Type-2 Fuzzistics for Symmetric Interval Type-2 Fuzzy Sets: Part 1, Forward Problems

Interval type-2 fuzzy sets (T2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). In this two-part paper, we focus on symmetric interval T2 FSs for which the centroid (which is an interval type-1 FS) provides a measure of its uncertainty. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, will be associated with the amount of uncertainty in such a T2 FS. The main purpose of this paper (Part 1) is to demonstrate that our intuition is correct and to quantify the centroid of a symmetric interval T2 FS, and consequently its uncertainty, with respect to such geometric properties. It is then possible, for the first time, to formulate and solve forward problems, i.e., to go from parametric interval T2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems. These solutions are used in Part 2 to solve some inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models (T2 fuzzistics)     

Type-2 Fuzzistics for Nonsymmetric Interval Type-2 Fuzzy Sets: Forward Problems

Interval type-2 fuzzy sets (IT2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). The centroid of an IT2 FS, which is an IT1 FS, provides a measure of the uncertainty in the IT2 FS. The main purpose of this paper is to quantify the centroid of a non-symmetric IT2 FS with respect to geometric properties of its FOU. This is very important because interval data collected from subjects about words suggests that the FOUs of most words are non-symmetrical. Using the results in this paper, it is possible to formulate and solve forward problems, i.e., to go from parametric non-symmetric IT2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems for non-symmetrical triangular, trapezoidal, Gaussian and shoulder FOUs.     

Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets

The focus of this paper is the linguistic weighted average (LWA), where the weights are always words modeled as interval type-2 fuzzy sets (IT2 FSs), and the attributes may also (but do not have to) be words modeled as IT2 FSs; consequently, the output of the LWA is an IT2 FS. The LWA can be viewed as a generalization of the fuzzy weighted average (FWA) where the type-1 fuzzy inputs are replaced by IT2 FSs. This paper presents the theory, algorithms, and an application of the LWA. It is shown that finding the LWA can be decomposed into finding two FWAs. Since the LWA can model more uncertainties, it should have wide applications in distributed and hierarchical decision-making.     

区间数的区间Ⅱ型模糊c均值聚类算法

针对区间数模糊c均值聚类算法存在模糊度指数m无法准确描述数据簇划分情况的问题,对点数据集合的区间Ⅱ型模糊c均值聚类算法进行拓展,将其扩展到区间型不确定数据的聚类中。同时,分析了区间数的区间Ⅱ型模糊c均值聚类算法的收敛性,以确定模糊度指数m1和m2的取值原则。基于合成数据和实测数据的仿真实验结果表明：区间数的区间Ⅱ型模糊c均值聚类算法比区间数的模糊c均值聚类算法的聚类效果好。     

Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to C-Means

In many pattern recognition applications, it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set. Uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms. Therefore, various types of uncertainty may be taken into account when performing several pattern recognition methods. When one performs clustering with fuzzy sets, fuzzy membership values express assignment availability of patterns for clusters. However, when one assigns fuzzy memberships to a pattern set, imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment. When one encounters fuzzy clustering, fuzzy membership design includes various uncertainties (e.g., distance measure, fuzzifier, prototypes, etc.). In this paper, we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final C-partition in the fuzzy C-means (FCM) algorithm. To design and manage uncertainty for fuzzifier m, we extend a pattern set to interval type-2 fuzzy sets using two fuzzifiers m₁ and m₂ which creates a footprint of uncertainty (FOU) for the fuzzifier m. Then, we incorporate this interval type-2 fuzzy set into FCM to observe the effect of managing uncertainty from the two fuzzifiers. We also provide some solutions to type-reduction and defuzzification (i.e., cluster center updating and hard-partitioning) in FCM. Several experimental results are given to show the validity of our method     

区间二型模糊子空间0阶TSK系统

人们倾向于使用少量的有代表性的特征来描述一条规则,而忽略极为次要的冗余的信息。经典的区间二型TSK(Takagi-Sugeno-Kang)模糊系统,在规则前件和后件部分会使用完整的数据特征空间,对于高维数据而言,易导致系统的复杂度增加和可解释性的损失。针对于此,提出了区间二型模糊子空间0阶TSK系统。在规则前件部分,使用模糊子空间聚类和网格划分相结合的方法生成稀疏的规整的规则中心,在规则后件部分,使用简化的0阶形式,从而得到规则语义更为简洁的区间二型模糊系统。在模拟和真实数据上的实验结果表明该方法分类效果良好,可解释性更好。     

Interval Type-2 Fuzzy Logic Systems Made Simple

To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way     

基于严格等价函数的区间二型模糊熵及其图像阈值分割方法

文中利用严格等价函数提出一种基于区间二型模糊熵的图像阈值分割方法.首先基于公理化定义,利用严格定价函数提出一种区间二型模糊熵的构建方法,由此可以得到多个不同的模糊熵计算表达式;然后通过理论分析给出了利用最小化模糊熵准则选取最优阈值的方法.实验结果表明,与现有的其他模糊阈值分割法和改进的2维Otsu法等相比,该方法的分割更加准确,运行时间更少,具有更广泛的适应性.     

A Fast Geometric Method for Defuzzification of Type-2 Fuzzy Sets

Generalized type-2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type-2 fuzzy set is too large. This paper presents a new method for defuzzifing a type-2 fuzzy set. The new much faster technique is based on geometric representations and operations. The results of a real world example contained in this paper show this new approach to be over 200 000 times faster than type-reduction. We present a new method for assessing the accuracy of the membership function of a type-2 fuzzy set. This method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set. We also discuss the differences between the new approach and type-reduction, identifying the origin of this massive improvement in execution speed.      

