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An Efficient Approximation of Concept Stability Using Low-Discrepancy Sampling

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Graph-Based Representation and Reasoning (ICCS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10872))

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Abstract

One key challenge in Formal Concept Analysis is the scalable and accurate computation of stability index as a means to identify relevant formal concepts. Unfortunately, most exact methods for computing stability have an algorithmic complexity that could be exponential w.r.t. the context size. While randomized approximate algorithms, such as Monte Carlo Sampling (MCS), can be good solutions in some situations, they frequently lead to the slow convergence problem with an inaccurate estimation of stability. In this paper, we introduce a new approximation method to estimate the stability using the low-discrepancy sampling (LDS) approach. To improve the convergence rate, LDS uses quasi-random sequence to distribute the sample points evenly across the power set of the concept intent (or extent). This helps avoid the clumping of samples and let all the areas of the sample space be duly represented. Our experiments on several formal contexts show that LDS can achieve faster convergence rate and better accuracy than MCS.

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Notes

  1. 1.

    The convergence rate quantifies how quickly the sampling error decreases with an increase in the number of samples.

  2. 2.

    Note that since both \(\sigma _{\text {in}}(c)\) and \(\sigma _{\text {ex}}(c)\) provide dual measurements, we generally use \(\sigma (c)\) throughout the rest of the paper.

  3. 3.

    Publicly available: http://fimi.ua.ac.be/data/.

  4. 4.

    Publicly available: https://pypi.python.org/pypi/concepts.

  5. 5.

    Publicly available: http://people.sc.fsu.edu/~jburkardt/py_src/sobol/sobol.html.

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Acknowledgment

We acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Authors and Affiliations

  1. Département d’informatique et d’ingénierie, Université du Québec en Outaouais, 101, rue St-Jean Bosco, Gatineau, Québec, J8X 3X7, Canada

    Mohamed-Hamza Ibrahim & Rokia Missaoui

Authors
  1. Mohamed-Hamza Ibrahim
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  2. Rokia Missaoui
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Correspondence to Mohamed-Hamza Ibrahim .

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Editors and Affiliations

  1. Edinburgh Napier University, Edinburgh, United Kingdom

    Peter Chapman

  2. Philipps University of Marburg, Marburg, Germany

    Dominik Endres

  3. University of Paris-Sud, Orsay, France

    Nathalie Pernelle

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Ibrahim, MH., Missaoui, R. (2018). An Efficient Approximation of Concept Stability Using Low-Discrepancy Sampling. In: Chapman, P., Endres, D., Pernelle, N. (eds) Graph-Based Representation and Reasoning. ICCS 2018. Lecture Notes in Computer Science(), vol 10872. Springer, Cham. https://doi.org/10.1007/978-3-319-91379-7_3

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  • DOI: https://doi.org/10.1007/978-3-319-91379-7_3

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