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Distance-Based Fuzzy-Rough Sets and Their Application to the Classification Problem

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Rough Sets (IJCRS 2024)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14839))

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Abstract

We propose distance-based fuzzy-rough sets (DBFR) that rely on distance functions for the granulation of the underlying universe. The classification problem is investigated from a fuzzy perspective and cast as a concept approximation problem. DBFRs are employed to facilitate the approximation process. The geometrical nature of approximation emerging due to the use of distance functions is investigated. Naive classifiers based on DBFRs are proposed and experimentally evaluated on benchmark datasets.

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Notes

  1. 1.

    Binary operations \(E_1\) and \(E_2\) on [0, 1] are called dual wrt negator \(\mathcal {N}\) if the duality identities \(\mathcal {N}(E_1(x,y)) = E_2(\mathcal {N}(x),\mathcal {N}(y))\) and \(\mathcal {N}(E_2(x,y)) = E_1(\mathcal {N}(x),\mathcal {N}(y))\) hold for all \(x,y \in [0,1]\). If \(\mathcal {N}\) is involutive, then \(E_1(x,y) := \mathcal {N}(E_2(\mathcal {N}(x),\mathcal {N}(y)))\) is called \(\mathcal {N}\)-dual of \(E_2\) and \(E_2(x,y) := \mathcal {N}(E_1(\mathcal {N}(x),\mathcal {N}(y)))\) and is called \(\mathcal {N}\)-dual of \(E_1\).

  2. 2.

    The actual theorem is much more general, but we only specify the part relevant to our work.

  3. 3.

    The points in the Fig. 2 are actually the feature vectors of the objects in the toy dataset. For convenience, we won’t explicate the distinction from here on.

  4. 4.

    By “y near class \(D_1\)”, it is meant that the feature vector of y, i.e., \(\phi (y)\) is near feature vector of objects in class \(D_1\), viz. blue points. As the feature space is equipped with a pseudo-metric, talking both “nearness” makes sense. For convenience, we won’t emphasize this technicality from here on.

  5. 5.

    \(\text {dist}(y,T) := \min _{x \in T} d(\phi (x), \phi (y))\).

  6. 6.

    Assuming that the pseudo-metric d is finite-valued which is typically true.

  7. 7.

    The decision boundary can be more finely traced by computing the label of objects with feature vectors separated by a smaller step-size, say, 0.01.

  8. 8.

    All datasets except “Diabetes” are taken from [1] and “Diabetes” dataset, originally from the National Institute of Diabetes and Digestive and Kidney Diseases, is taken from UCI Machine Learning’s Kaggle page.

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Acknowledgments

Authors would like to thank Dr. Balasubramanian Jayaram for drawing our attention to De Baets & Mesiar’s theorem.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India

    Amrit Kumar & Niladri Chatterjee

  2. School of Information Technology, School of Artificial Intelligence, Indian Institute of Technology Delhi, New Delhi, India

    Niladri Chatterjee

Authors
  1. Amrit Kumar
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  2. Niladri Chatterjee
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Correspondence to Amrit Kumar .

Editor information

Editors and Affiliations

  1. Saint Mary's University, Halifax, NS, Canada

    Mengjun Hu

  2. Ghent University, Ghent, Belgium

    Chris Cornelis

  3. California State University, San Bernardino, CA, USA

    Yan Zhang

  4. Saint Mary's University, Halifax, NS, Canada

    Pawan Lingras

  5. University of Warsaw, Warsaw, Poland

    Dominik Ślęzak

  6. University of Regina, Regina, SK, Canada

    JingTao Yao

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Kumar, A., Chatterjee, N. (2024). Distance-Based Fuzzy-Rough Sets and Their Application to the Classification Problem. In: Hu, M., Cornelis, C., Zhang, Y., Lingras, P., Ślęzak, D., Yao, J. (eds) Rough Sets. IJCRS 2024. Lecture Notes in Computer Science(), vol 14839. Springer, Cham. https://doi.org/10.1007/978-3-031-65665-1_9

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  • DOI: https://doi.org/10.1007/978-3-031-65665-1_9

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  • Online ISBN: 978-3-031-65665-1

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