Geometric Type-1 and Type-2 Fuzzy Logic Systems

This paper presents a novel approach to the representation of type-1 and type-2 fuzzy sets utilising computational geometry. To achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic. We provide new algorithms for various operations on type-1 and type-2 fuzzy sets and for defuzzification. Results of experiments indicate that this approach reduces the execution speed of these operations     

Parallel distributed genetic fuzzy rule selection

Genetic fuzzy rule selection has been successfully used to design accurate and compact fuzzy rule-based classifiers. It is, however, very difficult to handle large data sets due to the increase in computational costs. This paper proposes a simple but effective idea to improve the scalability of genetic fuzzy rule selection to large data sets. Our idea is based on its parallel distributed implementation. Both a training data set and a population are divided into subgroups (i.e., into training data subsets and sub-populations, respectively) for the use of multiple processors. We compare seven variants of the parallel distributed implementation with the original non-parallel algorithm through computational experiments on some benchmark data sets.     

Statistical confidence intervals for fuzzy data

The application of fuzzy sets theory to statistical confidence intervals for unknown fuzzy parameters is proposed in this paper by considering fuzzy random variables. In order to obtain the belief degrees under the sense of fuzzy sets theory, we transform the original problem into the optimization problems. We provide the computational procedure to solve the optimization problems. A numerical example is also provided to illustrate the possible application of fuzzy sets theory to statistical confidence intervals.     

Sine-square embedded fuzzy sets versus type-2 fuzzy sets

The uncertainty is an inherent part of real-world applications. Type-2 fuzzy sets minimize the effects of uncertainties that cannot be modeled using type-1 fuzzy sets. However, the computational complexity of the type-2 fuzzy sets is very high and it is more difficult than type-1 fuzzy sets to use and understand. This paper proposes sine-square embedded fuzzy sets and gives a comparison with type-2 and nonstationary fuzzy sets. The sine-square embedded fuzzy sets consist of type-1 fuzzy sets and the sine function. The footprint of uncertainty in the type-2 fuzzy sets is provided with amplitude and frequency of sine-square function in the proposed algorithm. The proposed sine-square embedded fuzzy sets are much simpler than the type-2 fuzzy sets and the nonstationary fuzzy sets. Two control applications that are chosen as position control of a dc motor and simulation of human lifting motion using five-segment human model are carried out to demonstrate the effectiveness of the proposed approach.     

Shadowed sets: representing and processing fuzzy sets

This study introduces a new concept of shadowed sets that can be regarded as a certain operational framework simplifying processing carried out with the aid of fuzzy sets and enhancing interpretation of results obtained therein. Some conceptual links between this idea and some others known in the literature are established. In particular, it is demonstrated how fuzzy sets can induce shadowed sets. Subsequently, shadowed sets reveal interesting conceptual and algorithmic relationships existing between rough sets and fuzzy sets. Detailed computational aspects of shadowed sets are discussed. Several illustrative examples are provided.     

Discrete Interval Type 2 Fuzzy System Models Using Uncertainty in Learning Parameters

Fuzzy system modeling (FSM) is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty. Conventional FSM techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system. However, since type 1 fuzzy sets express the belongingness of a crisp value x' of a base variable x in a fuzzy set A by a crisp membership value mu_A(x'), they cannot fully capture the uncertainties due to imprecision in identifying membership functions. Higher types of fuzzy sets can be a remedy to address this issue. Since, the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set, the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades. In this paper, we propose a black-box methodology that can identify robust type 2 Takagi-Sugeno, Mizumoto and Linguistic fuzzy system models with high predictive power. One of the essential problems of type 2 fuzzy system models is computational complexity. In order to remedy this problem, discrete interval valued type 2 fuzzy system models are proposed with type reduction. In the proposed fuzzy system modeling methods, fuzzy C-means (FCM) clustering algorithm is used in order to identify the system structure. The proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of FCM, known as the level of membership, and its variation over a specific set of values which generate the uncertainty associated with the system structure     

A Fuzzy Linguistic Methodology to Deal With Unbalanced Linguistic Term Sets

Many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects. In most of these problems, a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed. However, there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets that are not uniformly and symmetrically distributed. The use of linguistic variables implies processes of computing with words (CW). Different computational approaches can be found in the literature to accomplish those processes. The 2-tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed. In this paper, we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets. To do so, we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2-tuple fuzzy linguistic computational model to accomplish processes of CW with unbalanced term sets in a precise way and without loss of information.     

