Fuzzy C-regression Research Articles

On robust fuzzy c-regression models

One of the most popular clustering methods based on minimization of a criterion function is the fuzzy c-means one. Its generalization by application of hyperplane shaped prototypes of the clusters is known as the Fuzzy C-Regression Models (FCRM) method. Although with this generalization many new applications of clustering emerged, it appeared to be rather sensitive to poor initialization and to the presence of noise and outliers in data. In this paper we introduce a new objective function, using the Huber's M-estimators and the Yager's OWA operators to overcome the disadvantages of the approach considered. We derive and describe an algorithm for minimization of the objective function defined. We have called it the Fuzzy C-Ordered-Regression Models (FCORM) clustering algorithm. The algorithm is compared to a few other important reference ones. To this end experiments on synthetic data with various types of noise and different numbers of outliers are carried out. We investigate the methods performance in the conditions that can be encountered in signal analysis. Large-scale simulations demonstrate the competitiveness and usefulness of the method proposed.

向きに依存した不確実性を考慮したスイッチング回帰モデルと太陽光発電データの分析への応用

Switching regression is a data mining tool for revealing intrinsic non-linear dependencies among exploratory and objective variables, and Fuzzy c-Regression Models (FCRM) is an FCM-type switching regression algorithm. In this paper, FCRM is applied to a residential solar electric power analysis. Because meteorological observation of actual sunshine duration is often underestimated from the actual quantity of solar radiation, we often fail to reveal intrinsic relation among solar electric power and sunshine duration. In this paper, FCRM is modied for handling direction-dependent uncertain observations in an element-wise manner and its applicability is demonstrated with several experimental results.

A type-2 fuzzy expert system based on a hybrid inference method for steel industry

In this paper, a novel type-2 fuzzy expert system for prediction the amount of reagents in desulfurization process of a steel industry in Canada is developed. In this model, the new interval type-2 fuzzy c-regression clustering algorithm for structure identification phase of Takagi–Sugeno (T–S) systems is presented. Gaussian Mixture Model is used to generate partition matrix in clustering algorithm. Then, an interval type-2 hybrid fuzzy system, which is the combination of Mamdani and Sugeno method, is proposed. The new hybrid inference system uses fuzzy disjunctive normal forms and fuzzy conjunctive normal forms for aggregation of antecedents. A statistical test, which uses least square method, is implemented in order to select variables. In order to validate our method, we develop three system modeling techniques and compare the results with our proposed interval type-2 fuzzy hybrid expert system. These techniques are multiple regression, type-1 fuzzy expert system, and interval type-2 fuzzy TSK expert system. For tuning parameters of the system, adaptive-network-based fuzzy inference system is used. Finally, neural network is utilized in order to reduce error of the system. The results show that our proposed method has less error and high accuracy.

A type-2 fuzzy system model for reducing bullwhip effects in supply chains and its application in steel manufacturing

The purpose of this paper is to evaluate and reduce the bullwhip effect in fuzzy environments by means of type-2 fuzzy methodology. In order to reduce the bullwhip effect in a supply chain, we propose a new method for demand forecasting. First, the demand data of a real steel industry in Canada is clustered with an interval type-2 fuzzy c-regression clustering algorithm. Then, a novel interval type-2 fuzzy hybrid expert system is developed for demand forecasting. This system uses Fuzzy Disjunctive Normal Forms (FDNF) and Fuzzy Conjunctive Normal Forms (FCNF) for the aggregation of antecedents. An interval type-2 fuzzy order policy is developed to determine orders in the supply chain. Then, the results of the proposed method are compared with the type-1 fuzzy expert system as well as the type-1 fuzzy time series method in the literature. The results show that the bullwhip effect is significantly reduced; also, the system has less error and high accuracy.

Hydraulic turbine governing system identification using T–S fuzzy model optimized by chaotic gravitational search algorithm

Hydraulic turbine governing system (HTGS) is a complicated nonlinear system that controls the frequency and power output of hydroelectric generating unit (HGU). The modeling of HTGS is an important and difficult task, because some components, like hydraulic turbine and governor actuator, are with strong nonlinearity. In this paper, a novel Takagi–Sugeno (T–S) fuzzy model identification method based on chaotic gravitational search algorithm (CGSA) is proposed and applied in the modeling of HTGS. In the proposed method, fuzzy c-regression model clustering algorithm is used to partition the input space and identify the coarse antecedent membership function (MF) parameters at first. And then, a novel CGSA is proposed to search better MF parameters around the coarse results, in which chaotic search has been embedded in the iteration of basic GSA to search and replace the current best solution of GSA. The performance of the proposed fuzzy model identification method is validated by benchmark problems, and the results show that the accuracies of identified models have been improved significantly compared with the other existing models. Finally, the proposed approach has been applied to approximate the dynamic behaviors of HTGS of a HGU in a hydropower station of Jiangxi Province of China. The experimental results show that our approach can identify the HTGS satisfactorily with acceptable accuracy.

