Showing papers on "Membership function published in 2003"

Fuzzy association rules: general model and applications

Abstract: The theory of fuzzy sets has been recognized as a suitable tool to model several kinds of patterns that can hold in data. In this paper, we are concerned with the development of a general model to discover association rules among items in a (crisp) set of fuzzy transactions. This general model can be particularized in several ways; each particular instance corresponds to a certain kind of pattern and/or repository of data. We describe some applications of this scheme, paying special attention to the discovery of fuzzy association rules in relational databases.

A Fuzzy TOPSIS Method for Robot Selection

Abstract: A fuzzy TOPSIS method for robot selection is proposed, where the ratings of various alternatives versus various subjective criteria and the weights of all criteria are assessed in linguistic terms represented by fuzzy numbers. The values of objective criteria are converted into dimensionless indices to ensure compatibility between the values of objective criteria and the linguistic ratings of subjective criteria. The membership function of each weighted rating is developed by interval arithmetic of fuzzy numbers. To avoid complicated aggregation of fuzzy numbers, these weighted ratings are defuzzified into crisp values by the ranking method of mean of removals. A closeness coefficient is defined to determine the ranking order of alternatives by calculating the distances to both the ideal and negative-ideal solutions. A numerical example demonstrates the computational process of the proposed method.

A consensus-reaching process under intuitionistic fuzzy preference relations

Abstract: We extend the main idea of a fuzzy analysis of consensus—that is based on a concept of a distance from consensus—to a case when individual testimonies are individual intuitionistic fuzzy preference relations, as opposed to fuzzy preference relations commonly used. Intuitionistic fuzzy preference relations, that in addition to a membership degree (from [0, 1]) include a hesitation margin (concerning the membership degree), can better reflect the very imprecision of testimonies (expressing preferences) of the individuals during the consensus-reaching process. Our new solution, obtained as an interval-valued measure of a distance from consensus, better reflects both real human perception and a soft nature of consensus. © 2003 Wiley Periodicals, Inc.

Support vector learning for fuzzy rule-based classification systems

Abstract: To design a fuzzy rule-based classification system (fuzzy classifier) with good generalization ability in a high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, the support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high- (or even infinite) dimensional feature space. This paper investigates the connection between fuzzy classifiers and kernel machines, establishes a link between fuzzy rules and kernels, and proposes a learning algorithm for fuzzy classifiers. We first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function. Fuzzy inference on the IF-part of a fuzzy rule can be viewed as evaluating the kernel function. The kernel function is then proven to be a Mercer kernel if the reference functions meet a certain spectral requirement. The corresponding fuzzy classifier is named positive definite fuzzy classifier (PDFC). A PDFC can be built from the given training samples based on a support vector learning approach with the IF-part fuzzy rules given by the support vectors. Since the learning process minimizes an upper bound on the expected risk (expected prediction error) instead of the empirical risk (training error), the resulting PDFC usually has good generalization. Moreover, because of the sparsity properties of the SVMs, the number of fuzzy rules is irrelevant to the dimension of input space. In this sense, we avoid the "curse of dimensionality." Finally, PDFCs with different reference functions are constructed using the support vector learning approach. The performance of the PDFCs is illustrated by extensive experimental results. Comparisons with other methods are also provided.

Fuzzy least squares support vector machines for multiclass problems

Abstract: In least squares support vector machines (LS-SVMs), the optimal separating hyperplane is obtained by solving a set of linear equations instead of solving a quadratic programming problem. But since SVMs and LS-SVMs are formulated for two-class problems, unclassifiable regions exist when they are extended to multiclass problems.In this paper, we discuss fuzzy LS-SVMs that resolve unclassifiable regions for multiclass problems. We define a membership function in the direction perpendicular to the optimal separating hyperplane that separates a pair of classes. Using the minimum or average operation for these membership functions, we define a membership function for each class. Using some benchmark data sets, we show that recognition performance of fuzzy LS-SVMs with the minimum operator is comparable to that of fuzzy SVMs, but fuzzy LS-SVMs with the average operator showed inferior performance.

