Spherical Functions on Fuzzy Lie Group

Let G be a locally compact Lie group and its Lie algebra. We consider a fuzzy analogue of G, denoted by called a fuzzy Lie group. Spherical functions on are constructed and a version of the existence result of the Helgason-spherical function on G is then established on .

Simulation of Haloxylon Ammodendron Stand Basic Diameter Structure Based on Fuzzy Distribution Function

Based on the investigation data of 12 Haloxylon ammodendron plots in the south edge of Gurbantunggut Desert, Fuzzy distribution was introduced into the study of Haloxylon ammodendron base diameter structure fitting according to the consistency between the characteristics of Fuzzy distribution function and the distribution series of cumulative percentage of stand base diameter, and the fitting precision and effect of Fuzzy distribution function were discussed. The root mean square error RMSE and determination coefficient R2 values showed that Fuzzy-Γ1, Fuzzy-Γ2, Fuzzy-Γ3, Fuzzy-Γ4 had good fitting performance, among which Fuzzy-Γ1 had relatively high fitting precision, and its parameters were closely related to stand age and density, Fuzzy-Γ2 distribution function was the second, and Fuzzy-Γ4 distribution function had the worst fitting effect. By introducing a parameter c from the similarity of four distribution function formulas, a generalized Fuzzy distribution function Fuzzy-Γ5 is obtained. This function shows the highest fitting accuracy. Most of the values of parameter c are near 1 or 2, which shows that the diameter distribution is mainly approximate to Fuzzy-Γ1 and Fuzzy-Γ2.

Stability analysis for affine fuzzy system based on fuzzy Lyapunov functions

An analysis method based on the fuzzy Lyapunov functions is presented to analyze the stability of the continuous affine fuzzy systems. First, a method is introduced to deal with the consequent part of the fuzzy local model. Thus, the stability analysis method of the homogeneous fuzzy system can be used for reference. Stability conditions are derived in terms of linear matrix inequalities based on the fuzzy Lyapunov functions and the modified common Lyapunov functions, respectively. The results demonstrate that the stability result based on the fuzzy Lyapunov functions is less conservative than that based on the modified common Lyapunov functions via numerical examples. Compared with the method which does not expand the consequent part, the proposed method is simpler but its feasible region is reduced. Finally, in order to expand the application of the fuzzy Lyapunov functions, the piecewise fuzzy Lyapunov function is proposed, which can be used to analyze the stability for triangular or trapezoidal membership functions and obtain the stability conditions. A numerical example validates the effectiveness of the proposed approach.

Fuzzy smooth support vector machine with different smooth functions

Smooth support vector machine （SSVM） changs the normal support vector machine （SVM） into the unconstrained op- timization by using the smooth sigmoid function. The method can be solved under the Broyden-Fletcher-Goldfarb-Shanno （BFGS） algorithm and the Newdon-Armijio （NA） algorithm easily, however the accuracy of sigmoid function is not as good as that of polyno- mial smooth function. Furthermore, the method cannot reduce the influence of outliers or noise in dataset. A fuzzy smooth support vector machine （FSSVM） with fuzzy membership and polynomial smooth functions is introduced into the SVM. The fuzzy member- ship considers the contribution rate of each sample to the optimal separating hyperplane and makes the optimization problem more accurate at the inflection point. Those changes play a positive role on trials. The results of the experiments show that those FSSVMs can obtain a better accuracy and consume the shorter time than SSVM and lagrange support vector machine （LSVM）.

Semi-Continuity of Complex Fuzzy Functions

This paper introduces the concept of semi-continuity of complex fuzzy functions, and discusses some of their elementary properties, such as the sum of two complex fuzzy functions of type I upper (lower) semi-continuity is type I upper (lower) semi-continuous, and the opposite of complex fuzzy functions of type I upper (lower) semi-continuity is type I lower (upper) semi-continuous. Based on some assumptions on two complex fuzzy functions of type I upper (lower) semi-continuity, it is shown that their product is type I upper (lower) semi-continuous. The paper also investigates the convergence of complex fuzzy functions. In particular, sign theorem, boundedness theorem, and Cauchy's criterion for convergence are kept. In this paper the metrics introduced by Zhang Guangquan was used. This paper gives a contribution to the study of complex fuzzy functions, and extends the corresponding work of Zhang Guangquan.

