Exploring Zadeh's General Type-2 Fuzzy Logic Systems for Uncertainty Quantification
Yusuf Güven, Ata Köklü, Tufan Kumbasar
Abstract
This article introduces an exploration of general Type-2 (GT2) fuzzy logic systems (FLSs) via Zadeh's (Z) GT2 fuzzy set (FS) definition, with a strong emphasis on advancing uncertainty quantification (UQ). At the heart of our contribution is the introduction of Z-GT2-FLS, formed through the integration of Z-GT2-FS with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-plane representation. We show that the design flexibility of GT2-FLS is increased as it takes away the dependency of the secondary membership function definition from the primary membership function. For learning, we provide a solution to the curse of dimensionality problem alongside a method to seamlessly integrate deep learning (DL) optimizers. This article further presents a dual-focused Z-GT2-FLS within a DL framework, intending to learn Z-GT2-FLSs that are capable of achieving high-quality prediction intervals alongside high precision. In this context, we assign distinct roles for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha _{k}$</tex-math></inline-formula>-plane-associated interval type-2 FLSs through a composite loss function. In addition, we extend the application of Z-GT2-FLS to predictive distribution estimation, proposing a DL framework to learn the inverse cumulative distribution function by predicting entire quantile levels. We first reformulate the output of Z-GT2-FLS to represent a quantile level function, thereby offering flexibility in generating desired quantiles through <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-planes. For learning, we propose a simultaneous quantile learning method alongside an adaptation mechanism to enhance learning performance. Through comparative analyses, we show that the Z-GT2-FLS excels in UQ compared to its fuzzy and DL counterparts. The contributions of this study underscore the versatility and superior performance of Z-GT2-FLS, positioning it as a valuable tool for UQ.