Knowledge-Driven and Data-Driven Fuzzy Models for Predictive Mineral Potential Mapping

Abstract: In this paper, we describe new fuzzy models for predictive mineral potential mapping: (1) a knowledge-driven fuzzy model that uses a logistic membership function for deriving fuzzy membership values of input evidential maps and (2) a data-driven model, which uses a piecewise linear function based on quantified spatial associations between a set of evidential evidence features and a set of known mineral deposits for deriving fuzzy membership values of input evidential maps. We also describe a graphical defuzzification procedure for the interpretation of output fuzzy favorability maps. The models are demonstrated for mapping base metal deposit potential in an area in the south-central part of the Aravalli metallogenic province in the state of Rajasthan, western India. The data-driven and knowledge-driven models described in this paper predict potentially mineralized zones, which occupy less than 10% of the study area and contain at least 83% of the “model” and “validation” base metal deposits. A cross-validation of the favorability map derived from using one of the models with the favorability map derived from using the other model indicates a remarkable similarity in their results. Both models therefore are useful for predicting favorable zones to guide further exploration work.

Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge

Abstract: Just like rough set theory, fuzzy set theory addresses the topic of dealing with imperfect knowledge. Recent investigations have shown how both theories can be combined into a more flexible, more expressive framework for modelling and processing incomplete information in information systems. At the same time, intuitionistic fuzzy sets have been proposed as an attractive extension of fuzzy sets, enriching the latter with extra features to represent uncertainty (on top of vagueness). Unfortunately, the various tentative definitions of the concept of an ‘intuitionistic fuzzy rough set’ that were raised in their wake are a far cry from the original objectives of rough set theory. We intend to fill an obvious gap by introducing a new definition of intuitionistic fuzzy rough sets, as the most natural generalization of Pawlak's original concept of rough sets.

Fuzzy sets for words: a new beginning

Abstract: This paper begins with a delineation of two approaches to fuzzy sets, abstract mathematics and models for words. It demonstrates, by using Karl Popper's Falsificationism, the present approach to fuzzy sets (FSs) for words is scientifically incorrect. A new theory of fuzzy sets is then presented for words that is based on collecting data from people -person MFs-that reflect intra- and inter-levels of uncertainties about a word, and defines a word FS as the union of all such person fuzzy sets. It also demonstrates that intra-uncertainty about a word can be modeled using type-2 person fuzzy sets, and that inter-uncertainty about a word can be modeled by means of an equally weighted union of each person's type-2 fuzzy set. Finally, it proposes a methodology for obtaining a parsimonious parametric type-2 fuzzy set approximation to the aggregated type-2 person FSs. This new theory of fuzzy sets for words is testable and is therefore subject to refutation.

The canonical representation of multiplication operation on triangular fuzzy numbers

Abstract: The representation of multiplication operation on fuzzy numbers is very useful and important in the fuzzy system such as the fuzzy decision making. In this paper, we propose a new arithmetical principle and a new arithmetical method for the arithmetical operations on fuzzy numbers. The new arithmetical principle is the L−1-R−1 inverse function arithmetic principle. Based on the L−1-R−1 inverse function arithmetic principle, it is easy to interpret the multiplication operation with the membership functions of fuzzy numbers. The new arithmetical method is the graded multiple integrals representation method. Based on the graded multiple integrals representation method, it is easy to compute the canonical representation of multiplication operation on fuzzy numbers. Finally, the canonical representation is applied to a numerical example of fuzzy decision.

A neuro-fuzzy system modeling with self-constructing rule generationand hybrid SVD-based learning

Abstract: We propose an approach for neuro-fuzzy system modeling. A neuro-fuzzy system for a given set of input-output data is obtained in two steps. First, the data set is partitioned automatically into a set of clusters based on input-similarity and output-similarity tests. Membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster. Then, a fuzzy IF-THEN rule is extracted from each cluster to form a fuzzy rule-base. Second, a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule-base. To decrease the size of the search space and to speed up the convergence, we develop a hybrid learning algorithm which combines a recursive singular value decomposition-based least squares estimator and the gradient descent method. The proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points. Besides, it learns faster, consumes less memory, and produces lower approximation errors than other methods.

