In this paper, we introduce the concept of circular Pythagorean fuzzy set (value) (C-PFS(V)) as a new generalization of both circular intuitionistic fuzzy sets (C-IFSs) proposed by Atannassov and Pythagorean fuzzy sets (PFSs) proposed by Yager. A circular Pythagorean fuzzy set is represented by a circle that represents the membership degree and the non-membership degree and whose center consists of non-negative real numbers $\mu$ and $\nu$ with the condition $\mu^2+\nu^2\leq 1$. A C-PFS models the fuzziness of the uncertain information more properly thanks to its structure that allows modelling the information with points of a circle of a certain center and a radius. Therefore, a C-PFS lets decision makers to evaluate objects in a larger and more flexible region and thus more sensitive decisions can be made. After defining the concept of C-PFS we define some fundamental set operations between C-PFSs and propose some algebraic operations between C-PFVs via general $t$-norms and $t$-conorms. By utilizing these algebraic operations, we introduce some weighted aggregation operators to transform input values represented by C-PFVs to a single output value. Then to determine the degree of similarity between C-PFVs we define a cosine similarity measure based on radius. Furthermore, we develop a method to transform a collection of Pythagorean fuzzy values to a PFS. Finally, a method is given to solve multi-criteria decision making problems in circular Pythagorean fuzzy environment and the proposed method is practiced to a problem about selecting the best photovoltaic cell from the literature. We also study the comparison analysis and time complexity of the proposed method.

Distributed fuzzy neural networks (DFNNs) have attracted increasing attention recently due to their learning abilities in handling data uncertainties in distributed scenarios. However, it is challenging for DFNNs to handle cases in which the local data are non-independent and identically distributed (non-IID). In this paper, we propose a federated fuzzy neural network (FedFNN) with evolutionary rule learning (ERL) to cope with non-IID issues as well as data uncertainties. The FedFNN maintains a global set of rules in a server and a personalized subset of these rules for each local client. ERL is inspired by the theory of biological evolution; it encourages rule variations while activating superior rules and deactivating inferior rules for local clients with non-IID data. Specifically, ERL consists of two stages in an iterative procedure: a rule cooperation stage that updates global rules by aggregating local rules based on their activation statuses and a rule evolution stage that evolves the global rules and updates the activation statuses of the local rules. This procedure improves both the generalization and personalization of the FedFNN for dealing with non-IID issues and data uncertainties. Extensive experiments conducted on a range of datasets demonstrate the superiority of the FedFNN over state-of-the-art methods.

Mobile robots have been widely used in various aspects of human life. When a robot moves between different positions in the working area to perform the task, controlling motion to follow a pre-defined path is the primary task of a mobile robot. Furthermore, the robot must remain at its desired speed to cooperate with other agents. This paper presents a development of a motion controller, in which the fuzzy logic method is combined with a kinematic model of a differential drive robot. The simulation results are compared well with experimental results indicate that the method is effective and applicable for actual mobile robots.

Explainable Artificial Intelligence (XAI) is a paradigm that delivers transparent models and decisions, which are easy to understand, analyze, and augment by a non-technical audience. Fuzzy Logic Systems (FLS) based XAI can provide an explainable framework, while also modeling uncertainties present in real-world environments, which renders it suitable for applications where explainability is a requirement. However, most real-life processes are not characterized by high levels of uncertainties alone; they are inherently time-dependent as well, i.e., the processes change with time. In this work, we present novel Temporal Type-2 FLS Based Approach for time-dependent XAI (TXAI) systems, which can account for the likelihood of a measurement's occurrence in the time domain using (the measurement's) frequency of occurrence. In Temporal Type-2 Fuzzy Sets (TT2FSs), a four-dimensional (4D) time-dependent membership function is developed where relations are used to construct the inter-relations between the elements of the universe of discourse and its frequency of occurrence. The TXAI system manifested better classification prowess, with 10-fold test datasets, with a mean recall of 95.40\% than a standard XAI system (based on non-temporal general type-2 (GT2) fuzzy sets) that had a mean recall of 87.04\%. TXAI also performed significantly better than most non-explainable AI systems between 3.95\%, to 19.04\% improvement gain in mean recall. In addition, TXAI can also outline the most likely time-dependent trajectories using the frequency of occurrence values embedded in the TXAI model; viz. given a rule at a determined time interval, what will be the next most likely rule at a subsequent time interval. In this regard, the proposed TXAI system can have profound implications for delineating the evolution of real-life time-dependent processes, such as behavioural or biological processes.

