Paper:

# Fuzzy *c*-Regression Models for Fuzzy Numbers on a Graph

## Tatsuya Higuchi, Sadaaki Miyamoto, and Yasunori Endo

University of Tsukuba

1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

With the assumption that the vertices have numerical values. The aim of this paper is to construct regression models to estimate the values from their relationship on the graph by defining the vertex and the numerical value as an independent variable and a dependent variable, respectively. Given the condition that near vertices have close values, k-Nearest Neighbor regression models (KNN) has been proposed. However, the condition is not satisfied when some near vertices have different values. To overcome such difficulty, c-regression which classify data points int o some clusters has been proposed to improve performance of regression analysis. We moreover propose new c-regression models on a graph with fuzzy numbers on vertices and show some numerical examples.

*c*-Regression Models for Fuzzy Numbers on a Graph,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.20, No.4, pp. 521-534, 2016.

- [1] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Kluwer, 1981.
- [2] S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, Berlin, 2008.
- [3] R. J. Hathaway, and J. C. Bezdek, “Switching Regression Models and Fuzzy Clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, No.3, pp. 195-204, 1993.
- [4] H. J. Tang, “Sequential Extraction of Clusters in Hard and Fuzzy Regression Models,” Master’s Thesis, Graduate School of Systems and Information Engineering, 2012.
- [5] H. J. Tang, and S. Miyamoto, “Semi-supervised Sequential Kernel Regression Models with Pairwise Constraints,” V. Torra et al. (eds.), Modeling Decisions for Artificial Intelligence, LNAI 8234, Springer, Berlin, pp. 166-178, 2013.
- [6] C. J. Stone, “Consistent Nonparametric Regression,” The Annals of Statistics, Vol.5, No.4, pp. 595-620, 1977.
- [7] M. G. Kendall, “A course in multivariate analysis,” Hafner, London, 1957.
- [8] R. M. Mansfield, “PCR: Principal Component Regression Analysis,” J. of Marketing Research, Vol.15, No.3, pp. 471-472, 1978.
- [9] H. J. Zimmermann, “Fuzzy Set Theory and Its Applications,” Kluwer, Dordecht, 1991.
- [10] H. Tanaka, S. Vejima, and K. Asai, “Linear Regression Analysis with Fuzzy Model,” IEEE Trans. on Systems, Man, Cybernetics, Vol.12, pp. 903-907, 1982.
- [11] M. S. Yang and C. H. Ko, “On Cluster-wise Fuzzy Regression Analysis,” IEEE Trans. on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol.27, No.1, pp. 1-13, 1997.
- [12] M. S. Yang and C. H. Ko, “On A Class of Fuzzy c-numbers Clustering Procedures for Fuzzy Data,” Fuzzy Sets and Systems, Vol.84, No.1, pp. 49-60, 1996.
- [13] J. Shao, “Linear Model Selection by Cross-Validation,” J. of the American Statistical Association, Vol.88, No.422, pp. 486-494, 1993.
- [14] M. E. Newman, “Finding Community Structure in Networks Using the Eigenvectors of Matrices,” Physical review E., Vol.74, No.3, 036104. 2006.
- [15] http://www-personal.umich.edu/ mejn/netdata/adjnoun.zip [Accessed February, 2015]
- [16] V. D. Blondel, A. Gajardo, M. Heymans, P. Senellart, and P. Van Dooren, “A Measure of Similarity between graph vertices: Applications to synonym extraction and web searching,” SIAM review, Vol.46, No.4, pp. 647-666, 2004.
- [17] SentiWordNet, http://sentiwordnet.isti.cnr.it/ [Accessed February, 2015]