Hinging hyperplane based regression tree identified by fuzzy clustering and its application

Hierarchical fuzzy modeling techniques have great advantage since model accuracy and complexity can be easily controlled thanks to the transparent model structures. A novel tool for regression tree identification is proposed based on the synergistic combination of fuzzy c-regression clustering and the concept of hierarchical modeling. In a special case (c=2), fuzzy c-regression clustering can be used for identification of hinging hyperplane models. The proposed method recursively identifies a hinging hyperplane model that contains two linear submodels by partitioning operating region of one local linear model resulting a binary regression tree. Novel measures of model performance and complexity are developed to support the analysis and building of the proposed special model structure. Effectiveness of proposed model is demonstrated by benchmark regression datasets. Examples also demonstrate that the proposed model can effectively represent nonlinear dynamical systems. Thanks to the piecewise linear model structure the resulted regression tree can be easily utilized in model predictive control. A detailed application example related to the model predictive control of a water heater demonstrate that the proposed framework can be effectively used in modeling and control of dynamical systems.

A novel fuzzy c-regression model algorithm using a new error measure and particle swarm optimization

Abstract This paper presents a new algorithm for fuzzy c-regression model clustering. The proposed methodology is based on adding a second regularization term in the objective function of a Fuzzy C-Regression Model (FCRM) clustering algorithm in order to take into account noisy data. In addition, a new error measure is used in the objective function of the FCRM algorithm, replacing the one used in this type of algorithm. Then, particle swarm optimization is employed to finally tune parameters of the obtained fuzzy model. The orthogonal least squares method is used to identify the unknown parameters of the local linear model. Finally, validation results of two examples are given to demonstrate the effectiveness and practicality of the proposed algorithm.

SEQUENTIAL EXTRACTION OF FUZZY REGRESSION MODELS: LEAST SQUARES AND LEAST ABSOLUTE DEVIATIONS

Fuzzy c-regression models are known to be useful in real applications, but there are two drawbacks: strong dependency on the predefined number of clusters and sensitiveness against outliers or noises. To avoid these drawbacks, we propose sequential fuzzy regression models based on least absolute deviations which we call SFCRMLAD. This algorithm sequentially extracts one cluster at a time using a method of noise-detection, enabling the automatic determination of clusters and having robustness to noises. We compare this method with the ordinary fuzzy c-regression models based on least squares, fuzzy c-regression models based on least absolute deviations, and moreover sequential fuzzy regression models based on least squares. For this purpose we use a two-dimensional illustrative example whereby characteristics of the four methods are made clear. Moreover a simpler and more efficient algorithm of SFCRMLAD can be used for scalar input and output variables, while a general algorithm of SFCRMLAD uses linear programming solutions for multivariable input. By using the above example, we compare efficiency of different algorithms.

Comparison between Multiple Linear Regression and Fuzzy C-Regression Models towards Scale of Health in ICU

The multiple linear regression (MLR) model is well-known in analyzing linear model. Whereas, the new technique in clustering data, the fuzzy c-regression models (FCRM) are being widely used in analyzing the nonlinear model. The FCRM models are tested on simulated data and the FCRM models can approximate the given nonlinear system with a higher accuracy. A case study in scale of health at intensive care unit (ICU) using the two methods of modelling as mentioned above was carried out. The comparison between the MLR and FCRM models were done to find the better model by using the mean square error (MSE). After comparing the two models, it was found that the FCRM models appeared to be the better model, having a lower MSE. The MSE for MLR model is 498.29 whereas the MSE for FCRM models is 97.366. Keywords: Multiple linear regression (MLR) model; fuzzy c-regression models (FCRM); mean square error (MSE). 2010 Mathematics Subject Classification 62H12; 62M20.

A type-2 fuzzy c-regression clustering algorithm for Takagi–Sugeno system identification and its application in the steel industry

This paper proposes a new type-2 fuzzy c-regression clustering algorithm for the structure identification phase of Takagi–Sugeno (T–S) systems. We present uncertainties with fuzzifier parameter “ m”. In order to identify the parameters of interval type-2 fuzzy sets, two fuzzifiers “ m 1 ” and “ m 2 ” are used. Then, by utilizing these two fuzzifiers in a fuzzy c-regression clustering algorithm, the interval type-2 fuzzy membership functions are generated. The proposed model in this paper is an extended version of a type-1 FCRM algorithm [25], which is extended to an interval type-2 fuzzy model. The Gaussian Mixture model is used to create the partition matrix of the fuzzy c-regression clustering algorithm. Finally, in order to validate the proposed model, several numerical examples are presented. The model is tested on a real data set from a steel company in Canada. Our computational results show that our model is more effective for robustness and error reduction than type-1 NFCRM and the multiple-regression.