Energy Price Forecasting Through Novel Fuzzy Type-1 Membership Functions

Electricity price forecasting is a subset of energy and power forecasting that focuses on projecting commercial electricity market present and future prices.Electricity price forecasting have been a critical input to energy corporations’strategic decision-making systems over the last 15 years.Many strategies have been utilized for price forecasting in the past,however Artificial Intelligence Techniques(Fuzzy Logic and ANN)have proven to be more efficient than traditional techniques(Regression and Time Series).Fuzzy logic is an approach that uses membership functions(MF)and fuzzy inference model to forecast future electricity prices.Fuzzy c-means(FCM)is one of the popular clustering approach for generating fuzzy membership functions.However,the fuzzy c-means algorithm is limited to producing only one type of MFs,Gaussian MF.The generation of various fuzzy membership functions is critical since it allows for more efficient and optimal problem solutions.As a result,for the best and most improved results for electricity price forecasting,an approach to generate multiple type-1 fuzzy MFs using FCM algorithm is required.Therefore,the objective of this paper is to propose an approach for generating type-1 fuzzy triangular and trapezoidal MFs using FCM algorithm to overcome the limitations of the FCM algorithm.The approach is used to compute and improve forecasting accuracy for electricity prices,where Australian Energy Market Operator(AEMO)data is used.The results show that the proposed approach of using FCM to generate type-1 fuzzy MFs is effective and can be adopted.

RBF neural network regression model based on fuzzy observations

A fuzzy observations-based radial basis function neural network （FORBFNN） is presented for modeling nonlinear systems in which the observations of response are imprecise but can be represented as fuzzy membership functions. In the FORBFNN model, the weight coefficients of nodes in the hidden layer are identified by using the fuzzy expectation-maximization （ EM ） algorithm, whereas the optimal number of these nodes as well as the centers and widths of radial basis functions are automatically constructed by using a data-driven method. Namely, the method starts with an initial node, and then a new node is added in a hidden layer according to some rules. This procedure is not terminated until the model meets the preset requirements. The method considers both the accuracy and complexity of the model. Numerical simulation results show that the modeling method is effective, and the established model has high prediction accuracy.

Stabilization design for continuous-time Takagi-Sugeno fuzzy systems based on new relaxed conditions

This paper deals with the problem of stabilization design for a class of continuous-time Takagi-Sugeno（T-S）fuzzy systems.New stabilization conditions are derived based on a relaxed approach in which both fuzzy Lyapunov functions and staircase membership functions are used.Through the staircase membership functions approximating the continuous membership functions of the given fuzzy model,the information of the membership functions can be brought into the stabilization design of the fuzzy systems,thereby significantly reducing the conservativeness in the existing stabilization conditions of the T-S fuzzy systems.Unlike some previous fuzzy Lyapunov function approaches reported in the literature,the proposed stabilization conditions do not depend on the time-derivative of the membership functions that may be the main source of conservatism when considering fuzzy Lyapunov functions analysis.Moreover,conditions for the solvability of the controller design are written in the form of linear matrix inequalities,but not bilinear matrix inequalities,which are easier to be solved by convex optimization techniques.A simulation example is given to demonstrate the validity of the proposed approach.

Similarity measure design and similarity computation for discrete fuzzy data

The similarity computations for fuzzy membership function pairs were carried out.Fuzzy number related knowledge was introduced,and conventional similarity was compared with distance based similarity measure.The usefulness of the proposed similarity measure was verified.The results show that the proposed similarity measure could be applied to ordinary fuzzy membership functions,though it was not easy to design.Through conventional results on the calculation of similarity for fuzzy membership pair,fuzzy membership-crisp pair and crisp-crisp pair were carried out.The proposed distance based similarity measure represented rational performance with the heuristic point of view.Furthermore,troublesome in fuzzy number based similarity measure for abnormal universe of discourse case was discussed.Finally,the similarity measure computation for various membership function pairs was discussed with other conventional results.