During the COVID-19 pandemic, several countries have taken the approach of tiered restrictions which has remained a point of debate due to a lack of transparency. Using the dominance-based rough set approach, we identify patterns in the COVID-19 data pertaining to the UK government's tiered restrictions allocation system. These insights from the analysis are translated into "if-then" type rules, which can easily be interpreted by policy makers. The differences in the rules extracted from different geographical areas suggest inconsistencies in the allocations of tiers in these areas. We found that the differences delineated an overall north south divide in England, however, this divide was driven mostly by London. Based on our analysis, we demonstrate the usefulness of the dominance-based rough sets approach for investigating the fairness and explainabilty of decision making regarding COVID-19 restrictions. The proposed approach and analysis could provide a more transparent approach to localised public health restrictions, which can help ensure greater conformity to the public safety rules.

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $\epsilon$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e.g., width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the $\epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an $\alpha$-smooth Q-function in a $d$-dimensional feature space. We prove that, with $T$ episodes, scaling the width $m = \widetilde{\mathcal{O}}(T^{\frac{d}{2\alpha + d}})$ and the depth $L=\mathcal{O}(\log T)$ of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width $\Omega(\sqrt{T})$ is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the $\epsilon$-greedy exploration is not enough to ensure ``optimism''. Our analysis reformulates the temporal difference error in an $L^2(\mathrm{d}\mu)$-integrable space over a certain averaged measure $\mu$, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding $\epsilon$-greedy exploration in deep RL.

Understand the basic theory and implement fuzzy systems with skfuzzy library! Fuzzy Logic is a technique that can be used to model the human reasoning process in computers. It can be applied to several areas, such as: industrial automation, medicine, marketing, home automation, among others. A classic example is the use in industrial equipments, which can have the temperature automatically adjusted as the equipment heats up or cools down. Other examples of equipments are: vacuum cleaners (adjustment of suction power according to the surface and level of dirt), dishwashers and clothes washing machines (adjustment of the amount of water and soap to use), digital cameras (automatic focus setting), air conditioning (temperature setting according to the environment), and microwave (power adjustment according to the type of food).

Traditional machine learning techniques have been widely used to establish the trust management systems. However, the scale of training dataset can significantly affect the security performances of the systems, while it is a great challenge to detect malicious nodes due to the absence of labeled data regarding novel attacks. To address this issue, this paper presents a generative adversarial network (GAN) based trust management mechanism for Industrial Wireless Sensor Networks (IWSNs). First, type-2 fuzzy logic is adopted to evaluate the reputation of sensor nodes while alleviating the uncertainty problem. Then, trust vectors are collected to train a GAN-based codec structure, which is used for further malicious node detection. Moreover, to avoid normal nodes being isolated from the network permanently due to error detections, a GAN-based trust redemption model is constructed to enhance the resilience of trust management. Based on the latest detection results, a trust model update method is developed to adapt to the dynamic industrial environment. The proposed trust management mechanism is finally applied to secure clustering for reliable and real-time data transmission, and simulation results show that it achieves a high detection rate up to 96%, as well as a low false positive rate below 8%.

We provide performance guarantees for a variant of simulation-based policy iteration for controlling Markov decision processes that involves the use of stochastic approximation algorithms along with state-of-the-art techniques that are useful for very large MDPs, including lookahead, function approximation, and gradient descent. Specifically, we analyze two algorithms; the first algorithm involves a least squares approach where a new set of weights associated with feature vectors is obtained via least squares minimization at each iteration and the second algorithm involves a two-time-scale stochastic approximation algorithm taking several steps of gradient descent towards the least squares solution before obtaining the next iterate using a stochastic approximation algorithm.

This paper presents the application of an adaptive neuro-fuzzy inference system (ANFIS) to predict the generated electrical power in a combined cycle power plant. The ANFIS architecture is implemented in MATLAB through a code that utilizes a hybrid algorithm that combines gradient descent and the least square estimator to train the network. The Model is verified by applying it to approximate a nonlinear equation with three variables, the time series Mackey-Glass equation and the ANFIS toolbox in MATLAB. Once its validity is confirmed, ANFIS is implemented to forecast the generated electrical power by the power plant. The ANFIS has three inputs: temperature, pressure, and relative humidity. Each input is fuzzified by three Gaussian membership functions. The first-order Sugeno type defuzzification approach is utilized to evaluate a crisp output. Proposed ANFIS is cable of successfully predicting power generation with extremely high accuracy and being much faster than Toolbox, which makes it a promising tool for energy generation applications.