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A Review of Cluster Validation with an Example of Type-2 Fuzzy Application in R

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Book cover Advances in Type-2 Fuzzy Sets and Systems

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 301))

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Abstract

Interval valued type-2 fuzziness can be represented by means of membership functions obtained with upper and lower values of the level of fuzziness. These upper and lower values for the level of fuzziness in FCM algorithm were obtained in our previous studies. A particular application of Interval valued type-2 fuzziness is shown for cluster validity analysis in this chapter. For this purpose, we introduce a brief taxonomy for cluster validity indices to clarify the contribution of our novel approach. To provide reproducibility of our technique, the source code is written in freely available language ‘R’ and can be found on our web site.

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Notes

  1. 1.

    ‘R’ SW can be downloaded from http://cran.r-project.org/ web site.

  2. 2.

    See http://cran.r-project.org/web/packages/cluster/index.html.

  3. 3.

    http://www.stat.ucl.ac.be/ISdidactique/Rhelp/library/e1071/html/cmeans.html.

  4. 4.

    Euclidian distance is defined as square root of \( \sum\limits_{i} {(x_{i} - y_{i} )^{2} } \) and Manhattan is \( \sum\limits_{i} {abs(x_{i} - y_{i} )} . \)

  5. 5.

    Kim et al. [27], used similar intuition and suggested that the optimal number of clusters can be found by minimizing the change in cluster center with respect to the number of clusters.

  6. 6.

    \( l_{\infty } \) norm is defined as \( \lim_{p \to \infty } l_{p} \) where \( l_{p} \) is p-norm. Since p-norm is given as \( \left\| {v_{c}^{{m_{u} }} - v_{c}^{{m_{l} }} } \right\|_{p} = \sum\limits_{i = 1}^{nv} {((v_{c,i}^{{m_{u} }} )^{p} - (v_{c,i}^{{m_{l} }} )^{p} )^{\frac{1}{p}} } \), hence, \( \lim_{p \to \infty } \left\| {v_{c}^{{m_{u} }} - v_{c}^{{m_{l} }} } \right\|_{\infty } = \max_{i = 1}^{nv} \left| {v_{c,i}^{{m_{u} }} - v_{c,i}^{{m_{l} }} } \right| \).

  7. 7.

    Iris data is available in R SW. Wine data can be downloaded from http://archive.ics.uci.edu/ml/machine-learning-databases/wine/ manually or by using R SW as it is shown in “iris and wine data ex.r” script file.

  8. 8.

    http://www.ics.uci.edu/~mlearn/databases.html

  9. 9.

    This software can be downloaded from http://cran.r-project.org/. There are also several good documents related to this statistical computing environment and there are more than 3500 packages prepared already.

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Acknowledgments

This work was partially supported by the Natural Science and Engineering Research Council (NSERC) Grant (RPGIN 7698-05) to University of Toronto. Also, partial support is provided by Hacettepe University and TOBB Economics and Technology University. Their support is greatly appreciated.

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Authors and Affiliations

  1. Department of Economics, Hacettepe University, Beytepe, 06800, Ankara, Turkey

    Ibrahim Ozkan

  2. TOBB ETU, Ankara, Turkey

    I. Burhan Türkşen

  3. Department of Mechanical and Industrial Engineering, University of Toronto, Ontario, M5S3G8, Canada

    I. Burhan Türkşen

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  1. Ibrahim Ozkan
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Correspondence to Ibrahim Ozkan .

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Editors and Affiliations

  1. Ryerson University, Victoria Street 350, Toronto, M5B 2K3, Ontario, Canada

    Alireza Sadeghian

  2. Hughes Aircraft Electrical Engineering B, Department of Electrical Engineering - S, University of Southern California, 3740 McClintock Ave., Los Angeles, 90089-2564, California, USA

    Jerry M. Mendel

  3. Ryerson University, Victoria Street 350, Toronto, M5B 2K3, Ontario, Canada

    Hooman Tahayori

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Ozkan, I., Burhan Türkşen, I. (2013). A Review of Cluster Validation with an Example of Type-2 Fuzzy Application in R. In: Sadeghian, A., Mendel, J., Tahayori, H. (eds) Advances in Type-2 Fuzzy Sets and Systems. Studies in Fuzziness and Soft Computing, vol 301. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6666-6_14

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  • DOI: https://doi.org/10.1007/978-1-4614-6666-6_14

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