Fuzzy entropy design for non convex fuzzy set and application to mutual information

Fuzzy entropy was designed for non convex fuzzy membership function using well known Hamming distance measure.The proposed fuzzy entropy had the same structure as that of convex fuzzy membership case.Design procedure of fuzzy entropy was proposed by considering fuzzy membership through distance measure,and the obtained results contained more flexibility than the general fuzzy membership function.Furthermore,characteristic analyses for non convex function were also illustrated.Analyses on the mutual information were carried out through the proposed fuzzy entropy and similarity measure,which was also dual structure of fuzzy entropy.By the illustrative example,mutual information was discussed.

Fuzzy-GA based algorithm for optimal placement and sizing of distribution static compensator (DSTATCOM) for loss reduction of distribution network considering reconfiguration

This work presents a fuzzy based methodology for distribution system feeder reconfiguration considering DSTATCOM with an objective of minimizing real power loss and operating cost. Installation costs of DSTATCOM devices and the cost of system operation, namely, energy loss cost due to both reconfiguration and DSTATCOM placement, are combined to form the objective function to be minimized. The distribution system tie switches, DSTATCOM location and size have been optimally determined to obtain an appropriate operational condition. In the proposed approach, the fuzzy membership function of loss sensitivity is used for the selection of weak nodes in the power system for the placement of DSTATCOM and the optimal parameter settings of the DFACTS device along with optimal selection of tie switches in reconfiguration process are governed by genetic algorithm(GA). Simulation results on IEEE 33-bus and IEEE 69-bus test systems concluded that the combinatorial method using DSTATCOM and reconfiguration is preferable to reduce power losses to 34.44% for 33-bus system and to 45.43% for 69-bus system.

Delayed recognition:recent developments and a proposal to study this phenomenon as a fuzzy concept

Purpose:New developments in the study of delayed recognition are discussed.Design/methodology/approach:Based on these new developments a method is proposed to characterize delayed recognition as a fuzzy concept.Findings:A benchmark value of 0.333 corresponding with linear growth is obtained.Moreover,a case is discovered in which an expert found delayed recognition several years before citation analysis could discover this phenomenon.Research limitations:As all citation studies also this one is database dependent.Practical implications:Delayed recognition is turned into a fuzzy concept.Originality/value:The article presents a new way of studying delayed recognition.

Maximum power point tracking of a photovoltaic energy system using neural fuzzy techniques

In order to improve the output efficiency of a photovoltaic （PV） energy system, the real-time maximum power point （MPP） of the PV array should be tracked closely. The non-linear and time-variant characteristics of the photovoltaic array and the non-linear and non-minimum phase characteristics of a boost converter make it difficult to track the MPP as in traditional control strategies. A neural fuzzy controller （NFC） in conjunction with the reasoning capability of fuzzy logical systems and the learning capability of neural networks is proposed to track the MPP in this paper. A gradient estimator based on a radial basis function neural network is developed to provide the reference information to the NFC. With a derived learning algorithm, the parameters of the NFC are updated adaptively. Experimental results show that, compared with the fuzzy logic control algorithm, the proposed control algorithm provides much better tracking performance.

Derivatives and differentials for multiplicative intuitionistic fuzzy information

By using the unsymmetrical scale instead of the symmetrical scale,the multiplicative intuitionistic fuzzy sets(MIFSs) reflect our intuition more objectively.Each element in a MIFS is expressed by an ordered pair which is called a multiplicative intuitionistic fuzzy number(MIFN)and is based on the unbalanced scale(i.e.,Saaty’s 1-9 scale).In order to describe the derivatives and differentials for multiplicative intuitionistic fuzzy information more comprehensively,in this paper,we firstly propose two new basic operational laws for MIFNs,which are the subtraction law and the division law.Secondly,we describe the change values of MIFNs when considering them as variables,classify these change values based on the basic operational laws for MIFNs,and depict the convergences of sequences of MIFNs by the subtraction and division laws.Finally,we focus on the multiplicative intuitionistic fuzzy functions and derive some basic results related to their continuities,derivatives and differentials,and also give their application in selecting the configuration of a computer.

