Bee Colony Optimization metaheuristic for fuzzy membership functions tuning

The Role of Vagueness in the Numerical Translation of Verbal Probabilities: A Fuzzy Approach Franziska Bocklisch 1 (franziska.bocklisch@psychologie.tu-chemnitz.de) Steffen F. Bocklisch 1 (steffen.bocklisch@etit.tu-chemnitz.de) Martin R. K. Baumann 2 (martin.baumann@dlr.de) Agnes Scholz 1 (agnes.scholz@psychologie.tu-chemnitz.de) Josef F. Krems 1 (josef.krems@psychologie.tu-chemnitz.de) Wilhelm-Raabe-Str. 43, Chemnitz University of Technology, Germany Lilienthalplatz 7, German Aerospace Center Braunschweig, Germany Abstract The paper describes a general two-step procedure for the numerical translation of linguistic terms using parametric fuzzy potential membership functions. In an empirical study 121 participants estimated numerical values that correspond to 13 verbal probability expressions. Among the estimates are the most typical numerical equivalent and the minimal and maximal values that just correspond to the given linguistic terms. These values serve as foundation for the proposed fuzzy approach. Positions and shapes of the resulting membership functions suggest that the verbal probability expressions are not distributed equidistantly along the probability scale and vary considerably in symmetry, vagueness and overlap. The role of vagueness for further investigations in reasoning and decision making is discussed and relations to knowledge representation and working memory are highlighted. Keywords: verbal probability expressions; vagueness; fuzzy potential membership functions; knowledge representation; diagnostic reasoning; working memory Introduction Since the 1960s up to the present time researchers of different scientific areas have sustained an interest in studying the relationship between verbal and numerical probability expressions (Lichtenstein & Newman, 1967; Teigen & Brun, 2003; Smits & Hoorens, 2005). Among these are cognitive psychologists that inquire about the influence of uncertainty expressions on basic cognitive processes such as reasoning and decision making (Windschitl & Wells, 1996) as well as engineers, computer scientists and others that focus on the characterization (Zadeh, 1978, 2002) or on the treatment of uncertainty in applications such as medical decision support systems (Boegl, Adlassnig, Hayashi, Rothenfluh & Leitich, 2004). This broad interdisciplinary interest may be motivated by the essential role language plays in our daily life. Verbal probability terms, such as probably or thinkable are very widely used to express uncertainty about the occurrence of future events or about the degree of belief in hypotheses. For example, a typical statement that illustrates the use of linguistic terms in the conversation of stock market traders could be: “It is very unlikely that there will be a significant increase in the price of oil in the next month vice future.”. Several studies consistently show that people prefer words over numbers to express uncertainty (e.g. Wallsten, Budescu, Zwick & Kemp, 1993). This preference may be explained by the possibility of saying something about two different kinds of subjective uncertainty by using only one word. First, the stochastic uncertainty about the occurrence of an event (e.g. the probability of an increase of the oil price) and second, the vagueness of the event (e.g. what is meant by “a significant increase”). The understanding of these two kinds of uncertainty, their relations to each other and the way in which they influence human reasoning and decision making is crucial for any application that aims to support decision makers for example in medicine, business, risk management, marketing or politics. In our view, in order to contribute to the understanding of uncertainty, it is essential to first uncover the underlying relationship between word meaning and mathematical concepts such as subjective probability or fuzzy membership. Therefore, we propose a general two- step procedure for the numerical translation of verbal probability expressions based on (1) empirical estimates modelled by (2) fuzzy membership functions (Zadeh, 1965, Bocklisch & Bitterlich, 1994). The paper is structured as follows: first, we compare verbal and numerical probability expressions and discuss existing translation approaches. Second, we present our proposal that goes beyond other methodical issues and the results of an empirical investigation. Thereafter, the results are discussed and conclusions (e.g. for the construction of verbal probability scales for questionnaires) are highlighted. Further, potentialities of the fuzzy pattern classification method for reasoning and decision processes are pointed out. Verbal and Numerical Probabilities There is broad agreement concerning the different features of verbal and numerical expressions (see Teigen & Brun, 2003 for an overview). Numerical probabilities are commonly described as precise, unambiguous and especially useful for calculations. Additionally, the quality of numerical expressions can be evaluated and compared to predictions of normative models such as Bayes nets. Currently many researchers in the area of cognitive

