Showing papers on "Membership function published in 1992"

Fuzzy basis functions, universal approximation, and orthogonal least-squares learning

Abstract: Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules. >

Fuzzy systems are universal approximators

Abstract: The author proves that fuzzy systems are universal approximators. The Stone-Weierstrass theorem is used to prove that fuzzy systems with product inference, centroid defuzzification, and a Gaussian membership function are capable of approximating any real continuous function on a compact set to arbitrary accuracy. This result can be viewed as an existence theorem of an optimal fuzzy system for a wide variety of problems. >

Fuzzy min-max neural networks. I. Classification

Abstract: A supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described. Each fuzzy set is an aggregate (union) of fuzzy set hyperboxes. A fuzzy set hyperbox is an n-dimensional box defined by a min point and a max point with a corresponding membership function. The min-max points are determined using the fuzzy min-max learning algorithm, an expansion-contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining. The use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher-level decision making. The relationship between fuzzy sets and pattern classification is described. The fuzzy min-max classifier neural network implementation is explained, the learning and recall algorithms are outlined, and several examples of operation demonstrate the strong qualities of this new neural network classifier. >

Putting Rough Sets and Fuzzy Sets Together

Abstract: In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

Fuzzy Min-Max Neural Networks-Part 1 : Classification

Abstract: A supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described. Each fuzzy set is an aggregate (union) of fuzzy set hyperboxes. A fuzzy set hyperbox is an n-dimensional box defined by a min point and a max point with a corresponding membership function. The min-max points are determined using the fuzzy min-max learning algorithm, an expansionxontraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining. The use of a fuzzy set approach to pattern classification inherently provides degree of membership information that is extremely useful in higher level decision mak- ing. This paper will describe the relationship between fuzzy sets and pattern classification. It explains the fuzzy min-max classifier neural network implementation, it outlines the learning and recall algorithms, and it provides several examples of operation that demonstrate the strong qualities of this new neural network classifier. AmRN classification is a key element to many engi- P neering solutions. Sonar, radar, seismic, and diagnostic applications all require the ability to accurately classify a situation. Control, tracking, and prediction systems will often use classifiers to determine input-output relationships. Because of this wide range of applicability, pattern classification has been studied a great deal (13), (15), (19). This paper describes a neural network classifier that creates classes by aggregating several smaller fuzzy sets into a single fuzzy set class. This technique, introduced in (42) as an extension of earlier work (41), can learn pattern classes in a single pass through the data, it can add new pattern classes on the fly, it can refine existing pattern classes as new information is received, and it uses simple operations that allow for quick execution. Fuzzy min-max classification neural networks are built using hyperbox fuzzy sets. A hyperbox defines a region of the n-dimensional pattern space that has patterns with full class membership. A hyperbox is completely defined by its min point and its max point, and a membership function is defined with respect to these hyperbox min-max points. The min-max (hyperbox) membership function combination defines a fuzzy set, hyperbox fuzzy sets are aggregated to form a single fuzzy set class, and the resulting structure fits naturally into a neural network framework; hence this classification system is called a fuzzy min-max classification neural network. Learning in the fuzzy min-max classification neural network is performed by properly placing and adjusting hyperboxes in the pattern space.

Fuzzy Logic Controllers

Abstract: Fuzzy Set Theory, introduced by Zadeh in 1965 [77], has been the subject of much controversy and debate. In recent years, it has found many applications in a variety of fields. Among the most successful applications of this theory has been the area of Fuzzy Logic Control (FLC) initiated by the work of Mamdani and Assilian [36]. FLC has had considerable success in Japan, where many commercial products using this technology, have been built.

Topological and Fuzzy Rough Sets

Abstract: The approximation theory is studied via rough sets, fuzzy sets and topological spaces (more precisely, Frechet spaces). Rough set theory is a set theory via knowledge bases. This set theory is extended to fuzzy sets and Frechet topological spaces. By these results one can show that the classification preserves the approximation. We also showed that within the approximation theory, fuzzy set and Frechet topology are intrinsically equivalent notions. Finally, we show that even though approximation is a compromised solution, the three theories allow one to draw an exact solution whenever there are adequate approximations. This implies that these three approaches are good approximation theories.

System for accessing a database with an iterated fuzzy query notified by retrieval response

Abstract: A method and system are disclosed for evaluating imprecise database queries. At the time the imprecise query is executed, a membership function, representing the imprecise criteria of the query, is applied to entries of the databases. Data items are then accordingly identified depending on the results obtained from applying the membership function to entries of the database.

