二型模糊系统的规则提取算法

模糊规则提取是建立二型模糊系统需要解决的关键问题.提出一种改进的基于c均值模糊聚类算法(FCM)的二型模糊规则提取方法.该方法借助于二型模糊集主隶属度函数的期望与次隶属度函数值之间的联系,能克服已有算法忽略二型模糊集次隶属度函数对模糊聚类结果的影响.仿真实例表明.该算法能成功地提取二型模糊规则,比FCMV算法具有更好的性能和收敛性.     

Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets

In this study, a new approach for the formation of type-2 membership functions is introduced. The footprint of uncertainty is formed by using rectangular type-2 fuzzy granules and the resulting membership function is named as granular type-2 membership function. This new approach provides more degrees of freedom and design flexibility in type-2 fuzzy logic systems. Uncertainties on the grades of membership functions can be represented independently for any region in the universe of discourse and free of any functional form. So, the designer could produce nonlinear, discontinuous or hybrid membership functions in granular formation and therefore could model any desired discontinuity and nonlinearity. The effectiveness of the proposed granular type-2 membership functions is firstly demonstrated by simulations done on noise corrupted Mackey–Glass time series prediction. Secondly, flexible design feature of granular type-2 membership functions is illustrated by modeling a nonlinear system having dead zone with uncertain system parameters. The simulation results show that type-2 fuzzy logic systems formed by granular type-2 membership functions have more modeling capabilities than the systems using conventional type-2 membership functions and they are more robust to system parameter changes and noisy inputs.     

Genetic learning and performance evaluation of interval type-2 fuzzy logic controllers

Type-2 fuzzy sets, which are characterized by membership functions (MFs) that are themselves fuzzy, have been attracting interest. This paper focuses on advancing the understanding of interval type-2 fuzzy logic controllers (FLCs). First, a type-2 FLC is evolved using Genetic Algorithms (GAs). The type-2 FLC is then compared with another three GA evolved type-1 FLCs that have different design parameters. The objective is to examine the amount by which the extra degrees of freedom provided by antecedent type-2 fuzzy sets is able to improve the control performance. Experimental results show that better control can be achieved using a type-2 FLC with fewer fuzzy sets/rules so one benefit of type-2 FLC is a lower trade-off between modeling accuracy and interpretability.     

Polynomial regression interval-valued fuzzy systems

In recent years, the type-2 fuzzy sets theory has been used to model and minimize the effects of uncertainties in rule-base fuzzy logic system (FLS). In order to make the type-2 FLS reasonable and reliable, a new simple and novel statistical method to decide interval-valued fuzzy membership functions and probability type reduce reasoning method for the interval-valued FLS are developed. We have implemented the proposed non-linear (polynomial regression) statistical interval-valued type-2 FLS to perform smart washing machine control. The results show that our quadratic statistical method is more robust to design a reliable type-2 FLS and also can be extend to polynomial model.     

Bearing condition prediction considering uncertainty: An interval type-2 fuzzy neural network approach

Rolling-element bearings are critical components of rotating machinery. It is important to accurately predict in real-time the health condition of bearings so that maintenance practices can be scheduled to avoid malfunctions or even catastrophic failures. In this paper, an Interval Type-2 Fuzzy Neural Network (IT2FNN) is proposed to perform multi-step-ahead condition prediction of faulty bearings. Since the IT2FNN defines an interval type-2 fuzzy logic system in the form of a multi-layer neural network, it can integrate the merits of each, such as fuzzy reasoning to handle uncertainties and neural networks to learn from data. The interval type-2 fuzzy linguistic process in the IT2FNN enables the system to handle prediction uncertainties, since the type-2 fuzzy sets are such sets whose membership grades are type-1 fuzzy sets that can be used in failure prediction due to the difficult determination of an exact membership function for a fuzzy set. Noisy data of faulty bearings are used to validate the proposed predictor, whose performance is compared with that of a prevalent type-1 condition predictor called Adaptive Neuro-Fuzzy Inference System (ANFIS). The results show that better prediction accuracy can be achieved via the IT2FNN.     

Designing of a type-2 fuzzy logic filter for improving edge-preserving restoration of interlaced-to-progressive conversion

This paper focuses on recently advanced fuzzy models and the application of type-2 fuzzy sets in video deinterlacing. The final goal of the proposed deinterlacing algorithm is to exactly determine an unknown pixel value while preserving the edges and details of the image. To begin, we will discuss some artefacts of spatial, temporal, and spatio-temporal domain deinterlacing methods. In order to address the aforementioned issues, we adopted type-2 fuzzy sets concepts to design a weight evaluating approach. In the proposed method, the upper and lower fuzzy membership functions of the type-2 fuzzy logic filters are derived from the type-1 (or primary) fuzzy membership function. The weights from upper and lower membership functions are considered to be multiplied with the candidate deinterlaced pixels. Experimental results proved that the performance of the proposed method was superior, both objectively and subjectively to other different conventional deinterlacing methods. Moreover, the proposed method preserved the smoothness of the original image edges and produced a high-quality progressive image.     

