A Proposal of Visualization Method for Interpretable Fuzzy Model on Fusion Axes

Kosuke Yamamoto; Tomohiro Yoshikawa; Takeshi Furuhashi

doi:10.20965/jaciii.2006.p0121

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JACIII Vol.10 No.1 pp. 121-131

doi: 10.20965/jaciii.2006.p0121

(2006)

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A Proposal of Visualization Method for Interpretable Fuzzy Model on Fusion Axes

Kosuke Yamamoto, Tomohiro Yoshikawa, and Takeshi Furuhashi

Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

Received:

July 1, 2005

Accepted:

September 13, 2005

Published:

January 20, 2006

Keywords:

interpretability, visualization, fusion axis, fuzzy modeling

Abstract

Interpretability of fuzzy models has become one of the major topics in the field of fuzzy modeling. Visualization that makes input-output relationships interpretable is effective in extracting useful knowledge from unknown data. This paper presents visualization method that considers the visibility of fuzzy models. This method identifies clusters that have different statistical features, and projects the data to the “fusion axes”, which are linear combinations of the multiple input variables, considering the distribution of each cluster in the projected space. This paper applies the proposed method to artificial data and also to collected data from the mobile robot, and shows that the proposed method can extract useful knowledge from the obtained visible and interpretable models.

Cite this article as:

K. Yamamoto, T. Yoshikawa, and T. Furuhashi, “A Proposal of Visualization Method for Interpretable Fuzzy Model on Fusion Axes,” J. Adv. Comput. Intell. Intell. Inform., Vol.10 No.1, pp. 121-131, 2006.

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References

[1] J. Casillas, O. Cordón, F. Herrera, and L. Magdalena, “Interpretability Issues in Fuzzy Modeling,” Springer, 2003.
[2] S. Horikawa, T. Furuhashi, and Y. Uchikawa, “On Fuzzy Modeling Using Fuzzy Neural Networks with the Back-Propagation Algorithm,” IEEE Trans. on Neural Networks, Vol.3, No.5, pp. 801-806, 1992.
[3] M. Sugeno, and T. Yasukawa, “A Fuzzy Logic based Approach to Qualitative Modeling,” IEEE Trans. on Fuzzy System, Vol.1, No.1, pp. 7-31, 1993.
[4] D. Tikk, G. Biró, T. D. Gedeon, L. T. Kóczy, and J. D. Yang, “Improvements and Critique on Sugeno’s and Yasukawa’s Qualitative Modeling,” IEEE Trans. on Fuzzy Systems, Vol.10, No.5, pp. 596-606, 2002.
[5] M. Bikdash, “A highly interpretable form of Sugeno inference systems,” IEEE Trans. on Fuzzy Systems, Vol.7 No.6, pp. 686-696, 1999.
[6] A. K. Jain, R. Duin, and J. Mao, “Statistical Pattern Recognition: A Review,” IEEE Trans. on PAMI, Vol.22, No.1, pp. 4-37, 2000.
[7] K. Yamamoto, T. Yoshikawa, and T. Furuhashi, “A Proposal of Fuzzy Modeling on Fusion Axes Considering the Data Structure,” FUZZ-IEEE 2003, pp. 348-353, 2003.
[8] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.
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[10] G. E. P. Box, and G. M. Jenkins, “Time Series Analysis,” Forecasting and Control. San Francisco, CA, Holden Day, 1970.
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[12] D. Suizu, and S. Miyamoto, “Fuzzy c-Means Clustering using Mapping into High Dimensional Spaces,” Proc. of the 18th Fuzzy System Symposium, pp. 123-126, 2002 (In Japanese).
[13] E. Kim, M. Park, S. Kim, and M. Park, “A Transformed Input-Domain Approach to Fuzzy Modeling,” IEEE Trans. on Fuzzy Systems, Vol.6, No.4, pp. 596-605, 1998.
[14] R. J. Hathaway, and J. C. Bezdek, “Switching Regression Models and Fuzzy Clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, No.3, 1993.
[15] Y. Yam, P. Baranyi, and C. T. Yang, “Reduction of Fuzzy Rule Base Via Singular Value Decomposition,” IEEE Trans. of Fuzzy System, Vol.7, No.2, pp. 120-132, 1999.
[16] A. Hyvärinen, and E. Oja, “Independent Component Analysis: A Tutorial,”
http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb.
[17] T. Kohonen, “Self Organizing Map,” Heidelberg, Springer, 1995.
[18] T. A. Runkler, “Fuzzy Nonlinear Projection,” FUZZ-IEEE 2003, pp. 863-868, 2003.
[19] D. E. Gustafson, and W. C. Kessel, “Fuzzy Clustering with a Fuzzy Covariance Matrix,” Proc. IEEE CDC, pp. 761-766, 1979.
[20] Y. Morii, T. Furuhashi, K. Morikawa, and H. Itani, “A Study on Skill Acquisition from Teaching Information by Mobile Robot,” Proc. of IEEE Int. Symposium on Computational Intelligence in Robotics and Automation (CIRA2003), pp. 753-757, 2003.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] J. Casillas, O. Cordón, F. Herrera, and L. Magdalena, “Interpretability Issues in Fuzzy Modeling,” Springer, 2003.

