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On stability of a formal concept

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Abstract

In this paper we define and analyze stability of a formal concept. A stability index is based on the idea of a dependency in a dataset that can be reconstructed from different parts of the dataset. This idea, underlying various methods of estimating scientific hypotheses, is used here for estimating concept-based hypotheses. Combinatorial properties of stability indices, algorithmic comlpexity of their computation, as well as their dynamics with arrival of new examples are studied.

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References

  1. Carnap, R.: The Logical Foundations of Probability. University of Chicago Press, Chicago, IL (1962)

    Google Scholar 

  2. Efron, B.: The jackknife, the bootstrap and other resampling plans. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38. SIAM, Philadelphia (1982)

    Google Scholar 

  3. Finn, V.K.: On machine-oriented formalization of plausible reasoning in the style of F. Bacon–J.S. Mill. Semiot. Inform. 20, 35–101 (in Russian) (1983)

    MATH  Google Scholar 

  4. Finn, V.K.: Plausible reasoning in systems of JSM-type. Itogi Nauki Teh., Ser. Inform. 15, 54–101 (in Russian) (1991)

    Google Scholar 

  5. Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer (1999)

  6. Ganter, B., Kuznetsov, S.O.: Formalizing hypotheses with concepts. In: Mineau, G., Ganter, B. (eds.) Proc. 8th International Conference on Conceptual Structures, ICCS’00. Lecture Notes in Artificial Intelligence, vol. 1867, pp. 342–356 (2000)

  7. Ganter, B., Kuznetsov, S.O.: Pattern structures and their projections. In: Stumme, G., Delugach, H. (eds.) Proc. 9th Int. Conf. on Conceptual Structures, ICCS’01. Lecture Notes in Artificial Intelligence, vol. 2120, pp. 129-142 (2001)

  8. Kuznetsov, S.O.: Stability as an estimate of the degree of substantiation of hypotheses derived on the basis of operational similarity. Nauchn. Tekh. Inf., Ser. 2, no. 12, pp. 21–29 (in Russian) (1990)

  9. Kuznetsov, S.O.: Mathematical aspects of concept analysis. J. Math. Sci. 80(2), 1654–1698 (1996)

    Article  MATH  Google Scholar 

  10. Kuznetsov, S.O.: Learning of simple conceptual graphs from positive and negative examples. In: Zytkow, J., Rauch, J. (eds.) Proc. Principles of Data Mining and Knowledge Discovery, Third European Conference, PKDD’99. Lecture Notes in Artificial Intelligence, vol. 1704, pp. 384-392 (1999)

  11. Mitchell, T.: Machine Learning. McGraw-Hill, New York (1997)

    MATH  Google Scholar 

  12. Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Efficient minining of association rules using closed itemset lattices. J. Inf. Syst. 24, 25–46 (1999)

    Article  Google Scholar 

  13. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)

    Article  MATH  Google Scholar 

  14. Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, State University Higher School of Economics, Moscow, Russia

    Sergei O. Kuznetsov

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  1. Sergei O. Kuznetsov
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Correspondence to Sergei O. Kuznetsov.

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This work was supported by the project COMO (Concepts and Models) of the Deutsche Forschungsgemeinschaft (DFG) and Russian Foundation for Basic Research (RFBR).

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Kuznetsov, S.O. On stability of a formal concept. Ann Math Artif Intell 49, 101–115 (2007). https://doi.org/10.1007/s10472-007-9053-6

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  • DOI: https://doi.org/10.1007/s10472-007-9053-6

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