Type-2 fuzzy multi-intersection traffic signal control with differential evolution optimization

This paper proposes a multi-agent type-2 fuzzy logic control (FLC) method optimized by differential evolution (DE) for multi-intersection traffic signal control. Type-2 fuzzy sets can deal with models’ uncertainties efficiently because of its three-dimensional membership functions, but selecting suitable parameters of membership functions and rule base is not easy. DE is adopted to decide the parameters in the type-2 fuzzy system, as it is easy to understand, simple to implement and possesses low space complexity. In order to avoid the computational complexity, the expert rule base and the parameters of membership functions (MF) are optimized by turns. An eleven-intersection traffic network is studied in which each intersection is governed by the proposed controller. A secondary layer controller is set in every intersection to select the proper phase sequence. Furthermore, the communication among the adjacent intersections is implemented using multi-agent system. Simulation experiments are designed to compare communicative type-2 FLC optimized by DE with type-1 FLC, fixed-time signal control, etc. Experimental results indicate that our proposed method can enhance the vehicular throughput rate and reduce delay, queue length and parking rate efficiently.     

Type-1 and type-2 fuzzy inference systems as integration methods in modular neural networks for multimodal biometry and its optimization with genetic algorithms

We describe in this paper a comparative study between fuzzy inference systems as methods of integration in modular neural networks for multimodal biometry. These methods of integration are based on techniques of type-1 fuzzy logic and type-2 fuzzy logic. Also, the fuzzy systems are optimized with simple genetic algorithms with the goal of having optimized versions of both types of fuzzy systems. First, we considered the use of type-1 fuzzy logic and later the approach with type-2 fuzzy logic. The fuzzy systems were developed using genetic algorithms to handle fuzzy inference systems with different membership functions, like the triangular, trapezoidal and Gaussian; since these algorithms can generate fuzzy systems automatically. Then the response integration of the modular neural network was tested with the optimized fuzzy systems of integration. The comparative study of the type-1 and type-2 fuzzy inference systems was made to observe the behavior of the two different integration methods for modular neural networks for multimodal biometry.     

Type-2 Fuzzy Control for Driving State and Behavioral Decisions of Unmanned Vehicle

In this paper, interval type-2 fuzzy sets, fuzzy comprehensive evaluation and the fuzzy control rules are synthesized to realize the control of unmanned vehicle in driving state and behavioral decisions. Compared to the type-1 fuzzy set, type-2 fuzzy sets have more advantages in handling the model based on uncertainties, linguistic information because the membership functions are fuzzy sets. Different membership functions are established for each factor when the unmanned vehicle is driving at different speed intervals. In addition, a new evaluation method is developed to analyze unmanned vehicle’s driving state. Finally, a set of dynamic fuzzy rules are sorted out, which can be applied to the unmanned vehicle’s behavioral decision-making and provide a new idea to related research.      

Image segmentation based on histogram analysis utilizing the cloud model

Both the cloud model and type-2 fuzzy sets deal with the uncertainty of membership which traditional type-1 fuzzy sets do not consider. Type-2 fuzzy sets consider the fuzziness of the membership degrees. The cloud model considers fuzziness, randomness, and the association between them. Based on the cloud model, the paper proposes an image segmentation approach which considers the fuzziness and randomness in histogram analysis. For the proposed method, first, the image histogram is generated. Second, the histogram is transformed into discrete concepts expressed by cloud models. Finally, the image is segmented into corresponding regions based on these cloud models. Segmentation experiments by images with bimodal and multimodal histograms are used to compare the proposed method with some related segmentation methods, including Otsu threshold, type-2 fuzzy threshold, fuzzy C-means clustering, and Gaussian mixture models. The comparison experiments validate the proposed method.     