Fuzzy Model Identification With Enhanced Validity Criterion for Mechanical System Design

Model identification for machine system design, design optimization, and manufacturing planning is an important method that has high prediction accuracy and could become an essential stage in practical applications. In this paper, an effective fuzzy model identification algorithm for mechanical system design is developed. First, a fuzzy c-regression model clustering algorithm, in which hyperplane-shaped cluster representatives are utilized to provide a mathematical tool to partition the input–output space reasonably, is introduced. Then, an enhanced cluster validity criterion, in which the structural information hidden in the clusters can be reflected in the index, is proposed to choose the optimal number of clusters. In the proposed architecture, an improved Takagi–Sugeno fuzzy model is proposed to describe the system. Two illustrative examples under various conditions are provided, and their performances are indicated in comparison with other published works. In comparison to these fuzzy works, the proposed fuzzy model identification requires fewer fuzzy rules and a shorter tuning time.

Identification of the linear parts of nonlinear systems for fuzzy modeling

In direct approach to fuzzy modeling, structure identification is one of the most critical tasks. In modeling the nonlinear system, this fact is more crucial. In this paper, a new hybrid method is proposed to cluster the data located in the linear parts on the nonlinear systems. The proposed method can partition the input–output data in two groups: data located in the linear parts and data in the extrema. It is shown that the first group of data is suitable to be clustered by Fuzzy C-Regression Model (FCRM) clustering algorithm and the second group by Fuzzy C-Means (FCM). Then, based on the above findings, a new hybrid clustering algorithm is proposed. Finally, the proposed approach is tested and validated by several numerical examples of nonlinear functions.

Fuzzy functions

We first review the development of fuzzy system models from 'fuzzy rule bases' proposed by Zadeh (1965, 1975) to 'fuzzy functions' proposed by Turksen (2008) and further developed by Celikyilmaz and Turksen (2008a; 2008b; 2008c; 2009a; 2009b) in a variety of versions. Next, we also review a complementary development of 'fuzzy C-regression model' (FCRM) proposed by Hathaway and Bezdek (1993), as well as a 'combined FCM and FCRM algorithms' proposed by H?ppner and Klawonn (2003). Finally, various experimental assessments of such fuzzy function models are presented.

A new T–S fuzzy-modeling approach to identify a boiler–turbine system

In order to build accurate model for complicated nonlinear system in engineering, like boiler–turbine system, a novel fuzzy-modeling approach is proposed, which is based on a new fuzzy c-regression model (NFCRM) clustering algorithm and is able to determine the right number of rules automatically. In this method, NFCRM is applied to build the fuzzy structure and then identify the premise parameters; a new criterion is proposed to auto determine the number of rules in fuzzy modeling; after the fuzzy rules have been decided, orthogonal least square is exploited to identify the consequent parameters. Simulation examples are given to demonstrate the validity of the proposed modeling approach, and the results show the new approach is very efficient with high accuracy. Finally, the new approach is applied in fuzzy modeling of a typical boiler–turbine system successfully.

T–S fuzzy model identification based on a novel fuzzy c-regression model clustering algorithm

This paper proposes a novel approach for identification of Takagi–Sugeno (T–S) fuzzy model, which is based on a new fuzzy c-regression model (FCRM) clustering algorithm. The clustering prototype in fuzzy space partition is hyper-plane, so FCRM clustering technique is more suitable to be applied in premise parameters identification of T–S fuzzy model. A new FCRM clustering algorithm (NFCRMA) is presented, which is deduced from the fuzzy clustering objective function of FCRM with Lagrange multiplier rule, possessing integrative and concise structure. The proposed approach consists mainly of two steps: premise parameter identification and consequent parameter identification. The NFCRMA is utilized to partition the input–output data and identify the premise parameters, which can discover the real structure of the training data; on the other hand, orthogonal least square is exploited to identify the consequent parameters. Finally, some examples are given to verify the validity of the proposed modeling approach, and the results show the new approach is very efficient and of high accuracy.

