis provides a natural theoretical framework for this problem because it can guarantee maximal factorization while preserving specialization relationships. The framework can be useful for several software development scenarios within the class hierarchy life-cycle such as design from scratch using a set of class specifications, or a set of object examples, refactoring/reengineering from existing object code or from the observation of the actual use of the classes in applications and hierarchy evolution by incrementally adding new classes. The framework can take into account different levels of specification details and suggests a number of well-defined alternative designs. These alternatives can be viewed as normal forms for class hierarchies where each normal form addresses particular design goals. An overview of work in the area is presented by highlighting the formal concept analysis notions that are involved. One particularly difficult problem arises when taking associations between classes into account. Basic scaling has to be extended because the scales used for building the concept lattice are dependent on it. An approach is needed to treat this circularity in a well-defined manner. Possible solutions are discussed.