An Interval Type-2 Fuzzy Rough Set Model for Attribute Reduction

Rough set theory is a very useful tool for describing and modeling vagueness in ill-defined environments. Traditional rough set theory is restricted to crisp environments. However, nowadays, it has been extended to fuzzy environments, resulting in the development of the so-called fuzzy rough sets. Type-2 fuzzy sets possess many advantages over type-1 fuzzy sets, but for the general type-2 fuzzy sets, the computational complexity is severe. On the other hand, set-theoretic and arithmetic computations for the interval type-2 fuzzy sets are very simple. Motivated by the aforementioned accomplishments, in this paper, the concept of fuzzy rough sets is generalized to interval type-2 fuzzy environments. Subsequently, a method of attribute reduction within the interval type-2 fuzzy rough set framework is proposed. Lastly, the properties of the interval type-2 fuzzy rough sets are presented.     

Implementing Interval Type-2 Fuzzy Processors [Developmental Tools]

This article discusses some relevant methodological aspects for implementing IT2-FLS on hardware. First it describes a hardware architecture for an IT2-FP. In addition, it details particular considerations for the design of the different components of the architecture. As a study case, a simple implementation of an IT2-FP in FPGA technology is presented     

Calculating Functions of Interval Type-2 Fuzzy Numbers for Fault Current Analysis

In this work, functions of type-2 fuzzy numbers are analyzed. For the special case of interval type-2 fuzzy numbers, the type-2 membership function of the output variable is calculated using the lower and upper membership functions of the input variables and the vertex method. This procedure is used in an application where the type-2 fuzzy fault currents of an electric distribution system are calculated. The results are shown and the advantages of the approach are discussed     

Encoding Words Into Interval Type-2 Fuzzy Sets Using an Interval Approach

This paper presents a very practical type-2-fuzzistics methodology for obtaining interval type-2 fuzzy set (IT2 FS) models for words, one that is called an interval approach (IA). The basic idea of the IA is to collect interval endpoint data for a word from a group of subjects, map each subject's data interval into a prespecified type-1 (T1) person membership function, interpret the latter as an embedded T1 FS of an IT2 FS, and obtain a mathematical model for the footprint of uncertainty (FOU) for the word from these T1 FSs. The IA consists of two parts: the data part and the FS part. In the data part, the interval endpoint data are preprocessed, after which data statistics are computed for the surviving data intervals. In the FS part, the data are used to decide whether the word should be modeled as an interior, left-shoulder, or right-shoulder FOU. Then, the parameters of the respective embedded T1 MFs are determined using the data statistics and uncertainty measures for the T1 FS models. The derived T1 MFs are aggregated using union leading to an FOU for a word, and finally, a mathematical model is obtained for the FOU. In order that all researchers can either duplicate our results or use them in their research, the raw data used for our codebook examples, as well as a MATLAB M-file for the IA, have been put on the Internet at: http://sipi.usc.edu/$sim$mendel.      

A Recurrent Self-Evolving Interval Type-2 Fuzzy Neural Network for Dynamic System Processing

This paper proposes a recurrent self-evolving interval type-2 fuzzy neural network (RSEIT2FNN) for dynamic system processing. An RSEIT2FNN incorporates type-2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system. The antecedent parts in each recurrent fuzzy rule in the RSEIT2FNN are interval type-2 fuzzy sets, and the consequent part is of the Takagi-Sugeno-Kang (TSK) type with interval weights. The antecedent part of RSEIT2FNN forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself. The TSK-type consequent part is a linear model of exogenous inputs. The RSEIT2FNN initially contains no rules; all rules are learned online via structure and parameter learning. The structure learning uses online type-2 fuzzy clustering. For the parameter learning, the consequent part parameters are tuned by a rule-ordered Kalman filter algorithm to improve learning performance. The antecedent type-2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm. The RSEIT2FNN is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise-free and noisy conditions. Comparisons with type-1 recurrent fuzzy neural networks validate the performance of the RSEIT2FNN.     

基于二型模糊集推理的优化航路规划方法

航路规划是带约束的多目标优化问题,常用的优化算法是通过加权系数法把多目标优化问题转化为单目标优化问题。该固定的加权系数无法适应战场环境的变化,且无法满足不同专家对优化目标的个人偏好。针对以上问题,提出基于二型模糊集推理的优化航路规划方法。建立航行器复杂约束层次表达模型,采用改进的Per-C方法,利用不同专家对优化目标的偏好信息以及航路约束值实现模糊推理,求取航路模糊代价。将模糊推理应用于A*搜索代价计算过程,最终实现优化的多目标航路规划方法。实验结果表明,该方法能够准确反映各专家对优化目标的偏好,具有较强的灵活性和通用性。     

Wireless Sensor Network Lifetime Analysis Using Interval Type-2 Fuzzy Logic Systems

Extending the lifetime of the energy constrained wireless sensor networks is a crucial challenge in sensor network research. In this paper, we present a novel approach based on fuzzy logic systems to analyze the lifetime of a wireless sensor network. We demonstrate that a type-2 fuzzy membership function (MF), i.e., a Gaussian MF with uncertain standard deviation (std) is most appropriate to model a single node lifetime in wireless sensor networks. In our research, we study two basic sensor placement schemes: square-grid and hex-grid. Two fuzzy logic systems (FLSs): a singleton type-1 FLS and an interval type-2 FLS are designed to perform lifetime estimation of the sensor network. We compare our fuzzy approach with other nonfuzzy schemes in previous papers. Simulation results show that FLS offers a feasible method to analyze and estimate the sensor network lifetime and the interval type-2 FLS in which the antecedent and the consequent membership functions are modeled as Gaussian with uncertain std outperforms the singleton type-1 FLS and the nonfuzzy schemes.