A Shape Preserving Adaptive Elastic Net Framework with a Hybrid Local and Global Dictionary for Fuzzy Nonlinear Regression

Document Type : Research article

Authors

School of Mathematics and Computer Science, Damghan University, Damghan, Iran

Abstract

Fuzzy nonlinear regression learns functional relationships from imprecise observations, yet common estimators suffer from two persistent difficulties: shrinkage rules that cannot discard redundant basis functions, and endpoint fits that may violate the defining properties of a fuzzy number. This paper develops a shape-preserving adaptive elastic net framework for fuzzy responses with crisp or fuzzy inputs. The fuzzy response is decomposed at every membership level into a center and a nonnegative radius, and each component is regressed on a hybrid dictionary joining a polynomial tail, Gaussian atoms with adaptively rescaled dispersions, and a stacked local smoothing pilot atom. An adaptive elastic net penalty with data driven weights selects and shrinks the dictionary, all tuning constants are chosen by an internal cross validation that respects the stacking structure, and a non expansive projection restores nestedness of the estimated level sets. We prove existence and uniqueness of the estimator, convergence of its coordinate descent solver, and that the shape repair can never increase the Hausdorff estimation error. Monte Carlo experiments on four simulated benchmarks and a real clinical dataset in ten dimensions compare the proposal with ridge regularized fuzzy regression, local linear smoothing, and kernel smoothing under a nested cross validation protocol. The proposed method improves on the ridge estimator uniformly, attains the best cross validation error and bias in three simulated designs and the best mean cross validation error on the real data, is statistically tied with the best local smoother elsewhere, and alone guarantees valid fuzzy outputs.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 12, Issue 1
June 2027
Pages 90-108

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History

  • Receive Date: 15 August 2026
  • Revise Date: 08 September 2026
  • Accept Date: 23 September 2026
  • Publish Date: 02 November 2026

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