Robust TSK fuzzy modeling for function approximation with outliers

The Takagi-Sugeno-Kang (TSK) type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications. Most approaches for modeling TSK fuzzy rules define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions. Besides, training data sets algorithms often contain outliers, which seriously affect least-square error minimization clustering and learning algorithms. A robust TSK fuzzy modeling approach is presented. In the approach, a clustering algorithm termed as robust fuzzy regression agglomeration (RFRA) is proposed to define fuzzy subspaces in a fuzzy regression manner with robust capability against outliers. To obtain a more precision model, a robust fine-tuning algorithm is then employed. Various examples are used to verify the effectiveness of the proposed approach. From the simulation results, the proposed robust TSK fuzzy modeling indeed showed superior performance over other approaches     

Hybrid robust support vector machines for regression with outliers

In this study, a hybrid robust support vector machine for regression is proposed to deal with training data sets with outliers. The proposed approach consists of two stages of strategies. The first stage is for data preprocessing and a support vector machine for regression is used to filter out outliers in the training data set. Since the outliers in the training data set are removed, the concept of robust statistic is not needed for reducing the outliers’ effects in the later stage. Then, the training data set except for outliers, called as the reduced training data set, is directly used in training the non-robust least squares support vector machines for regression (LS-SVMR) or the non-robust support vector regression networks (SVRNs) in the second stage. Consequently, the learning mechanism of the proposed approach is much easier than that of the robust support vector regression networks (RSVRNs) approach and of the weighted LS-SVMR approach. Based on the simulation results, the performance of the proposed approach with non-robust LS-SVMR is superior to the weighted LS-SVMR approach when the outliers exist. Moreover, the performance of the proposed approach with non-robust SVRNs is also superior to the RSVRNs approach.     

Bayesian zero-order TSK fuzzy system modeling

Different from the existing TSK fuzzy system modeling methods, a novel zero-order TSK fuzzy modeling method called Bayesian zero-order TSK fuzzy system (B-ZTSK-FS) is proposed from the perspective of Bayesian inference in this paper. The proposed method B-ZTSK-FS constructs zero-order TSK fuzzy system by using the maximum a posteriori (MAP) framework to maximize the corresponding posteriori probability. First, a joint likelihood model about zero-order TSK fuzzy system is defined to derive a new objective function which can assure that both antecedents and consequents of fuzzy rules rather than only their antecedents of the most existing TSK fuzzy systems become interpretable. The defined likelihood model is composed of three aspects: clustering on the training set for antecedents of fuzzy rules, the least squares (LS) error for consequent parameters of fuzzy rules, and a Dirichlet prior distribution for fuzzy cluster memberships which is considered to not only automatically match the “sum-to-one” constraints on fuzzy cluster memberships, but also make the proposed method B-ZTSK-FS scalable for large-scale datasets by appropriately setting the Dirichlet index. This likelihood model indeed indicates that antecedent and consequent parameters of fuzzy rules can be linguistically interpreted and simultaneously optimized by the proposed method B-ZTSK-FS which is based on the MAP framework with the iterative sampling algorithm, which in fact implies that fuzziness and probability can co-jointly work for TSK fuzzy system modeling in a collaborative rather than repulsive way. Finally, experimental results on 28 synthetic and real-world datasets are reported to demonstrate the effectiveness of the proposed method B-ZTSK-FS in the sense of approximation accuracy, interpretability and scalability.     

Hybrid compensation control for affine TSK fuzzy control systems

The paper proposes a way of designing state feedback controllers for affine Takagi-Sugeno-Kang (TSK) fuzzy models. In the approach, by combining two different control design methodologies, the proposed controller is designed to compensate all rules so that the desired control performance can appear in the overall system. Our approach treats all fuzzy rules as variations of a nominal rule and such variations are individually dealt with in a Lyapunov sense. Previous approaches have proposed a similar idea but the variations are dealt with as a whole in a robust control sense. As a consequence, when fuzzy rules are distributed in a wide range, the stability conditions may not be satisfied. In addition, the control performance of the closed-loop system cannot be anticipated in those approaches. Various examples were conducted in our study to demonstrate the effectiveness of the proposed control design approach. All results illustrate good control performances as desired.     

