Boosting Sparsity in Gram Matrix of Fuzzy Regression Models through Radial Basis Functions

Document Type : Research Article

Authors

Department of Mathematics and Statistics, Faculty of Energy and Data science, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran

Abstract

The sparsity of the Gram matrix in linear regression can influence the model's accuracy. Sparse matrices reduce computational complexity and improve generalization by minimizing overfitting. This advantage is particularly beneficial in high-dimensional data where the number of features exceeds the number of observations. This paper explores the integration of Radial Basis Functions (RBFs) in developing sparse Gram matrix fuzzy regression models. RBFs are powerful tools for function approximation, defined by their dependence on the distance from a center point, which allows for flexible modeling of nonlinear relationships. The focus will be on compactly supported RBF kernels, which facilitate sparsity in the Gram matrix, thereby improving computational efficiency and memory usage. By leveraging the properties of RBFs, particularly their ability to localize influence and reduce dimensionality, we aim to enhance the performance of fuzzy regression models. This study will present theoretical insights and empirical results demonstrating how the adoption of RBFs can lead to significant improvements in model accuracy and computational speed, making them a valuable asset in the field of fuzzy regression analysis.

Keywords

Main Subjects

Full Text

 

Article PDF

References

[1] M. Arefi, A. H. Khammar, Nonlinear prediction of fuzzy regression model based on quantile loss function, Soft Computing, 1–11, (2023).
[2] M. Arefi, Quantile fuzzy regression based on fuzzy outputs and fuzzy parameters, Soft Computing, 24(1), 311–320, (2020).
[3] Z. Behdani, M. Darehmiraki, Theil-Sen Estimators for fuzzy regression model, Iranian Journal of Fuzzy Systems, 21(3), 177–192, (2024).
[4] Z. Behdani, M. Darehmiraki, Enhancing Kernel Ridge Regression Models with Compact Support Wendland Functions, Analytical and Numerical Solutions for Nonlinear Equations, 9(1), 34–47, (2025).
[5] Z. Behdani, M. Darehmiraki, Neutrosophic fuzzy regression: A linear programming approach, Iranian Journal of Operations Research, 15(1), 1–11, (2024).
[6] Q. Cai, Z. Hao, X. Yang, Gaussian kernel-based fuzzy inference systems for high dimensional regression, Neurocomputing, 77(1), 197–204, (2012).
[7] W. Chung, A Fuzzy Convex Nonparametric Least Squares Method with Different Shape Constraints, International Journal of Fuzzy Systems, 1–15, (2023).
[8] G. E. del Pino, H. Galaz, Statistical applications of the inverse gram matrix: A revisitation, Brazilian Journal of Probability and Statistics, 177–196, (1995).
[9] P. Diamond, R. Korner, Extended fuzzy linear models and least squares estimates, Computers & Mathematics with Applications, 33(9), 15–32, (1997).
[10] P. Drineas, M. W. Mahoney, Approximating a gram matrix for improved kernel-based learning, In: International Conference on Computational Learning Theory, 323–337, (2005).
[11] A. I. Iacob, C. C. Popescu, Regression using partially linearized gaussian fuzzy data, In: Informatics Engineering and Information Science: International Conference, ICIEIS 2011, Kuala Lumpur, Malaysia, November 14-16, 2011. Proceedings, Part II, 584–595, (2011).
[12] E. J. Kansa, Radial basis functions: achievements and challenges, WIT Transactions on Modelling and Simulation, 61, 3–22, (2015).
[13] A. H. Khammar, M. Arefi, M. G. Akbari, A general approach to fuzzy regression models based on different loss functions, Soft Computing, 25(2), 835–849, (2021).
[14] J. M. Leski, Fuzzy c-ordered-means clustering, Fuzzy Sets and Systems, 286, 114–133, (2016).
[15] Y. Li, X. He, X. Liu, Fuzzy multiple linear least squares regression analysis, Fuzzy Sets and Systems, 459, 118–143, (2023).
[16] D. A. Savic, W. Pedrycz, Evaluation of fuzzy linear regression models, Fuzzy sets and systems, 39(1), 51–63, (1991).
[17] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in computational Mathematics, 4(1), 389–396, (1995).
[18] H. Wendland, Scattered data approximation, Cambridge, UK: Cambridge University Press, (2005).
[19] K. Wiktorowicz, T. Krzeszowski, Approximation of two-variable functions using high-order Takagi-Sugeno fuzzy systems, sparse regressions, and metaheuristic optimization, Soft Computing, 24(20), 15113–15127, (2020).
[20] Z. Wu, Compactly supported positive definite radial functions, Advances in Computational Mathematics, 4(1), 283–292, (1995).
[21] M. S. Yang, C. H. Ko, On a class of fuzzy c-numbers clustering procedures for fuzzy data, Fuzzy sets and systems, 84(1), 49–60, (1996).
[22] L. A. Zadeh, Toward extended fuzzy logic first step, Fuzzy sets and systems, 160(21), 3175–3181, (2009). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 9, Issue 2
September 2025
Pages 161-175

Files

History

  • Receive Date: 18 June 2025
  • Revise Date: 11 September 2025
  • Accept Date: 13 September 2025
  • Publish Date: 14 September 2025

Share

How to cite

Statistics