Hybrid robust approach for TSK fuzzy modeling with outliers

This study proposes a hybrid robust approach for constructing Takagi–Sugeno–Kang (TSK) fuzzy models with outliers. The approach consists of a robust fuzzy C-regression model (RFCRM) clustering algorithm in the coarse-tuning phase and an annealing robust back-propagation (ARBP) learning algorithm in the fine-tuning phase. The RFCRM clustering algorithm is modified from the fuzzy C-regression models (FCRM) clustering algorithm by incorporating a robust mechanism and considering input data distribution and robust similarity measure into the FCRM clustering algorithm. Due to the use of robust mechanisms and the consideration of input data distribution, the fuzzy subspaces and the parameters of functions in the consequent parts are simultaneously identified by the proposed RFCRM clustering algorithm and the obtained model will not be significantly affected by outliers. Furthermore, the robust similarity measure is used in the clustering process to reduce the redundant clusters. Consequently, the RFCRM clustering algorithm can generate a better initialization for the TSK fuzzy models in the coarse-tuning phase. Then, an ARBP algorithm is employed to obtain a more precise model in the fine-tuning phase. From our simulation results, it is clearly evident that the proposed robust TSK fuzzy model approach is superior to existing approaches in learning speed and in approximation accuracy.

A New Cluster Validity Criterion for Fuzzy C-Regression Models Clustering and Its Application to Fuzzy Model Identification

In this paper, a new cluster validity criterion for fuzzy c-regression models (FCRM) clustering algorithm with affine linear functional cluster representatives is proposed. The proposed cluster validity criterion calculates the overall compactness and separateness of the FCRM partition and then determines the appropriate number of clusters. Besides, its application to fuzzy model identification is discussed. A TS fuzzy model identification algorithm is proposed to extract compact number of IF-THEN rules from data. Two simulation examples are provided to demonstrate the potential of the proposed cluster validity criterion and the accuracy of the constructed T-S fuzzy model.

Affine Takagi-Sugeno fuzzy modelling algorithm by fuzzy c -regression models clustering with a novel cluster validity criterion

An effective approach is developed to establish affine Takagi-Sugeno (T-S) fuzzy model for a given nonlinear system from its input–output data. Firstly, the fuzzy c-regression model (FCRM) clustering technique is applied to partition the product space of the given input–output data into hyper-plan-shaped clusters. Each cluster is essentially a basis of the fuzzy rule that describes the system behaviour, and the number of clusters is just the number of fuzzy rules. Particularly, a novel cluster validity criterion for FCRM is set up to choose the appropriate number of clusters (rules). Once the number of clusters is determined, the consequent parameters of each IF-THEN rule are directly obtained from the functional cluster representatives (affine linear functions). The antecedent fuzzy sets of each IF-THEN fuzzy rule are acquired by projecting the fuzzy partitions matrix U onto the axes of individual antecedent variable to obtain point-wise defined fuzzy sets and to approximate these point-wise defined fuzzy sets by normal bell-shaped membership functions. Additionally, a check and repartition algorithm is suggested to prevent the inappropriate premise structure where separate regions of data shared the same regression model. Finally, the gradient descent algorithm is included to adjust the fuzzy model precisely. An affine T-S fuzzy model with compact IF-THEN rules could thus be generated systematically. Several simulation examples are provided to demonstrate the accuracy and effectiveness of the affine T-S fuzzy modelling algorithm.

A time-domain-constrained fuzzy clustering method and its application to signal analysis

This paper introduces a new fuzzy clustering method with time-domain-constraints which is used to signal analysis. Proposed method makes it possible to include natural constraints for signal analysis using fuzzy clustering, that is, the neighboring samples of signal belong to the same cluster. This method can be called time-domain-constrained fuzzy clustering. This paper introduces two approaches to include the above kind of constraints. The first approach leads to the time-domain-constrained fuzzy c-regression models method. The second approach leads to the ɛ -insensitive version of the above method, which results in additional robustness for outliers and non-Gaussian noise. Finally, simulations on synthetic as well as real-life signals are realized to evaluate the performance of the time-domain-constrained fuzzy clustering methods. A comparison with the traditional fuzzy c-regression models is also made. Large-scale simulations demonstrate the competitiveness of the proposed methods for signal analysis with respect to the traditional fuzzy clustering methods.

Epsilon-insensitive fuzzy c-regression models: introduction to epsilon-insensitive fuzzy modeling.

This paper introduces a new epsilon-insensitive fuzzy c-regression models (epsilonFCRM), that can be used in fuzzy modeling. To fit these regression models to real data, a weighted epsilon-insensitive loss function is used. The proposed method make it possible to exclude an intrinsic inconsistency of fuzzy modeling, where crisp loss function (usually quadratic) is used to match real data and the fuzzy model. The epsilon-insensitive fuzzy modeling is based on human thinking and learning. This method allows easy control of generalization ability and outliers robustness. This approach leads to c simultaneous quadratic programming problems with bound constraints and one linear equality constraint. To solve this problem, computationally efficient numerical method, called incremental learning, is proposed. Finally, examples are given to demonstrate the validity of introduced approach to fuzzy modeling.