Designing an algorithm for evaluating decision-making units based on neural weighted function

Fuzzy systems have gained more and more attention from researchers and practitioners of various fields. In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the defuzzification process. Several analytic methods have been proposed for this problem, but in this paper, firstly the researchers introduce a novel parametric distance between fuzzy numbers and secondly suggest a new approach to the problem of defuzzification, using this distance. This defuzzification can be used as a crisp approximation with respect to fuzzy quantity. By considering this and with benchmark between fuzzy numbers, we can present a method for evaluating. The method can effectively evaluate various fuzzy numbers and their images and overcome the shortcomings of the previous techniques.     

DD-HDS: A method for visualization and exploration of high-dimensional data.

Mapping high-dimensional data in a low-dimensional space, for example, for visualization, is a problem of increasingly major concern in data analysis. This paper presents data-driven high-dimensional scaling (DD-HDS), a nonlinear mapping method that follows the line of multidimensional scaling (MDS) approach, based on the preservation of distances between pairs of data. It improves the performance of existing competitors with respect to the representation of high-dimensional data, in two ways. It introduces (1) a specific weighting of distances between data taking into account the concentration of measure phenomenon and (2) a symmetric handling of short distances in the original and output spaces, avoiding false neighbor representations while still allowing some necessary tears in the original distribution. More precisely, the weighting is set according to the effective distribution of distances in the data set, with the exception of a single user-defined parameter setting the tradeoff between local neighborhood preservation and global mapping. The optimization of the stress criterion designed for the mapping is realized by "force-directed placement" (FDP). The mappings of low- and high-dimensional data sets are presented as illustrations of the features and advantages of the proposed algorithm. The weighting function specific to high-dimensional data and the symmetric handling of short distances can be easily incorporated in most distance preservation-based nonlinear dimensionality reduction methods.     

A fast method for computing the centroid of a type-2 fuzzy set

Type reduction does the work of computing the centroid of a type-2 fuzzy set. The result is a type-1 fuzzy set from which a corresponding crisp number can then be obtained through defuzzification. Type reduction is one of the major operations involved in type-2 fuzzy inference. Therefore, making type reduction efficient is a significant task in the application of type-2 fuzzy systems. Liu introduced a horizontal slice representation, called the α-plane representation, and proposed a type-reduction method for a type-2 fuzzy set. By exploring some useful properties of the α-plane representation and of the type reduction for interval type-2 fuzzy sets, a fast method is developed for computing the centroid of a type-2 fuzzy set. The number of computations and comparisons involved is greatly reduced. Convergence in each iteration can then speed up, and type reduction can be done much more efficiently. The effectiveness of the proposed method is analyzed mathematically and demonstrated by experimental results.     

Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures

We describe the representation of a fuzzy subset in terms of its crisp level sets. We then generalize these level sets to the case of interval valued fuzzy sets and provide for a representation of an interval valued fuzzy set in terms of crisp level sets. We note that in this representation while the level sets are crisp the memberships are still intervals. Once having this representation we turn to its role in the extension principle and particularly to the extension of measures of uncertainty of interval valued fuzzy sets. Two types of extension of uncertainty measures are investigated. The first, based on the level set representation, leads to extensions whose values for the measure of uncertainty are themselves fuzzy sets. The second, based on the use of integrals, results in extensions whose value for the uncertainty of an interval valued fuzzy sets is an interval.     

Learning indistinguishability from data

 In this paper we revisit the idea of interpreting fuzzy sets as representations of vague values. In this context a fuzzy set is induced by a crisp value and the membership degree of an element is understood as the similarity degree between this element and the crisp value that determines the fuzzy set. Similarity is assumed to be a notion of distance. This means that fuzzy sets are induced by crisp values and an appropriate distance function. This distance function can be described in terms of scaling the ordinary distance between real numbers. With this interpretation in mind, the task of designing a fuzzy system corresponds to determining suitable crisp values and appropriate scaling functions for the distance. When we want to generate a fuzzy model from data, the parameters have to be fitted to the data. This leads to an optimisation problem that is very similar to the optimisation task to be solved in objective function based clustering. We borrow ideas from the alternating optimisation schemes applied in fuzzy clustering in order to develop a new technique to determine our set of parameters from data, supporting the interpretability of the fuzzy system.     

Fuzzy relational classifier trained by fuzzy clustering

A novel approach to nonlinear classification is presented, in the training phase of the classifier, the training data is first clustered in an unsupervised way by fuzzy c-means or a similar algorithm. The class labels are not used in this step. Then, a fuzzy relation between the clusters and the class identifiers is computed. This approach allows the number of prototypes to be independent of the number of actual classes. For the classification of unseen patterns, the membership degrees of the feature vector in the clusters are first computed by using the distance measure of the clustering algorithm. Then, the output fuzzy set is obtained by relational composition. This fuzzy set contains the membership degrees of the pattern in the given classes. A crisp decision is obtained by defuzzification, which gives either a single class or a "reject" decision, when a unique class cannot be selected based on the available information. The principle of the proposed method is demonstrated on an artificial data set and the applicability of the method is shown on the identification of live-stock from recorded sound sequences. The obtained results are compared with two other classifiers.     