This paper deals with the principal component analysis (PCA) optimization and fuzzy theory modeling techniques to evaluate and predict the main significant mechanical properties of the studied composite samples such as adhesion between the rubber matrix and the flax nonwoven structure, cyclic stress ( C s ( 80 % ) ), elongation at break and breaking strength in two directions, longitudinal ( 0 ° ) and transverse ( 90 ° ) . In the studied experimental design of interest, the investigated input and output parameters were tested and reduced using the PCA technique with three input parameters: the size of tire rubber particles, the flax nonwoven areal density and the duration of thermo-press molding. Thus, a factorial design is applied to objectively study the contributions of the selected inputs on the mechanical behavior of the investigated composite samples. Subsequent to this, the developed fuzzy models are compared to predict the flax fiber-reinforced rubber composites’ performance based on the experimental results obtained by their mechanical characterization. Membership functions ( Triangular , Trapezoïdal , Gaussian , Π-shaped , etc.) and fuzzy rules were constructed such that the fuzzy model can precisely predict the mechanical properties of the composites. The results show that the Π -shaped membership function gives a better fit of experimental findings because of its spline-based curve shape and is a product of the smf and zmf membership functions that represent polynomial-based curves. Although the Gaussian membership functions and bell membership functions achieve smoothness, they are unable to specify asymmetric membership functions compared to the Π -shaped membership function, which is important in certain applications. Furthermore, compared to the experimental properties, the studied theoretical performances of the reinforced flax/rubber composites can be accurately predicted in the desired field of interest. Hence, the significant coefficient of determination (R 2 ) values relative to the breaking strength, B s ( M D ) and cyclic strength, C s ( 80 % ) , are ranged from 90.1% to 96.6% and from71.4% to 77.6% respectively. Indeed, the theoretical models fit accurately the investigated data. By classifying the accuracy values of the best fuzzy models, a set of test case experiments were conducted so as to validate the theoretical findings. Three selected fuzzy membership functions ( Gauss2mf , Gbellmf , and Gaussmf ) seem accurate to evaluate and predict sufficiently the flax fiber reinforced rubber composite performance.

A hybrid approach composed by different types of fuzzy systems, such as the Type-1 Fuzzy Logic System (T1FLS), Interval Type-2 Fuzzy Logic System (IT2FLS) and Generalized Type-2 Fuzzy Logic System (GT2FLS) for the dynamic adaptation of the alpha and beta parameters of a Bee Colony Optimization (BCO) algorithm is presented. The objective of the work is to focus on the BCO technique to find the optimal distribution of the membership functions in the design of fuzzy controllers. We use BCO specifically for tuning membership functions of the fuzzy controller for trajectory stability in an autonomous mobile robot. We add two types of perturbations in the model for the Generalized Type-2 Fuzzy Logic System to better analyze its behavior under uncertainty and this shows better results when compared to the original BCO. We implemented various performance indices; ITAE, IAE, ISE, ITSE, RMSE and MSE to measure the performance of the controller. The experimental results show better performances using GT2FLS then by IT2FLS and T1FLS in the dynamic adaptation the parameters for the BCO algorithm.

Because of the degradation of classification accuracy that is caused by the uncertainty of pixel class and classification decisions of high-resolution remote-sensing images, we proposed a supervised classification method that is based on an interval type-2 fuzzy membership function for high-resolution remote-sensing images. We analyze the data features of a high-resolution remote-sensing image and construct a type-1 membership function model in a homogenous region by supervised sampling in order to characterize the uncertainty of the pixel class. On the basis of the fuzzy membership function model in the homogeneous region and in accordance with the 3σ criterion of normal distribution, we proposed a method for modeling three types of interval type-2 membership functions and analyze the different types of functions to improve the uncertainty of pixel class expressed by the type-1 fuzzy membership function and to enhance the accuracy of classification decision. According to the principle that importance will increase with a decrease in the distance between the original, upper, and lower fuzzy membership of the training data and the corresponding frequency value in the histogram, we use the weighted average sum of three types of fuzzy membership as the new fuzzy membership of the pixel to be classified and then integrated into the neighborhood pixel relations, constructing a classification decision model. We use the proposed method to classify real high-resolution remote-sensing images and synthetic images. Additionally, we qualitatively and quantitatively evaluate the test results. The results show that a higher classification accuracy can be achieved with the proposed algorithm.