Multi-attribute classification using fuzzy integral

Abstract: Fuzzy set theory can provide a suitable framework for pattern classification, because of the inherent fuzziness involved in the definition of a class or a cluster. Fuzzy set theory is discussed based on a fuzzy pattern matching procedure, where partial matching values with respect to a given attribute are combined. This approach is closely related to a statistical approach to pattern classification. A new method based on a fuzzy integral and possibility theory is presented. A critical examination of the statistical approach and the supervised learning process is outlined. Experimental test results on real data are presented. >

Fuzzy data processing method and data smoothing filter

Abstract: A unique data processing method and a fuzzy smoothing filter are disclosed which employ a three-dimensional elliptic membership function based on the fuzzy logic to remove noises from data including a sequence of measured points for evaluating the linearity of the measured points. The evaluation is carried out by superimposing the center of an ellipse represented by an elliptic membership function on a substantial central portion of inputted data, evaluating a factor representing the linearity of the inputted data by summing the degrees of membership derived at respective inputted data, calculating the ratio of the factor to a factor representing an ideal linearity of the inputted data, rotating the ellipse by a predetermined angular distance to derive the ratio at that position, repeating the rotating step until the peak is found, and determining the ratio when the peak is found as the linearity of the inputted data. The fuzzy smoothing filter is comprised of a data input for inputting data to be smoothed, a multi-dimensional membership function generator for calculating a degree of membership for the inputted data, a calculator for deriving an angle of the data and calculating the linearity of the data, smoothing processors coupled to receive the angle and the linearity of the data from the calculator for executing a smoothing operation in a plurality of modes, a selector for selecting one mode from among the plurality of smoothing modes, and a store for holding smoothing filter parameters supplied to the calculator and the selector.

Fuzzy model reference learning control

Abstract: A learning controller that is developed by synthesizing several basic ideas from fuzzy set and control theory, self-organizing control, and conventional adaptive control is introduced. A learning mechanism that observes the plant outputs and adjusts the rules in a direct fuzzy controller so that the overall system behaves like a reference model is used. The effectiveness of this fuzzy model reference learning controller is evaluated by comparing its performance to that of a self-organizing controller for a cart and pendulum system. >

Gaussian membership functions are most adequate in representing uncertainty in measurements

Abstract: In rare situations, like fundamental physics, we perform experiments without knowing what their results will be. In the majority of real-life measurement situations, we more or less know beforehand what kind of results we will get. Of course, this is not the precise knowledge of the type 'the result will be between alpha - beta and alpha + beta,' because in this case, we would not need any measurements at all. This is usually a knowledge that is best represented in uncertain terms, like 'perhaps (or 'most likely', etc.) the measured value x is between alpha - beta and alpha + beta.' Traditional statistical methods neglect this additional knowledge and process only the measurement results. So it is desirable to be able to process this uncertain knowledge as well. A natural way to process it is by using fuzzy logic. But, there is a problem; we can use different membership functions to represent the same uncertain statements, and different functions lead to different results. What membership function do we choose? In the present paper, we show that under some reasonable assumptions, Gaussian functions mu(x) = exp(-beta(x(exp 2))) are the most adequate choice of the membership functions for representing uncertainty in measurements. This representation was efficiently used in testing jet engines to airplanes and spaceships.

Combining estimates using fuzzy sets

Abstract: A system for obtaining a estimate of a physical parameter. The system is a multiple point system, in that estimates are obtained from a number of distributed sensors at a single observation time, or alternatively, from a single sensor at a number of observation times. Each sensor point is associated with a processor that calculates an estimate pair, consisting of an estimate of the parameter and a variance. Each estimate pair is used to construct a membership function. The membership functions are combined, and the combined function is optimized to determine a final estimate.

Current-mode analog fuzzy hardware with voltage input interface and normalization locked loop

Abstract: A voltage-input current-output membership function circuit (MFC) and a normalization locked loop (NLL) are proposed. They are useful building blocks for current-mode analog fuzzy hardware. The voltage-input current-output MFC consists of two-source-coupled-type operational transconductance amplifiers (OTAs). The MFC is used in the input parts of the analog fuzzy hardware system. The fuzzy hardware system can execute the singleton fuzzy control algorithm. In the algorithm, the weighted average operation is processed. When the weighted average operation is directly realized by analog circuits, a divider must be implemented. The NLL circuit, which can process the weighted average operation without the divider, is implemented using a one-source-coupled OTA. The proposed circuits were designed by using 2- mu m CMOS design rules and their operations were confirmed using SPICE simulations. >

Nonlinear Membership Functions in Multiobjective Fuzzy Optimization of Mechanical and Structural Systems

Abstract: An application of fuzzy mathematical programming techniques to multiple objective design problems is presented. Two examples dealing with the multiobjective design of mechanical and structural systems are considered. The concept of a Pareto-optimal and fuzzy Pareto-optimal solution is discussed, and it is shown that the resulting formulation yields Pareto-optimal solutions. The fundamental assumption in fuzzy mathematical programming applications involving the use of linear membership functions is critically examined. Several nonlinear shapes for the membership functions of the fuzzy sets are proposed, consistent with varying perceptions of the designer, and are analyzed to determine their impact on the overall design process. These shapes correspond to what we define as the coefficient of membership satiation. It is seen that optimum designs for both examples are strongly influenced by the sign of the membership satiation coefficient.