Zadeh's Type-2 Fuzzy Logic Systems: Precision and High-Quality Prediction Intervals

TL;DR

This work addresses uncertainty quantification in high-risk tasks by revisiting General Type-2 Fuzzy Logic Systems through Zadeh's GT2-FS definition and integrating an alpha-plane representation to decouple SMF shape from the PMF. It introduces a DL-enabled framework for learning dual-focused Z-GT2-FLSs that deliver accurate point predictions while producing High-Quality Prediction Intervals (HQ-PIs). Empirical results on high-dimensional datasets demonstrate that Z-GT2-FLS achieves superior or competitive RMSE with HQ-PIs and often tighter intervals than MJ-GT2-FLS and IT2-FLS baselines, despite fewer learnable parameters. The approach offers strong potential for reliable uncertainty quantification in complex, high-dimensional decision tasks, with scalable learning through parameterization tricks and dimension-aware design choices.

Abstract

General Type-2 (GT2) Fuzzy Logic Systems (FLSs) are perfect candidates to quantify uncertainty, which is crucial for informed decisions in high-risk tasks, as they are powerful tools in representing uncertainty. In this paper, we travel back in time to provide a new look at GT2-FLSs by adopting Zadeh's (Z) GT2 Fuzzy Set (FS) definition, intending to learn GT2-FLSs that are capable of achieving reliable High-Quality Prediction Intervals (HQ-PI) alongside precision. By integrating Z-GT2-FS with the \(α\)-plane representation, we show that the design flexibility of GT2-FLS is increased as it takes away the dependency of the secondary membership function from the primary membership function. After detailing the construction of Z-GT2-FLSs, we provide solutions to challenges while learning from high-dimensional data: the curse of dimensionality, and integrating Deep Learning (DL) optimizers. We develop a DL framework for learning dual-focused Z-GT2-FLSs with high performances. Our study includes statistical analyses, highlighting that the Z-GT2-FLS not only exhibits high-precision performance but also produces HQ-PIs in comparison to its GT2 and IT2 fuzzy counterparts which have more learnable parameters. The results show that the Z-GT2-FLS has a huge potential in uncertainty quantification.

Figures (4)

• Figure 1: Illustrations of a GT2-FS with an $\alpha$ - plane
• Figure 2: White Wine dataset: Notched box-and-whisker plots
• Figure 3: Parkinson dataset: Notched box-and-whisker plots
• Figure 4: AIDS dataset: Notched box-and-whisker plots

Theorems & Definitions (2)

• Definition 1
• Definition 2