Fuzzy optical metrology

When operating in the real world, the conventional fuzzy approach is to extract crisp data then fuzzify it. This paper explores the concept of measuring fuzzy set memberships directly and obtaining the underlying crisp values by defuzzification. Both speed and accuracy advantages of fuzzy over conventional, crisp metrology are noted     

A Screen Space Quality Method for Data Abstraction

The rendering of large data sets can result in cluttered displays and non‐interactive update rates, leading to time consuming analyses. A straightforward solution is to reduce the number of items, thereby producing an abstraction of the data set. For the visual analysis to remain accurate, the graphical representation of the abstraction must preserve the significant features present in the original data. This paper presents a screen space quality method, based on distance transforms, that measures the visual quality of a data abstraction. This screen space measure is shown to better capture significant visual structures in data, compared with data space measures. The presented method is implemented on the GPU, allowing interactive creation of high quality graphical representations of multivariate data sets containing tens of thousands of items.     

Genetic algorithm-based optimal fuzzy controller design in the linguistic space

In this paper, a genetic algorithm (GA) based optimal fuzzy controller design is proposed. The design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule. The inputs of the controller, after scaling, are utilized by the index function for computing the output linguistic value. This linguistic value can then be used to map the suitable fuzzy control actions. This proposed novel fuzzy control rule has crisp input and fuzzified output characteristics. The index function plays a role in mapping the desired fuzzy sets for defuzzification resulting in a controlled hypersurface in the linguistic space formed by the input fuzzy variables. Two types of index functions, both linear and nonlinear, are introduced for controlling systems with different degrees of nonlinearity. The parameters of the index function are obtained by applying a simple GA with a suitable fitness function. Various controlled systems result in various parameter sets depending on their dynamics. Under the acquired optimal parameter set the optimal index function can be used to generate the desired control actions. Several simulation examples are given to verify the performance of the proposed GA-based fuzzy controller.     

Divergence and fuzziness measures

In an axiomatic way a divergence between fuzzy sets is introduced which extends the symmetric difference between crisp sets. Any fuzzy measure of the divergence between two fuzzy sets weighs their “distance”. The distance between a fuzzy set and the family of crisp sets is fuzziness measure.     

Fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems

In this paper, we study the fuzzy reasoning based on a new fuzzy rough set. First, we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms. Then, based on the approximation operators above, we propose the fuzzy reasoning based on the new fuzzy rough set. By means of the above fuzzy reasoning based on the new fuzzy rough set, for a given premise, we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets. Furthermore, through the defuzzification of the lower and upper approximations, we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification. Then, from the above interval, a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly. At last, we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems, and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the CRI method and the III method.     

Advantages of an alternative form of fuzzy logic

A specific implementation of fuzzy logic is described. This implementation uses multiplication rather than finding the minimum to determine the conjunction between antecedents. The fuzzy implication is also determined by the product operator. Crisp membership functions are used for the output fuzzy sets. Thus the product of the antecedents is then multiplied by a crisp output action for that rule and the sum of products determines the net output of the system. Advantages of this method over the “standard” methods include elimination of the defuzzification step, direct control of the shape of the input-to-output mapping surface, and an analytic formulation that can be easily implemented in software or hardware. Conventional controllers are shown to be a special case of the method. The method is also equivalent to a certain class of neural networks and, as such, can be trained to optimum values of the output actions of the system. The method is illustrated with some examples     

An image understanding system using attributed symbolic representation and inexact graph-matching

This paper presents a powerful image understanding system that utilizes a semantic-syntactic (or attributed-synibolic) representation scheme in the form of attributed relational graphs (ARG's) for comprehending the global information contents of images. Nodes in the ARG represent the global image features, while the relations between those features are represented by attributed branches between their corresponding nodes. The extraction of ARG representation from images is achieved by a multilayer graph transducer scheme. This scheme is basically a rule-based system that uses a combination of model-driven and data-driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial-domain representation into a global representation. Further analysis and inter-pretation of the imagery data is performed on the extracted ARG representation. A distance measure between images is defined in terms of the distance between their respective ARG representations. The distance between two ARG's and the inexact matching of their respective components are calculated by an efficient dynamic programming technique. The system handles noise, distortion, and ambiguity in real-world images by two means, namely, through modeling and embedding them into the transducer's mapping rules, as well as through the appropriate cost of error-transformation for the inexact matching of the ARG image representation. Two illustrative experiments are presented to demonstrate some capabilities of the proposed system. Experiment I deals with locating objects in multiobject scenes, while Experiment II is concerned with target detection in SAR images.     