Novel Active Contour Model for Image Segmentation Based on Local Fuzzy Gaussian Distribution Fitting

A novel active contour model is proposed, which incorporates local information distributions in a fuzzy energy function to effectively deal with the intensity inhomogeneity. Moreover, the proposed model is convex with respect to the variable which is used for extracting the contour. This makes the model independent on the initial condition and suitable for an automatic segmentation. Furthermore, the energy function is minimized in a computationally efficient way by calculating the fuzzy energy alterations directly. Experiments are carried out to prove the performance of the proposed model over some existing methods. The obtained results confirm the efficiency of the method.

Research on trend prediction of component stock in fuzzy time series based on deep forest

With the continuous development of machine learning and the increasing complexity of financial data analysis,it is more popular to use models in the field of machine learning to solve the hot and difficult problems in the financial industry.To improve the effectiveness of stock trend prediction and solve the problems in time series data processing,this paper combines the fuzzy affiliation function with stock-related technical indicators to obtain nominal data that can widely reflect the constituent stocks in the case of time series changes by analysing the S&P 500 index.Meanwhile,in order to optimise the current machine learning algorithm in which the setting and adjustment of hyperparameters rely too much on empirical knowledge,this paper combines the deep forest model to train the stock data separately.The experimental results show that(1)the accuracy of the extreme random forest and the accuracy of the multi-grain cascade forest are both higher than that of the gated recurrent unit(GRU)model when the un-fuzzy index-adjusted dataset is used as features for input,(2)the accuracy of the extreme random forest and the accuracy of the multigranular cascade forest are improved by using the fuzzy index-adjusted dataset as features for input,(3)the accuracy of the fuzzy index-adjusted dataset as features for inputting the extreme random forest is improved by 18.89% compared to that of the un-fuzzy index-adjusted dataset as features for inputting the extreme random forest and(4)the average accuracy of the fuzzy index-adjusted dataset as features for inputting multi-grain cascade forest increased by 5.67%.

Stability and stabilization of discrete T-S fuzzy time-delay system based on maximal overlapped-rules group

The problems of stability and stabilization for the discrete Takagi-Sugeno（T-S） fuzzy time-delay system are investigated.By constructing a discrete piecewise Lyapunov-Krasovskii function（PLKF） in each maximal overlapped-rules group（MORG）,a new sufficient stability condition for the open-loop discrete T-S fuzzy time-delay system is proposed and proved.Then the systematic design of the fuzzy controller is investigated via the parallel distributed compensation control scheme,and a new stabilization condition for the closed-loop discrete T-S fuzzy time-delay system is proposed.The above two sufficient conditions only require finding common matrices in each MORG.Compared with the common Lyapunov-Krasovskii function（CLKF） approach and the fuzzy Lyapunov-Krasovskii function（FLKF） approach,these proposed sufficient conditions can not only overcome the defect of finding common matrices in the whole feasible region but also largely reduce the number of linear matrix inequalities to be solved.Finally,simulation examples show that the proposed PLKF approach is effective.

On the No-arbitrage Principle and Option Pricing in a Fuzzy Market

Discuss the no-arbitrage principle in a fuzzy market and present a model for pricing an option. Get a fuzzy price for the contingent claim in a market involving fuzzy elements, whose level set can be seen as the possible price level interval with given belief degree. Use fuzzy densit） function and fuzzy mean as evidence for such model. Also give an example for comparing the result of the model in this article and that of another pricing method.

Modeling Multisource-heterogeneous Information Based on Random Set and Fuzzy Set Theory

This paper presents a new idea, named as modeling multisensor-heterogeneous information, to incorporate the fuzzy logic methodologies with mulitsensor-multitarget system under the framework of random set theory. Firstly, based on strong random set and weak random set, the unified form to describe both data (unambiguous information) and fuzzy evidence (uncertain information) is introduced. Secondly, according to signatures of fuzzy evidence, two Bayesian-markov nonlinear measurement models are proposed to fuse effectively data and fuzzy evidence. Thirdly, by use of "the models-based signature-matching scheme", the operation of the statistics of fuzzy evidence defined as random set can be translated into that of the membership functions of relative point state variables. These works are the basis to construct qualitative measurement models and to fuse data and fuzzy evidence.

FUZZY APPROACHING SET AND FUZZYAPPROACHING FUNCTIONAL MAPPING

By establishing the concepts of fuzzy approaching set and fuzzy approaching functional mapping and making research on them, a new method for time series prediction is introduced.