基于熵准则的发酵过程TSK模糊建模

提出了一种基于熵准则函数的TSK模糊系统建模方法.不同于传统的基于MSE经验误差最小的准则函数,该准则函数能从训练样本的整体分布结构来进行参数学习,有效地避免了由于过学习而导致泛化能力差的缺点.将其应用于复杂的发酵过程建模,结果表明新方法具有良好的预测精度、泛化能力和鲁棒性.为解决发酵过程建模中试验数据含有噪音,导致模型预测精度下降的问题提供了一条研究思路.     

Adaptive and robust controller design for uncertain nonlinear systems via fuzzy modeling approach

The issue for designing robust adaptive stabilizing controllers for nonlinear systems in Takagi-Sugeno fuzzy model with both parameter uncertainties and external disturbances is studied in this paper. It is assumed that the parameter uncertainties are norm-bounded and may be of some structure properties and that the external disturbances satisfy matching conditions and, besides, are also norm-bounded, but the bounds of the external disturbances are not necessarily known. Two adaptive controllers are developed based on linear matrix inequality technique and it is shown that the controllers can guarantee the state variables of the closed loop system to converge, globally, uniformly and exponentially, to a ball in the state space with any pre-specified convergence rate. Furthermore, the radius of the ball can also be designed to be as small as desired by tuning the controller parameters. The effectiveness of our approach is verified by its application in the control of a continuous stirred tank reactor.     

TSK fuzzy model with minimal parameters

In this paper, an efficient genetic algorithm (GA) to generate a simple and well defined TSK model is proposed. The approach is derived from the use of the improved Strength Pareto Evolutionary Algorithm (SPEA-2), where the genes of the chromosome are arranged into control genes and parameter genes. These genes are in a hierarchical form so that the control genes can manipulate the parameter genes in a more effective manner. In our approach, we first apply the back-propagation algorithm to optimize the parameters of the model (parameters of membership functions and fuzzy rules), then we apply the SPEA-2 to optimize the number of fuzzy rules, the number of parameters and to fine tune these parameters.Two well-known dynamic benchmarks are used to evaluate the performance of the proposed algorithm. Simulation results show that our modeling approach outperforms some methods proposed in previous work.     

A fuzzy modeling method via Enhanced Objective Cluster Analysis for designing TSK model

This paper proposes a fuzzy modeling method via Enhanced Objective Cluster Analysis to obtain the compact and robust approximate TSK fuzzy model. In our approach, the Objective Cluster Analysis algorithm is introduced. In order to obtain more compact and more robust fuzzy rule prototypes, this algorithm is enhanced by introducing the Relative Dissimilarity Measure and the new consistency criterion to represent the similarity degree between the clusters. By these additional criteria, the redundant clusters caused by iterations are avoided; the subjective influence from human judgment for clustering is weakened. Moreover the clustering results including the number of clusters and the cluster centers are considered as the initial condition of the premise parameters identification. Thus the traditional iteration modeling procedure for determining the number of rules and identifying parameters is changed into one-off modeling, which significantly reduces the burden of computation. Furthermore the decomposition errors and the approximation errors resulted from premise parameters identification by Fuzzy c-Means clustering are decreased. For the consequence parameters identification, the Stable Kalman Filter algorithm is adopted. The performance of the proposed modeling method is evaluated by the example of Box–Jenkins gas furnace. The simulation results demonstrate the power of our model.     

TSK fuzzy modeling for tool wear condition in turning processes: An experimental study

This paper presents an experimental study for turning process in machining by using Takagi-Sugeno-Kang (TSK) fuzzy modeling to accomplish the integration of multi-sensor information and tool wear information. It generates fuzzy rules directly from the input-output data acquired from sensors, and provides high accuracy and high reliability of the tool wear prediction over a wide range of cutting conditions. The experimental results show its effectiveness and satisfactory comparisons relative to other artificial intelligence methods.     