模糊滑动模态控制系统的性质分析

根据滑动模态原理,将模糊控制系统的输入量简化为广义跟踪误差的一个超平面,并基于三角形的非线性发语言变量的隶属度,分析了模糊控制系统的某些性质,表明在系统稳定性、稳态误差等指标方面,模糊控制器优于一般的ＰＩＤ控制器。     

基于密度峰值的加权犹豫模糊聚类算法

由于人们对事物认知的局限性和信息的不确定性,在对决策问题进行聚类分析时,传统的模糊聚类不能有效解决实际场景中的决策问题,因此有学者提出了有关犹豫模糊集的聚类算法。现有的层次犹豫模糊K均值聚类算法没有利用数据集本身的信息来确定距离函数的权值,且簇中心的计算复杂度和空间复杂度都是指数级的,不适用于大数据环境。针对上述问题,文中提出了一种基于密度峰值思想的加权犹豫模糊聚类算法(WHFDP),首先给出了犹豫模糊元素集的补齐方法,并结合变异系数理论给出了新的距离函数权重计算公式,然后利用密度峰值选取簇中心,不仅降低了簇中心计算的复杂度,而且提高了对不同规模以及任意形状数据集的适应性,算法的时间复杂度和空间复杂度也降为多项式级,最后采用典型数据集进行仿真实验,证明了所提算法的有效性。     

A narrow band interval type-2 fuzzy approach for image segmentation

Traditional fuzzy sets capture vagueness through precise numeric membership degrees. This poses a dilemma of excessive precision in describing uncertain phenomenon. Interval type-2 fuzzy sets have shown its effectiveness in handling uncertainties in comparison to the traditional fuzzy sets. In this paper, the interval type-2 fuzzy approach is introduced into the framework of active contour model, which effectively segment images with large uncertainties. However, the computational cost is largely increased by employing the interval type-2 fuzzy set. Therefore, we try to update the pixels within a narrow band region near the contour boundary for reducing the computational cost caused by employing the interval type-2 fuzzy set. Moreover, both spatial and gray constraints are taken into consideration when calculating the fuzzy membership value to retain more image details. Experimental results on synthetic and real images show that the proposed method is effective and efficient, and is relatively independent of initial conditions.     

Title Paper: Natural computing: A problem solving paradigm with granular information processing

Natural computing, inspired by biological course of action, is an interdisciplinary field that formalizes processes observed in living organisms to design computational methods for solving complex problems, or designing artificial systems with more natural behaviour. Based on the tasks abstracted from natural phenomena, such as brain modelling, self-organization, self-repetition, self evaluation, Darwinian survival, granulation and perception, nature serves as a source of inspiration for the development of computational tools or systems that are used for solving complex problems. Nature inspired main computing paradigms used for such development include artificial neural networks, fuzzy logic, rough sets, evolutionary algorithms, fractal geometry, DNA computing, artificial life and granular or perception-based computing. Information granulation in granular computing is an inherent characteristic of human thinking and reasoning process performed in everyday life. The present article provides an overview of the significance of natural computing with respect to the granulation-based information processing models, such as neural networks, fuzzy sets and rough sets, and their hybridization. We emphasize on the biological motivation, design principles, application areas, open research problems and challenging issues of these models.     

Analysis and efficient implementation of a linguistic fuzzy c-means

The paper is concerned with a linguistic fuzzy c-means (FCM) algorithm with vectors of fuzzy numbers as inputs. This algorithm is based on the extension principle and the decomposition theorem. It turns out that using the extension principle to extend the capability of the standard membership update equation to deal with a linguistic vector has a huge computational complexity. In order to cope with this problem, an efficient method based on fuzzy arithmetic and optimization has been developed and analyzed. We also carefully examine and prove that the algorithm behaves in a way similar to the FCM in the degenerate linguistic case. Synthetic data sets and the iris data set have been used to illustrate the behavior of this linguistic version of the FCM.     

