Jerry M. Mendel

This article introduces the fuzzy-system kernel machines —a class of machine learning models based on the connection between fuzzy inference systems and kernel machines. For the connection, we observed a relationship between the representer theorem of kernel methods and the functional representation of nonsingleton fuzzy systems. We found that the nonsingleton kernel on fuzzy sets —a kernel defined in this article—is the core element allowing this two-way connection perspective. Consequently, a fuzzy system trained with the kernel method can be regarded as a kernel machine, whereas a kernel machine trained with a nonsingleton kernel on fuzzy sets can be interpreted as a fuzzy system. We conducted several experiments in supervised classification to understand the generalization power and properties of the proposed fuzzy-system kernel machines.

This paper is about explainable AI (XAI) for rule-based fuzzy systems [that can be expressed generically, as y(x) = f(x)]. It explains why it is not valid to explain the output of Mamdani or TSK rule-based fuzzy systems using IF-THEN rules, and why it is valid to explain the output of such rule-based fuzzy systems as an association of the compound antecedents of a small subset of the original larger set of rules, using a phrase such as These linguistic antecedents are symptomatic of this output. Importantly, it provides a novel multi-step approach to obtain such a small subset of rules for three kinds of fuzzy systems, and illustrates it by means of a very comprehensive example. It also explains why the choice for antecedent membership function shapes may be more critical for XAI than before XAI, why Linguistic Approximation and similarity are essential for XAI, and, it provides a way to estimate the quality of the explanations.

The articles in this special section are dedicated to the memory of Prof. Robert John, one of the pioneers of type-2 fuzzy sets and systems, who passed away during the preparation of this issue.

This paper aims to provide a new perspective on how the deployment of General Type-2 (GT2) fuzzy sets affects the mapping of a class of Fuzzy Logic Controllers (FLCs). It is shown that an -plane represented GT2-FLC is easily designed via baseline type-1 and interval type-2 FLCs; and two Design Parameters (DPs). The DPs are the total number of -planes and the tuning parameter of the secondary membership function which are interpreted as sensitivity and shape DPs, respectively. We provide a clear understanding and interpretation of the sensitivity and shape DPs on controller performance through various comparative analyses. We present design approaches on how to tune the shape DP by providing a tradeoff between robustness and performance. We also propose two online scheduling mechanisms to tune the shape DP. We explore the effect of the sensitivity DP on the GT2-FLC and provide practical insights on how to tune the sensitivity DP. We present an algorithm for tuning the sensitivity DP that provides a compromise between computational time and sensitivity. We validate our analyses, interpretations and design methods with experimental results conducted on a drone. We believe that the paper provides clear explanations on the role of DPs on the performance, robustness, sensitivity and computational time of GT2-FLCs.

Improving the efficiency of type-reduction algorithms continues to attract research interest. Recently, there have been some new type-reduction approaches claiming that they are more efficient than the well-known algorithms such as the enhanced Karnik-Mendel (EKM) and the enhanced iterative algorithm with stopping condition (EIASC). In a previous paper, we found that the computational efficiency of an algorithm is closely related to the platform, and how it is implemented. In computer science, the dependence on languages is usually avoided by focusing on the complexity of algorithms (using big O notation). In this paper, the main contribution is the proposal of two novel type-reduction algorithms. Also, for the first time, a comprehensive study on both existing and new type-reduction approaches is made based on both algorithm complexity and practical computational time under a variety of programming languages. Based on the results, suggestions are given for the preferred algorithms in different scenarios depending on implementation platform and application context.

This paper provides new application-independent perspectives about the performance potential of an intuitionistic (I-) fuzzy system over a (classical) TSK fuzzy system. It does this by extending sculpting the state space works from a TSK fuzzy system to an I- fuzzy system. It demonstrates that, for piecewise-linear membership functions (trapezoids and triangles), an I-fuzzy system always has significantly more first-order rule partitions of the state space-the coarse sculpting of the state space-than does a TSK fuzzy system, and that some I-fuzzy systems also have more second-order rule partitions of the state space-the fine sculpting of the state space-than does a TSK fuzzy system. It is the author's conjecture that, for piecewise-linear membership functions (trapezoids and triangles): It is the always-significantly greater coarse (and possibly fine) sculpting of the state space that provides an I-fuzzy system with the potential to outperform a TSK fuzzy system; and, that a type-1 I-fuzzy system has the potential to outperform an interval type-2 fuzzy system.

There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This article proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: first, train an initial global model using all training data; second, identify from the initial global model the patches that contribute the most to the learning error, and train a (local) patch model for each such patch; and, third, update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on one-dimensional (1-D)/2-D/3-D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.

This paper provides a novel and better understanding of the performance potential of a non-singleton (NS) fuzzy system over a singleton (S) fuzzy system. It does this by extending sculpting the state space works from S to NS fuzzification, and it demonstrates that uncertainties about measurements, modeled by NS fuzzification: (1) fire more rules more often, manifested by a reduction (increase) in the sizes of first-order rule partitions for those partitions associated with the firing of a smaller (larger) number of rules-the course sculpting of the state space; (2) may lead to an increase or decrease in the number of type-1 (T1) and interval type-2 (IT2) first-order rule partitions, that now contain rule pairs that can never occur for S fuzzification-a new rule crossover phenomenon-discovered using partition theory; and, (3) may lead to a decrease, the same number, or an increase in the number of second-order rule partitions, all of which are system dependent-the fine sculpting of the state space. The author's conjecture that: It is the additional control of the course sculpting of the state space, accomplished by pre-filtering and the max-min (or max-product) composition, that provides a NS T1 or IT2 fuzzy system with the potential to outperform a S T1 or IT2 system, when measurements are uncertain.