Knowledge acquisition in incomplete fuzzy information systems via the rough set approach

Abstract: Machine learning can extract desired knowledge from training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete and incomplete data sets. If attribute values are known as possibility distributions on the domain of the attributes, the system is called an incomplete fuzzy information system. Learning from incomplete fuzzy data sets is usually more difficult than learning from complete data sets and incomplete data sets. In this paper, we deal with the problem of producing a set of certain and possible rules from incomplete fuzzy data sets based on rough sets. The notions of lower and upper generalized fuzzy rough approximations are introduced. By using the fuzzy rough upper approximation operator, we transform each fuzzy subset of the domain of every attribute in an incomplete fuzzy information system into a fuzzy subset of the universe, from which fuzzy similarity neighbourhoods of objects in the system are derived. The fuzzy lower and upper approximations for any subset of the universe are then calculated and the knowledge hidden in the information system is unravelled and expressed in the form of decision rules.

An immuno-fuzzy approach to anomaly detection

Abstract: This paper presents a new technique for generating a set of fuzzy rules that can characterize the non-self space (abnormal) using only self (normal) samples. Because, fuzzy logic can provide a better characterization of the boundary between normal and abnormal, it can increase the accuracy in solving the anomaly detection problem. Experiments with synthetic and real data sets are performed in order to show the applicability of the proposed approach and also to compare with other works reported in the literature.

An intuitionistic fuzzy set based approach to intelligent data analysis: an application to medical diagnosis

Abstract: We propose a new approach for medical diagnosis by employing intuitionistic fuzzy sets (cf. Atanassov [1]; [2]) which because of additional degree of freedom in comparison with fuzzy sets (Zadeh [14]), can be viewed as their generalization. Employing intuitionistic fuzzy sets, we can simply and adequately express a hesitation concerning the objects considered - both patients and illnesses. Solution is obtained by looking for the smallest distance (cf. Szmidt and Kacprzyk [8], [11]) between symptoms that are characteristic for a patient and symptoms describing illnesses considered. We point out advantages of this new technique over the method proposed by De, Biswas and Roy [4] where intuitionistic fuzzy sets were also applied but the max-min-max composition of intuitionistic fuzzy relations was used instead of taking into account all, unchanged symptom values as proposed in this article.

Application of Fuzzy Linear Programming in Production Planning

Abstract: In this paper, a new fuzzy linear programming based methodology using a specific membership function, named as modified logistic membership function is proposed. The modified logistic membership function is first formulated and its flexibility in taking up vagueness in parameters is established by an analytical approach. This membership function is tested for its useful performance through an illustrative example by employing fuzzy linear programming. The developed methodology of FLP has provided a confidence in applying to real life industrial production planning problem. This approach of solving industrial production planning problem can have feed back within the decision maker, the implementer and the analyst. In such case this approach can be called as IFLP (Interactive Fuzzy Linear Programming). There is a possibility to design the self organizing of fuzzy system for the mix products selection problem in order to find the satisfactory solution. The decision maker, the analyst and the implementer can incorporate their knowledge and experience to obtain the best outcome.

Predicting the directionality of probability words from their membership functions

Abstract: Teigen and Brun have suggested that distinct from their numerical implications, most probability phrases are either positive or negative, in that they encourage one to think of reasons why the target event will or will not occur. We report two experiments testing our hypotheses that (a) the direction of a phrase can be predicted from properties of its membership function, and (b) this relation is invariant across contexts, and (c) —originally formulated by Teigen and Brun (1999)—that strong modifiers intensify phrase directionality. For each phrase, participants encoded membership functions by judging the degree to which it described the numerical probabilities 0.0, 0.1, …, 1.0, and also completed sentences including the target phrase. The types of reasons given in the sentence completion task were used to determine the phrase's directionality. The results support our hypotheses (a) and (b) regarding the relation between directionality and the membership functions, but we found only limited support for hypothesis (c) regarding the effects of modifiers on directionality. A secondary goal, to validate an efficient method of encoding membership functions, was also achieved. Copyright © 2003 John Wiley & Sons, Ltd.