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.     

Granular computing based on fuzzy similarity relations

Rough sets and fuzzy rough sets serve as important approaches to granular computing, but the granular structure of fuzzy rough sets is not as clear as that of classical rough sets since lower and upper approximations in fuzzy rough sets are defined in terms of membership functions, while lower and upper approximations in classical rough sets are defined in terms of union of some basic granules. This limits further investigation of the existing fuzzy rough sets. To bring to light the innate granular structure of fuzzy rough sets, we develop a theory of granular computing based on fuzzy relations in this paper. We propose the concept of granular fuzzy sets based on fuzzy similarity relations, investigate the properties of the proposed granular fuzzy sets using constructive and axiomatic approaches, and study the relationship between granular fuzzy sets and fuzzy relations. We then use the granular fuzzy sets to describe the granular structures of lower and upper approximations of a fuzzy set within the framework of granular computing. Finally, we characterize the structure of attribute reduction in terms of granular fuzzy sets, and two examples are also employed to illustrate our idea in this paper.     

Interval type-2 neuro-fuzzy system with implication-based inference mechanism

Neuro-fuzzy systems have been proved to be an efficient tool for modelling real life systems. They are precise and have ability to generalise knowledge from presented data. Neuro-fuzzy systems use fuzzy sets – most commonly type-1 fuzzy sets. Type-2 fuzzy sets model uncertainties better than type-1 fuzzy sets because of their fuzzy membership function. Unfortunately computational complexity of type reduction in general type-2 systems is high enough to hinder their practical application. This burden can be alleviated by application of interval type-2 fuzzy sets. The paper presents an interval type-2 neuro-fuzzy system with interval type-2 fuzzy sets both in premises (Gaussian interval type-2 fuzzy sets with uncertain fuzziness) and consequences (trapezoid interval type-2 fuzzy set). The inference mechanism is based on the interval type-2 fuzzy Łukasiewicz, Reichenbach, Kleene-Dienes, or Brouwer–Gödel implications. The paper is accompanied by numerical examples. The system can elaborate models with lower error rate than type-1 neuro-fuzzy system with implication-based inference mechanism. The system outperforms some known type-2 neuro-fuzzy systems.     

Fuzzy logic microcontroller

We propose an architecture dedicated mainly to medium-range applications that demand computational power combined with low cost for the resulting hardware system (chip and board). Our architecture is a 16-bit processor with dedicated instructions and hardware for efficient support of fuzzy logic. To make the architecture effective for control applications developed with a traditional approach or with fuzzy logic, we equipped the processor with a microcontroller's general features. Our design accounts for application characteristics to provide efficient hardware support for fuzzy logic. To achieve this we first analyzed fuzzy control algorithms and derived a general model for fuzzy computation. In defining the model, we considered the large spectrum of possible inference methods, fuzzification and defuzzification mechanisms, and the operators used in control applications. On this basis, we defined the instruction set that supports this computational model and a proper architectural solution. We tested the system (composed of the software model and its hardware support) by simulating different sets of general-purpose and fuzzy control benchmarks     

一种改进的模糊连接点聚类算法

传统的模糊连接点FJP聚类算法采用基于欧氏距离的最大 最小合成运算法生成传递闭包,该方法所生成的传递闭包存在失真问题,即包含有较多错误的数据关联信息,最终造成算法聚类精度低且计算时间长。针对以上问题,提出一种改进的模糊连接点聚类算法：先用组合核函数计算数据集的模糊相似度矩阵,提高算法对数据非线性特征的辨识能力,并用大顶堆存储之;然后遍历传递闭包矩阵中的空元素,用堆顶的桥元素填充传递闭包的空元素,直至生成传递闭包。在测试数据集上的实验结果表明,本文算法的平均聚类精度较传统FJP算法有20%以上的提升,显著改善了传递闭包的失真问题;另外,在大型数据集上的计算效率亦优于传统FJP算法的,说明本文改进FJP算法的思路是有效的、可行的。