Hybrid bee colony optimization for examination timetabling problems

Fuzzy systems generally work based on expert knowledge base. Fuzzy expert knowledge base incorporates human knowledge through fuzzy rules and fuzzy membership functions. In designing fuzzy models, a major difficulty in the identification of an optimized fuzzy rules as well as membership function shape and type of an individual rule. In fact, it is difficult and time consuming for an expert to define a complete rule set for a complex system having a large number of parameters. In this paper, we proposed a flexible encoding method for evolutionary algorithm to discover parameters of fuzzy rule, design of a suitable fuzzy models and controller for a particular environment. In Evolutionary Fuzzy System (EFS), evolutionary algorithms are adapted for finding the optimal fuzzy rule sets including the number of rules inside it and selecting the membership function shape and type of each individual rule in two different ways respectively. The benefits of this methodology are illustrated for the modeling and control of nonlinear system that shows better performance than existing fuzzy expert systems.

A PSO-aided neuro-fuzzy classifier employing linguistic hedge concepts

In this paper, a bee colony optimization (BCO) algorithm is adapted to the current control strategy of power factor (PF) correction for a Sheppard-Taylor converter. The control approach is developed to eliminate the input current harmonics in the converter. The BCO algorithm is a new population-based search algorithm which is one of the metaheuristic techniques based on swarm intelligence. This algorithm depends on modeling the natural behavior of real honey bees. In this approach, duty cycles are generated by the BCO algorithm for the Sheppard-Taylor converter switch to obtain unity PF (UPF) and lower total harmonic distortion of the input current. The duty cycles for half a line period are calculated and stored in a lookup table. By synchronizing the memory with the line, near-unity PFs can be achieved in an operating point. The feedforward is also used into the control algorithm with the maximum value of the input voltage. The implementation of feedforward improves the converter performance. The input current of the converter is operated in continuous conduction mode. The simulation and experimental results show that the proposed control strategy works well and the UPF can be achieved with wide input voltage, load, and parameter variation. The results are compatible with the International Electrotechnical Commission (IEC) 61000-3-2 current harmonic standard.

Temperature and humidity are two highly coupled variables in a control system, which need to be decoupled for effective control. Moreover, the coupling problem may get more severe and the two control loops may produce a strong interference to each other that can cause system instability when the humidity is measured by dry-and-wet bulb method. In this study, a control method based on fuzzy-neural-network was studied for solving the coupling problem. The shape of membership function can be adjusted in time by using a wavelet basis as the fuzzy membership function. An effective real-time decoupling control system for temperature and humidity could be realized by neural network fuzzy inference. Decoupling control tests were conducted in a control room with 1.6 m × 1.0 m × 4.0 m. The results show that the performance of the control system on dynamic response speed, stability, and anti-jamming have been improved after decoupling.

We present in this paper a novel method (called PID design through numerical optimization) for the design of adaptive fuzzy PID controllers to achieve optimal control performance. By applying the numerical optimization, the fuzzy PID design problem is transferred to a numerical optimization problem. First, a fuzzy parameter tuner is built to generate initial PID parameters, including positions and shapes of fuzzy membership functions and scaling factors. Then the gradient-based Sequential Quadratic programming (SQP) algorithm is employed to iteratively minimize the cost function by iterating the parameters. Unlike conventional PID design, this method is capable of reaching the desired control performance, such as minimum overshoot or long term stability. Another unique feature is that the optimized adaptive PID controller is applicable to a wide variety of time-varying and nonlinear systems. The proposed method is applied to the induction motor control problem and the experimental results are presented to demonstrate its effectiveness and performance.

The paper is concerned with the application of reinforcement learning techniques to feedback control of nonlinear systems using adaptive fuzzy logic systems (FLS). Even if a good model of the nonlinear system is known, it is often difficult to formulate a control law. The work in this paper addresses this problem by showing how an adaptive FLS can cope with nonlinearities through reinforcement learning with no preliminary off-line learning phase required. The reinforcement learning rules for finding proper fuzzy rules and tuning membership functions (MFs) do not assume that there is a supervisor to decide whether the current control action is correct. Instead, the FLS is indirectly told about the affect of its control action on the system performance. The learning is performed online based on a binary reinforcement signal from a critic without knowing the nonlinearity appearing in the system, and so is called direct reinforcement adaptive learning (DRAL). The learning algorithm is derived from Lyapunov stability analysis, so that both system tracking stability and error convergence can be guaranteed in the closed-loop system.