A type-2 neural fuzzy system learned through type-1 fuzzy rules and its FPGA-based hardware implementation

This paper first proposes a type-2 neural fuzzy system (NFS) learned through its type-1 counterpart (T2NFS-T1) and then implements the built IT2NFS-T1 in a field-programmable gate array (FPGA) chip. The antecedent part of each fuzzy rule in the T2NFS-T1 uses interval type-2 fuzzy sets, while the consequent part uses a Takagi-Sugeno-Kang (TSK) type with interval combination weights. The T2NFS-T1 uses a simplified type-reduction operation to reduce system training time and hardware implementation cost. Given a training data set, a TSK type-1 NFS is first learned through structure and parameter learning. The built type-1 fuzzy logic system (FLS) is then extended to a type-2 FLS, where highly overlapped type-1 fuzzy sets are merged into interval type-2 fuzzy sets to reduce the total number of fuzzy sets. Finally, the rule consequent and antecedent parameters in the T2NFS-T1 are tuned using a hybrid of the gradient descent and rule-ordered recursive least square (RLS) algorithms. Simulation results and comparisons with various type-1 and type-2 FLSs verify the effectiveness and efficiency of the T2NFS-T1 for system modeling and prediction problems. A new hardware circuit using both parallel-processing and pipeline techniques is proposed to implement the learned T2NFS-T1 in an FPGA chip. The T2NFS-T1 chip reduces the hardware implementation cost in comparison to other type-2 fuzzy chips.     

Type-2 fuzzy Gaussian mixture models

This paper presents a new extension of Gaussian mixture models (GMMs) based on type-2 fuzzy sets (T2 FSs) referred to as T2 FGMMs. The estimated parameters of the GMM may not accurately reflect the underlying distributions of the observations because of insufficient and noisy data in real-world problems. By three-dimensional membership functions of T2 FSs, T2 FGMMs use footprint of uncertainty (FOU) as well as interval secondary membership functions to handle GMMs uncertain mean vector or uncertain covariance matrix, and thus GMMs parameters vary anywhere in an interval with uniform possibilities. As a result, the likelihood of the T2 FGMM becomes an interval rather than a precise real number to account for GMMs uncertainty. These interval likelihoods are then processed by the generalized linear model (GLM) for classification decision-making. In this paper we focus on the role of the FOU in pattern classification. Multi-category classification on different data sets from UCI repository shows that T2 FGMMs are consistently as good as or better than GMMs in case of insufficient training data, and are also insensitive to different areas of the FOU. Based on T2 FGMMs, we extend hidden Markov models (HMMs) to type-2 fuzzy HMMs (T2 FHMMs). Phoneme classification in the babble noise shows that T2 FHMMs outperform classical HMMs in terms of the robustness and classification rate. We also find that the larger area of the FOU in T2 FHMMs with uncertain mean vectors performs better in classification when the signal-to-noise ratio is lower.     

Type-2 fuzzy sets applied to multivariable self-organizing fuzzy logic controllers for regulating anesthesia

In this paper, novel interval and general type-2 self-organizing fuzzy logic controllers (SOFLCs) are proposed for the automatic control of anesthesia during surgical procedures. The type-2 SOFLC is a hierarchical adaptive fuzzy controller able to generate and modify its rule-base in response to the controller's performance. The type-2 SOFLC uses type-2 fuzzy sets derived from real surgical data capturing patient variability in monitored physiological parameters during anesthetic sedation, which are used to define the footprint of uncertainty (FOU) of the type-2 fuzzy sets. Experimental simulations were carried out to evaluate the performance of the type-2 SOFLCs in their ability to control anesthetic delivery rates for maintaining desired physiological set points for anesthesia (muscle relaxation and blood pressure) under signal and patient noise. Results show that the type-2 SOFLCs can perform well and outperform previous type-1 SOFLC and comparative approaches for anesthesia control producing lower performance errors while using better defined rules in regulating anesthesia set points while handling the control uncertainties. The results are further supported by statistical analysis which also show that zSlices general type-2 SOFLCs are able to outperform interval type-2 SOFLC in terms of their steady state performance.     

An evolving recurrent interval type-2 intuitionistic fuzzy neural network for online learning and time series prediction

The prediction of time series has both the theoretical value and practical significance in reality. However, since the high nonlinear and noises in the time series, it is still an open problem to tackle with the uncertainties and fuzziness in the forecasting process. In this article, an evolving recurrent interval type-2 intuitionistic fuzzy neural network (eRIT2IFNN) is proposed for time series prediction and regression problems. The eRIT2IFNN employs interval type-2 intuitionistic fuzzy sets to enhance the modeling of uncertainties by intuitionistic evaluation and noise tolerance of the system. In the eRIT2IFNN, the antecedent part of each fuzzy rule is defined using intuitionistic interval type-2 fuzzy sets, and the consequent realizes the Takagi–Sugeno–Kang type fuzzy inference mechanism. In order to utilize the prior knowledge including intuitionistic information, a local internal feedback is established by feeding the rule firing strength of each rule to itself eRIT2IFNN is fully adaptive to the evolving of sequence data by online learning of structure and parameters. A modified density-based clustering is implemented for the structure learning, where both densities and membership degrees are involved to determine the fuzzy rules. Performance of eRIT2IFNN is evaluated using a set of benchmark problems and compared with existing fuzzy inference systems. Moreover, the eRIT2IFNN is tested for identification of dynamics under both noise-free and noisy environments. Finally, a group of practical financial price-tracking problems including high-frequency data of financial future, commodity future and precious metal are used for the evaluation of the proposed inference system.     

