Fuzzy and rough formal concept analysis: a survey
Formal Concept Analysis (FCA) is a mathematical technique that has been extensively applied to Boolean data in knowledge discovery, information retrieval, web mining, etc. applications. During the past years, the research on extending FCA theory to cope with imprecise and incomplete information made significant progress. In this paper, we give a systematic overview of the more than 120 papers published between 2003 and 2011 on FCA with fuzzy attributes and rough FCA. We applied traditional FCA as a text-mining instrument to 1072 papers mentioning FCA in the abstract. These papers were formatted in pdf files and using a thesaurus with terms referring to research topics, we transformed them into concept lattices. These lattices were used to analyze and explore the most prominent research topics within the FCA with fuzzy attributes and rough FCA research communities. FCA turned out to be an ideal metatechnique for representing large volumes of unstructured texts.
A novel approach to triclustering of a three-way binary data is proposed. Tricluster is defined in terms of Triadic Formal Concept Analysis as a dense triset of a binary relation Y , describing relationship between objects, attributes and conditions. This definition is a relaxation of a triconcept notion and makes it possible to find all triclusters and triconcepts contained in triclusters of large datasets. This approach generalizes the similar study of concept-based biclustering.
This paper addresses the important problem of efficiently mining numerical data with formal concept analysis (FCA). Classically, the only way to apply FCA is to binarize the data, thanks to a so-called scaling procedure. This may either involve loss of information, or produce large and dense binary data known as hard to process. In the context of gene expression data analysis, we propose and compare two FCA-based methods for mining numerical data and we show that they are equivalent. The first one relies on a particular scaling, encoding all possible intervals of attribute values, and uses standard FCA techniques. The second one relies on pattern structures without a priori transformation, and is shown to be more computationally efficient and to provide more readable results. Experiments with real-world gene expression data are discussed and give a practical basis for the comparison and evaluation of the methods.
Concept discovery is a Knowledge Discovery in Databases (KDD) research field that uses human-centered techniques such as Formal Concept Analysis (FCA), Biclustering, Triclustering, Conceptual Graphs etc. for gaining insight into the underlying conceptual structure of the data. Traditional machine learning techniques are mainly focusing on structured data whereas most data available resides in unstructured, often textual, form. Compared to traditional data mining techniques, human-centered instruments actively engage the domain expert in the discovery process. This volume contains the contributions to CDUD 2011, the International Workshop on Concept Discovery in Unstructured Data (CDUD) held in Moscow. The main goal of this workshop was to provide a forum for researchers and developers of data mining instruments working on issues with analyzing unstructured data. We are proud that we could welcome 13 valuable contributions to this volume. The majority of the accepted papers described innovative research on data discovery in unstructured texts. Authors worked on issues such as transforming unstructured into structured information by amongst others extracting keywords and opinion words from texts with Natural Language Processing methods. Multiple authors who participated in the workshop used methods from the conceptual structures field including Formal Concept Analysis and Conceptual Graphs. Applications include but are not limited to text mining police reports, sociological definitions, movie reviews, etc.
The definition of a phoneme as a fuzzy set of minimal speech units from the model database is proposed. On the basis of this definition and the Kullback-Leibler minimum information discrimination principle the novel phoneme recognition algorithm has been developed as an enhancement of the phonetic decoding method. The experimental results in the problems of isolated vowels recognition and word recognition in Russian are presented. It is shown that the proposed method is characterized by the increase of recognition accuracy and reliability in comparison with the phonetic decoding method
Concept Relation Discovery and Innovation Enabling Technology (CORDIET), is a toolbox for gaining new knowledge from unstructured text data. At the core of CORDIET is the C-K theory which captures the essential elements of innovation. The tool uses Formal Concept Analysis (FCA), Emergent Self Organizing Maps (ESOM) and Hidden Markov Models (HMM) as main artifacts in the analysis process. The user can define temporal, text mining and compound attributes. The text mining attributes are used to analyze the unstructured text in documents, the temporal attributes use these document’s timestamps for analysis. The compound attributes are XML rules based on text mining and temporal attributes. The user can cluster objects with object-cluster rules and can chop the data in pieces with segmentation rules. The artifacts are optimized for efficient data analysis; object labels in the FCA lattice and ESOM map contain an URL on which the user can click to open the selected document.
Formal Concept Analysis (FCA) is an unsupervised clustering technique and many scientific papers are devoted to applying FCA in Information Retrieval (IR) research. We collected 103 papers published between 2003-2009 which mention FCA and information retrieval in the abstract, title or keywords. Using a prototype of our FCA-based toolset CORDIET, we converted the pdf-files containing the papers to plain text, indexed them with Lucene using a thesaurus containing terms related to FCA research and then created the concept lattice shown in this paper. We visualized, analyzed and explored the literature with concept lattices and discovered multiple interesting research streams in IR of which we give an extensive overview. The core contributions of this paper are the innovative application of FCA to the text mining of scientific papers and the survey of the FCA-based IR research.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.