Fuzzy Regression Analysis and Its Applications

Fuzzy regression analysis extends classical regression by allowing model parameters or observations to be represented as fuzzy sets rather than precise values. This approach accommodates vagueness, ambiguity and imprecision inherent in many real-world data sets. By replacing crisp coefficients with fuzzy numbers or membership functions, fuzzy regression captures the range of possible relationships between independent and dependent variables under uncertainty. Over recent decades the field has diversified to include interval type-2 and hesitant fuzzy frameworks, soft nonlinear and quantile-based formulations, as well as probabilistic extensions. Applications span environmental modelling, economic forecasting, risk assessment, quality control and decision support in engineering and finance. Key advantages include robustness to outliers, enhanced interpretability of uncertainty bounds and flexibility in model fitting. Contemporary challenges centre on optimal selection of membership functions, computational efficiency in high-dimensional settings and integration with machine-learning techniques for complex nonlinear relationships.

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In the probabilistic hesitant fuzzy linear regression model, input–output variables are treated as probabilistic hesitant fuzzy elements, combining probability values with hesitant fuzzy sets to retain more information in multicriteria decision-making. A linear programming formulation estimates symmetric triangular fuzzy parameters, and comparative studies demonstrate improved ranking stability and decision accuracy in complex systems where experts express hesitation in assessments.

A hesitant fuzzy linear regression framework has been proposed to address scenarios in which experts provide multiple possible membership degrees rather than a single value. Parameters are estimated via linear programming under symmetric triangular fuzzy numbers, yielding a model that better reflects indecision. Applications to multicriteria decision-making illustrate enhanced flexibility compared with classical fuzzy linear regression and established ranking tools.

To tackle multicollinearity among explanatory variables, a ridge fuzzy regression model introduces an α-level estimation algorithm. By incorporating α-cuts into the estimation procedure, the model controls the influence of highly correlated predictors within fuzzy linear regression. Simulation experiments and empirical case studies confirm that the ridge approach maintains predictive accuracy across varying degrees of correlation and response spread.

Fuzzy Regression Analysis and Its Applications publication trend

The graph below shows the total number of articles in fuzzy regression analysis and its applications across all publications each year (not limited to Nature Index journals).

Technical terms

Fuzzy regression: A regression methodology in which coefficients or observations are expressed as fuzzy sets to model uncertainty and imprecision.

Fuzzy set: A collection of elements with degrees of membership between zero and one, representing gradual transitions rather than crisp inclusion or exclusion.

Membership function: A mapping that assigns to each element a value in [0, 1], indicating its degree of belonging to a fuzzy set.

Hesitant fuzzy set: A fuzzy set in which the membership degree of an element is represented by a finite set of possible values, reflecting hesitation or indecision.

α-level (α-cut): A crisp set derived from a fuzzy set by including all elements whose membership degree is at least α, used to simplify computations.

Triangular fuzzy number: A fuzzy number characterised by a triplet (l, m, u) defining a piecewise linear membership function with lower limit l, peak m and upper limit u.

Multicollinearity: A statistical phenomenon in which explanatory variables in a regression model are highly correlated, leading to unstable parameter estimates.

References

1. A flexible soft nonlinear quantile-based regression model. Fuzzy Optimization and Decision Making (2025).
2. Hesitant Fuzzy Linear Regression Model for Decision Making. Symmetry (2021).
3. Ridge Fuzzy Regression Modelling for Solving Multicollinearity. Mathematics (2020).
4. Quadrilateral Interval Type‐2 Fuzzy Regression Analysis for Data Outlier Detection. Mathematical Problems in Engineering (2019).
5. Making Group Decisions within the Framework of a Probabilistic Hesitant Fuzzy Linear Regression Model. Sensors (2022).