Web shopping expert using new interval type-2 fuzzy reasoning

Finding a product with high quality and reasonable price online is a difficult task due to uncertainty of Web data and queries. In order to handle the uncertainty problem, the Web Shopping Expert, a new type-2 fuzzy online decision support system, is proposed. In the Web Shopping Expert, a fast interval type-2 fuzzy method is used to directly use all rules with type-1 fuzzy sets to perform type-2 fuzzy reasoning efficiently. The parameters of type-2 fuzzy sets are optimized by a least square method. The Web Shopping Expert based on the interval type-2 fuzzy inference system provides reasonable decisions for online users.     

Experimental study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic

Uncertainty is an inherent part in control systems used in real world applications. The use of new methods for handling incomplete information is of fundamental importance. Type-1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems. Type-2 fuzzy sets that are used in type-2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom. This paper deals with the design of control systems using type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements, environmental noise, etc. The experimental results are divided in two classes, in the first class, simulations of a feedback control system for a non-linear plant using type-1 and type-2 fuzzy logic controllers are presented; a comparative analysis of the systems’ response in both cases was performed, with and without the presence of uncertainty. For the second class, a non-linear identification problem for time-series prediction is presented. Based on the experimental results the conclusion is that the best results are obtained using type-2 fuzzy systems.     

Interval type-2 fuzzy logic based antenatal care system using phonocardiography

The assessment of fetal wellbeing depends heavily on variations in fetal heart rate (FHR) patterns. The variations in FHR patterns are very complex in nature thus its reliable interpretation is very difficult and often leads to erroneous diagnosis. We propose a new method for evaluation of fetal health status based on interval type-2 fuzzy logic through fetal phonocardiography (fPCG). Type-2 fuzzy logic is a powerful tool in handling uncertainties due to extraneous variations in FHR patterns through its increased fuzziness of relations. Four FHR parameters are extracted from each fPCG signal for diagnostic decision making. The membership functions of these four inputs and one output are chosen as a range of values so as to represent the level of uncertainty. The fuzzy rules are constructed based on standard clinical guidelines on FHR parameters. Experimental clinical tests have shown very good performance of the developed system in comparison with the FHR trace simultaneously recorded through standard fetal monitor. Statistical evaluation of the developed system shows 92% accuracy. With the proposed method we hope that, long-term and continuous antenatal care will become easy, cost effective, reliable and efficient.     

Ⅱ-型模糊集截集的定义及特征研究

经典模糊集的截集概念是经典模糊集合与经典集合联系的桥梁,对于II-型模糊集,该文在分析II-型模糊集、区间值II-型模糊集、I-型模糊集以及经典集合之间关系的基础上,定义了II-型模糊集的截集概念,分析了II-型模糊集截集的特征,仿真证明了II-型模糊集截集的有效性,为基于II-型模糊集的决策、聚类等实际应用提供了新的方法。     

On the influence of using fuzzy extensions in linguistic fuzzy rule-based regression systems

Fuzzy Rule-Based Systems, FRBSs, are powerful tools to address regression problems. They can model the relationship between inputs and outputs by linguistic concepts. However, those FRBSs which are based on the conventional Type-1 fuzzy sets may not be able to handle some difficulties of real-world applications. In such situations, using novel representations of fuzzy sets seems like a good idea. Different extensions of fuzzy sets usually help to provide more precise models in the real-world problems. In this study, the influence of using fuzzy extensions in improving the efficiency of linguistic fuzzy rule-based regression models is investigated. For this purpose, a conventional Type-1 Mamdani FRBS is adapted to the three extensions of fuzzy sets, namely Interval Type-2, Intuitionistic, and Interval Type-2 Intuitionistic fuzzy sets. A two-pass method is proposed to define membership (non-membership) functions of these fuzzy sets; this method is based on the 3-tuples representation of the standard Type-1 membership functions. Wang and Mendel’s rule learning method is adapted to extract fuzzy rules from regression data. In order to tune the membership functions up to different extents, three evolutionary extensions are also presented for each type of the proposed FRBSs. Individual, internal, and external comparisons of the proposed FRBSs were done using 22 real-world regression datasets and statistical tests. Experimental results confirm that all the three proposed FRBSs outperform the classical Type-1 framework; furthermore, the Interval Type-2 Intuitionistic FRBS is the superior system so that an appropriate tuning of its parameters makes it the most accurate model.     

关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义,使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合,并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型：离散、半连通、连通及复合型,并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support,CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合,并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后,探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.