任务间共享和特有结构分解的多任务TSK模糊系统建模

现有的多任务Takagi-Sugeno-Kang (TSK) 模糊建模方法更注重利用任务间的相关性信息,而忽略了单个任务的特殊性。针对此问题,本文提出了一种考虑所有任务之间的共享结构和特有结构的TSK模糊系统多任务建模新方法。该方法将后件参数分解为共享参数和特有参数两个分量,既充分利用了任务间共享信息,又有效地保留了单个任务的特性。最后,本文利用增广拉格朗日乘子法(ALM)求解该最优化问题。实验结果表明,该方法比现有的模型获得了更好的表现。     

Multi-objective optimization of TSK fuzzy models

In this paper we propose a hybrid algorithm to optimize the structure of TSK type fuzzy model using backpropagation (BP) learning algorithm and non-dominated sorting genetic algorithm (NSGA-II). In a first step, BP algorithm is used to optimize the parameters of the model (parameters of membership functions and fuzzy rules). NSGA-II is used in a second phase, to optimize the number of fuzzy rules and to fine tune the parameters. A well known benchmark is used to evaluate performances of the proposed modelling approach, and compare it with other modelling approaches.     

Hybrid approach for the design of robust control systems

This paper addresses the problem of robust controller design for stable finite dimensional linear time-invariant systems (stable FDLTIS) within the framework of model reference schemes. The proposed methodology applies to both continuous and discrete-time systems and employs two distinct design stages: (1) design of the feedforward controller and (2) design of the feedback filter. Algebraic and topological concepts are utilized to tackle several important design issues including stability, performance and robustness. Finally, two examples are given to demonstrate the power of the design method.     

CPBUM neural networks for modeling with outliers and noise

In this study, CPBUM neural networks with annealing robust learning algorithm (ARLA) are proposed to improve the problems of conventional neural networks for modeling with outliers and noise. In general, the obtained training data in the real applications maybe contain the outliers and noise. Although the CPBUM neural networks have fast convergent speed, these are difficult to deal with outliers and noise. Hence, the robust property must be enhanced for the CPBUM neural networks. Additionally, the ARLA can be overcome the problems of initialization and cut-off points in the traditional robust learning algorithm and deal with the model with outliers and noise. In this study, the ARLA is used as the learning algorithm to adjust the weights of the CPBUM neural networks. It tunes out that the CPBUM neural networks with the ARLA have fast convergent speed and robust against outliers and noise than the conventional neural networks with robust mechanism. Simulation results are provided to show the validity and applicability of the proposed neural networks.     

A fuzzy index for detecting spatiotemporal outliers

The detection of spatial outliers helps extract important and valuable information from large spatial datasets. Most of the existing work in outlier detection views the condition of being an outlier as a binary property. However, for many scenarios, it is more meaningful to assign a degree of being an outlier to each object. The temporal dimension should also be taken into consideration. In this paper, we formally introduce a new notion of spatial outliers. We discuss the spatiotemporal outlier detection problem, and we design a methodology to discover these outliers effectively. We introduce a new index called the fuzzy outlier index, FoI, which expresses the degree to which a spatial object belongs to a spatiotemporal neighbourhood. The proposed outlier detection method can be applied to phenomena evolving over time, such as moving objects, pedestrian modelling or credit card fraud.     

GSETSK: a generic self-evolving TSK fuzzy neural network with a novel Hebbian-based rule reduction approach

Takagi–Sugeno–Kang (TSK) fuzzy systems have been widely applied for solving function approximation and regression-centric problems. Existing dynamic TSK models proposed in the literature can be broadly classified into two classes. Class I TSK models are essentially fuzzy systems that are limited to time-invariant environments. Class II TSK models are generally evolving systems that can learn in time-variant environments. This paper attempts to address the issues of achieving compact, up-to-date fuzzy rule bases and interpretable knowledge bases in TSK models. It proposes a novel rule pruning method which is simple, computationally efficient and biologically plausible. This rule pruning algorithm applies a gradual forgetting approach and adopts the Hebbian learning mechanism behind the long-term potentiation phenomenon in the brain. It also proposes a merging approach which is used to improve the interpretability of the knowledge bases. This approach can prevent derived fuzzy sets from expanding too many times to protect their semantic meanings. These two approaches are incorporated into a generic self-evolving Takagi–Sugeno–Kang fuzzy framework (GSETSK) which adopts an online data-driven incremental-learning-based approach.Extensive experiments were conducted to evaluate the performance of the proposed GSETSK against other established evolving TSK systems. GSETSK has also been tested on real world dataset using the high-way traffic flow density and Dow Jones index time series. The results are encouraging. GSETSK demonstrates its fast learning ability in time-variant environments. In addition, GSETSK derives an up-to-date and better interpretable fuzzy rule base while maintaining a high level of modeling accuracy at the same time.     