A trainable transparent universal approximator for defuzzificationin Mamdani-type neuro-fuzzy controllers

A novel technique of designing application specific defuzzification strategies with neural learning is presented. The proposed neural architecture considered as a universal defuzzification approximator is validated by showing the convergence when approximating several existing defuzzification strategies. The method is successfully tested with fuzzy controlled reverse driving of a model truck. The transparent structure of the universal defuzzification approximator allows us to analyze the generated customized defuzzification method using the existing theories of defuzzification. The integration of universal defuzzification approximator instead of traditional methods in Mamdani-type fuzzy controllers can also be considered as an addition of trainable nonlinear noise to the output of the fuzzy rule inference before calculating the defuzzified crisp output. Therefore, nonlinear noise trained specifically for a given application shows a grade of confidence on the rule base, providing an additional opportunity to measure the quality of the fuzzy rule base. The possibility of modeling a Mamdani-type fuzzy controller as a feedforward neural network with the ability of gradient descent training of the universal defuzzification approximator and antecedent membership functions fulfil the requirement known from multilayer preceptrons in finding solutions to nonlinear separable problems     

结合多特征和模糊偏好关系的高分辨率遥感影像分割

高空间分辨率(简称高分辨率)遥感影像除光谱特征外,还包含丰富的纹理特征,为了实现高分辨率遥感影像的高精度分割,提出结合多特征和模糊偏好关系的分割方法.首先,通过像素光谱测度定义多种统计特征,根据定义的各个特征提取特征影像并分别实现影像分割,利用其结果构建模糊决策矩阵;然后,基于像素定义特征间的模糊偏好关系矩阵,计算不同特征对最终分割决策的权重,并对模糊决策矩阵加权以突出优势特征,抑制劣势特征;最后,通过反模糊化决策矩阵得到最优影像分割结果.对合成影像和真实高分辨率遥感影像的分割结果进行定性和定量评价,结果表明,合成影像的分割总精度为99.8%,Kappa值为0.998,说明所提出的算法通过结合各特征的优势部分能够获得高精度的分割结果.     

基于指数型模糊数的多属性决策模型及其应用

针对模糊多属性决策问题,给出一种基于指数型模糊数的多属性决策模型。一方面,通过定义指数型模糊数的期望,以实现属性权重向量的解模糊化处理;另一方面,根据三元区间数理论和指数型模糊数的截集信息,定义指数型模糊数上一种新的距离度量,以计算各备选方案与正、负理想方案之间的距离。根据模糊理想点思想,基于指数型模糊数的期望和距离的定义,给出一种指数型模糊数上的Topsis多属性决策方法。将该模型应用于一个具体实例,其结果证实了该方法的有效性。     

A survey of defuzzification strategies

Defuzzification is an important operation in the theory of fuzzy sets. It transforms a fuzzy set information into a numeric data information. This operation along with the operation of fuzzification is critical to the design of fuzzy systems as both of these operations provide nexus between the fuzzy set domain and the real-valued scalar domain. We need the synergy of both of these domains to solve many of our ill-posed problems effectively. In this paper, we address the problem of defuzzification, we present merits and demerits of various defuzzification strategies that are used in the theory and practice, and in design and implementation of applications involving fuzzy theory, fuzzy control, and fuzzy rule base, and fuzzy inference-based systems. We also present in this paper a simple and yet a novel defuzzification mechanism. © 2001 John Wiley & Sons, Inc.     

A Novel Navigation Method for Autonomous Mobile Vehicles

This paper presents a novel navigation method for Autonomous Mobile Vehicle in unknown environments. The proposed navigator consists of an Obstacle Avoider (OA), a Goal Seeker (GS), a Navigation Supervisor (NS) and an Environment Evaluator (EE). The fuzzy actions inferred by the OA and the GS are weighted by the NS using the local and global environmental information and fused through fuzzy set operation to produce a command action, from which the final crisp action is determined by defuzzification. The EE tunes the supports of the fuzzy sets for the OA and the NS; therefore, the capability of the navigation method is enhanced. Simulation shows that the navigator is able to perform successful navigation task in various unknown or partially known environments, and it has satisfactory ability in tackling moving obstacles. More importantly, it has smooth action and exceptionally good robustness to sensor noise.     

Different averages of a fuzzy set with an application to vessel segmentation

Image segmentation is a major problem in image processing, particularly in medical image analysis. A great number of segmentation procedures produce intermediate gray-scale images that can be understood as fuzzy sets. Additionally, some segmentation procedures tend to leave free tuning parameters (very influential in the final binary image) for the user. These different binary images can be easily aggregated (into a fuzzy set) by making use of fuzzy set theory. In any case, a single binary image is required so our interest is to associate a crisp set to a given fuzzy set in an intelligent and unsupervised manner. The main idea of this paper is to define the averages of a given fuzzy set by using different definitions of the mean of a random compact set. In particular, the average distance of Baddeley-Molchanov and the mean of Vorob'ev have been used. A theoretical study of some new definitions of fuzzy set averages has been performed. In particular, these averages have been obtained for L-R fuzzy numbers. Finally, we present a medical image application, that of retinal vessel detection. Some recent segmentation procedures have been revisited and modified using these new averages. The experimental results are very promising.     

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。