This paper presents the analysis of measurement uncertainty propagation in algorithms for the synthesis of one fault risk index for diagnostic applications. The reference application is non-invasive diagnostic of three-phase induction motor drives. This work focuses on algorithms for fault index calculation based on neuro-fuzzy networks and wavelet decomposition of acquired data. The impact on the index of the shape of fuzzy membership functions, the number of training epochs and is first analyzed. Then the propagation of uncertainty through the neuron-fuzzy network is assessed with a numerical approach.

This paper presents an algorithm of new fuzzy membership function for power quality evaluation. Firstly, the six membership functions are presented in this paper to show the main characters as well as the differences. Then, the six functions are applied in the power quality evaluation in Gl method. Thirdly, a new fuzzy membership functions is constructed through weighting and combining six common membership functions by means of unbiased variance, which improves the stability and the accuracy of the final result. In the end, according to the power quality data of a power supply bureau, the new fuzzy membership function is applied in the evaluation to illustrate the rationality of this method in this paper.

Clustering is a basic data mining operation that groups data points with similar inherent structure. Among clustering techniques, Density Peak Clustering (DPC) is notable for detecting clusters of arbitrary shapes without requiring the number of clusters in advance. However, DPC suffers from parameter-sensitive local density estimation, inadequate handling of noise and boundary points, and rigid binary cluster assignments. To overcome these limitations, fuzzy logic is often embedded in DPC, but the selection of the most appropriate membership function is still an open research question. In this paper, we propose an enhanced DPC variant with three methodological improvements: (i) substituting cutoff distance-based density estimation with a K-Nearest Neighbors (KNN) functioned kernel to achieve stable and robust density estimation, (ii) adding a noise parameter Lambda ( λ ) to better identify and distinguish noise and boundary points, and (iii) using fuzzy membership functions to allow for probabilistic and soft cluster assignments of data points. Comparative analyses of four fuzzy membership functions (i.e., Gaussian, Combination Gaussian, Trapezoidal, and Triangular) were performed on real, synthetic, and high-dimensional datasets. The experimental results indicate that the proposed algorithm with the Gaussian fuzzy membership function outperforms other membership functions and traditional clustering algorithms including the original DPC, FKNN, and DPC-DBFN in terms of clustering accuracy, stability, adaptability, and noise resistance, and is therefore part of our final suggested method. The proposed method overall relieves original DPC shortcomings by strengthening adaptability, accuracy, and applicability to complex real-world clustering tasks. • Substituting standard cutoff-distance-based kernel with KNN for enhanced clustering. • Introducing λ parameter to manage noise and boundary points for separating clusters. • Introducing fuzzy neighborhood kernel for soft cluster membership assignment. • Investigating the impact of several fuzzy membership functions on clustering. • Proposed strategy outperforms original DPC, FKNN, and DPC-DBFN on datasets.

비선형성이 강한 컨테이너 크레인은 작업 시에 호이스트 와이어로프의 길이와 화물의 질량 변화로 인해 더욱 복잡한 동역학적 특성을 나타낸다. 이 같은 복잡한 비선형시스템을 다루기 위해 퍼지로직이 종종 사용되는데, 특히 각 퍼지 규칙의 결론부를 상태 방정식으로 표현하는 T-S 퍼지모델이 대표적인 방법이다. 본 논문에서는 T-S 퍼지모델을 이용하여 호이스트 와이어로프의 길이나, 화물의 질량이 변화하는 환경에서도 컨테이너 크레인의 동특성을 표현할 수 있는 퍼지모델을 얻는 방법을 제안한다. 이때, 퍼지모델의 소속함수 파라미터는 RCGA가 결합된 모델조정기법을 통해 최적으로 조정된다. 이렇게 구현한 퍼지모델과 컨테이너 크레인 비선형시스템의 개루프 응답을 비교하여 그 유효성을 확인한다. In this paper, we focuses on the development of Takagi-Sugeno (T-S) fuzzy modeling in a nonlinear container crane system. A T-S fuzzy model is characterized by fuzzy "if-then" rules which represent the locally input-output relationship whose consequence part is described by a state space equation as subsystem. The T-S fuzzy model in container cranes first obtains a few number of linear models according to operation conditions and blends these conditions using fuzzy membership functions. Parameters of the membership functions are adjusted by a RCGA to have same dynamic characteristics with nonlinear system of a container crane. Simulations are given to illustrate the performance of T-S fuzzy model.