A two-stage evolutionary process for designing TSK fuzzy rule-basedsystems

Nowadays, fuzzy rule-based systems are successfully applied to many different real-world problems. Unfortunately, relatively few well-structured methodologies exist for designing and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. Takagi-Sugeno-Kang (TSK) fuzzy rule-based systems were enunciated in order to solve this design problem because they are usually identified using numerical data. In this paper we present a two-stage evolutionary process for designing TSK fuzzy rule-based systems from examples combining a generation stage based on a (mu, lambda)-evolution strategy, in which the fuzzy rules with different consequents compete among themselves to form part of a preliminary knowledge base, and a refinement stage in which both the antecedent and consequent parts of the fuzzy rules in this previous knowledge base are adapted by a hybrid evolutionary process composed of a genetic algorithm and an evolution strategy to obtain the final Knowledge base whose rules cooperate in the best possible way. Some aspects make this process different from others proposed until now: the design problem is addressed in two different stages, the use of an angular coding of the consequent parameters that allows us to search across the whole space of possible solutions, and the use of the available knowledge about the system under identification to generate the initial populations of the Evolutionary Algorithms that causes the search process to obtain good solutions more quickly. The performance of the method proposed is shown by solving two different problems: the fuzzy modeling of some three-dimensional surfaces and the computing of the maintenance costs of electrical medium line in Spanish towns. Results obtained are compared with other kind of techniques, evolutionary learning processes to design TSK and Mamdani-type fuzzy rule-based systems in the first case, and classical regression and neural modeling in the second.     

A new approach to fuzzy modeling

This paper proposes a new approach to fuzzy modeling. The suggested fuzzy model can express a given unknown system with a few fuzzy rules as well as Takagi and Sugeno's model (1985), because it has the same structure as that of Takagi and Sugeno's model. It is also as easy to implement as Sugeno and Yasukawa's model (1993) because its identification mimics the simple identification procedure of Sugeno and Yasukawa's model. The suggested algorithm is composed of two steps: coarse tuning and fine tuning. In coarse tuning, fuzzy C-regression model (FCRM) clustering is used, which is a modified version of fuzzy C-means (FCM). In fine tuning, gradient descent algorithm is used to precisely adjust parameters of the fuzzy model instead of nonlinear optimization methods used in other models. Finally, some examples are given to demonstrate the validity of this algorithm     

An energy-gain bounding approach to robust fuzzy identification

A novel method for the robust identification of interpretable fuzzy models, based on the criterion that identification errors are least sensitive to data uncertainties and modelling errors, is suggested. The robustness of identification errors towards unknown disturbances (data uncertainties, modelling errors, etc.) is achieved by bounding (i.e. minimizing) the maximum possible value of energy-gain from disturbances to the identification errors. The solution of energy-gain bounding problem, being robust, shows an improved performance of the identification method. The flexibility of the proposed framework is shown by designing the variable learning rate identification algorithms in both deterministic and stochastic frameworks.     

并行集成具有高可解释的TSK模糊分类器

针对分层Takagi-Sugeno-Kang (TSK)模糊分类器可解释性差,以及当增加或删除一个TSK模糊子分类器时Boosting模糊分类器需要重新训练所有TSK模糊子分类器等问题,提出一种并行集成具有高可解释的TSK模糊分类器EP-Q-TSK.该集成模糊分类器每个TSK模糊子分类器可以使用最小学习机(LLM)被并行地快速构建.作为一种新的集成学习方式,该分类器利用每个TSK模糊子分类器的增量输出来扩展原始验证数据空间,然后采用经典的模糊聚类算法FCM获取一系列代表性中心点,最后利用KNN对测试数据进行分类.在标准UCI数据集上,分别从分类性能和可解释性两方面验证了EP-Q-TSK的有效性.     

A formal approach to fuzzy modeling

A formalism for coding fuzzy models of dynamical systems is presented. It is shown that the formalism is rich enough to capture the performance of arbitrary conventional discrete time dynamical systems whose transition maps are polynomials with rational coefficients. The proof of this fact provides a constructive algorithm for generating fuzzy models to arbitrarily closely approximate an arbitrary map on a compact set. Our modeling formalism highlights the similarities between fuzzy systems and hybrid control systems. We hope to be able to exploit these similarities by extending results from the area of hybrid systems to the fuzzy domain and vice versa