[2] [2] S. Horikawa, T. Furuhashi, and Y. Uchikawa, “On Fuzzy Modeling Using Fuzzy Neural Networks with the Back-Propagation Algorithm,” IEEE Trans. on Neural Networks, Vol.3, No.5, pp. 801-806, 1992.

[3] [3] M. Sugeno, and T. Yasukawa, “A Fuzzy Logic based Approach to Qualitative Modeling,” IEEE Trans. on Fuzzy System, Vol.1, No.1, pp. 7-31, 1993.

[4] [4] D. Tikk, G. Biró, T. D. Gedeon, L. T. Kóczy, and J. D. Yang, “Improvements and Critique on Sugeno’s and Yasukawa’s Qualitative Modeling,” IEEE Trans. on Fuzzy Systems, Vol.10, No.5, pp. 596-606, 2002.

[5] [5] M. Bikdash, “A highly interpretable form of Sugeno inference systems,” IEEE Trans. on Fuzzy Systems, Vol.7 No.6, pp. 686-696, 1999.

[6] [6] A. K. Jain, R. Duin, and J. Mao, “Statistical Pattern Recognition: A Review,” IEEE Trans. on PAMI, Vol.22, No.1, pp. 4-37, 2000.

[7] [7] K. Yamamoto, T. Yoshikawa, and T. Furuhashi, “A Proposal of Fuzzy Modeling on Fusion Axes Considering the Data Structure,” FUZZ-IEEE 2003, pp. 348-353, 2003.

[8] [8] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.

[9] [9] S. Miyamoto, and M. Mukaidono, “Fuzzy C-means as a Regularization and Maximum Entropy Approach,” IFSA’97, Vol.2, pp. 86-92, 1997.

[10] [10] G. E. P. Box, and G. M. Jenkins, “Time Series Analysis,” Forecasting and Control. San Francisco, CA, Holden Day, 1970.

[11] [11] S. Akaho, “EM Algorithm – Application to Clustering and Recent Development,” Journal of Japan Society of Fuzzy Theory and Systems, Vol.12, No.5, pp. 594-602, 2000 (In Japanese).

[12] [12] D. Suizu, and S. Miyamoto, “Fuzzy c-Means Clustering using Mapping into High Dimensional Spaces,” Proc. of the 18th Fuzzy System Symposium, pp. 123-126, 2002 (In Japanese).

[13] [13] E. Kim, M. Park, S. Kim, and M. Park, “A Transformed Input-Domain Approach to Fuzzy Modeling,” IEEE Trans. on Fuzzy Systems, Vol.6, No.4, pp. 596-605, 1998.

[14] [14] R. J. Hathaway, and J. C. Bezdek, “Switching Regression Models and Fuzzy Clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, No.3, 1993.

[15] [15] Y. Yam, P. Baranyi, and C. T. Yang, “Reduction of Fuzzy Rule Base Via Singular Value Decomposition,” IEEE Trans. of Fuzzy System, Vol.7, No.2, pp. 120-132, 1999.

[16] [16] A. Hyvärinen, and E. Oja, “Independent Component Analysis: A Tutorial,”
http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb.

[17] [17] T. Kohonen, “Self Organizing Map,” Heidelberg, Springer, 1995.

[18] [18] T. A. Runkler, “Fuzzy Nonlinear Projection,” FUZZ-IEEE 2003, pp. 863-868, 2003.

[19] [19] D. E. Gustafson, and W. C. Kessel, “Fuzzy Clustering with a Fuzzy Covariance Matrix,” Proc. IEEE CDC, pp. 761-766, 1979.

[20] [20] Y. Morii, T. Furuhashi, K. Morikawa, and H. Itani, “A Study on Skill Acquisition from Teaching Information by Mobile Robot,” Proc. of IEEE Int. Symposium on Computational Intelligence in Robotics and Automation (CIRA2003), pp. 753-757, 2003.