Clustering methods need to be robust if they are to be useful in practice. In this paper, we analyze several popular robust clustering methods and show that they have much in common. We also establish a connection between fuzzy set theory and robust statistics, and point out the similarities between robust clustering methods and statistical methods such as the weighted least-squares technique, the M estimator, the minimum volume ellipsoid algorithm, cooperative robust estimation, minimization of probability of randomness, and the epsilon contamination model. By gleaning the common principles upon which the methods proposed in the literature are based, we arrive at a unified view of robust clustering methods. We define several general concepts that are useful in robust clustering, state the robust clustering problem in terms of the defined concepts, and propose generic algorithms and guidelines for clustering noisy data. We also discuss why the generalized Hough transform is a suboptimal solution to the robust clustering problem.
This paper proposes a new approach to fuzzy modeling. The suggested fuzzy model can express a given unknown system with a few fuzzy rules as well as Takagi and Sugeno's model (1985), because it has the same structure as that of Takagi and Sugeno's model. It is also as easy to implement as Sugeno and Yasukawa's model (1993) because its identification mimics the simple identification procedure of Sugeno and Yasukawa's model. The suggested algorithm is composed of two steps: coarse tuning and fine tuning. In coarse tuning, fuzzy C-regression model (FCRM) clustering is used, which is a modified version of fuzzy C-means (FCM). In fine tuning, gradient descent algorithm is used to precisely adjust parameters of the fuzzy model instead of nonlinear optimization methods used in other models. Finally, some examples are given to demonstrate the validity of this algorithm.
This paper addresses the structure and an associated online learning algorithm of a feedforward multilayer neural net for realizing the basic elements and functions of a fuzzy controller. The proposed fuzzy adaptive learning control network (FALCON) can be contrasted with traditional fuzzy control systems in network structure and learning ability. An online structure/parameter learning algorithm, FALCON-ART, is proposed for constructing FALCON dynamically. It combines backpropagation for parameter learning and fuzzy ART for structure learning. FALCON-ART partitions the input state space and output control space using irregular fuzzy hyperboxes according to the data distribution. In many existing fuzzy or neural fuzzy control systems, the input and output spaces are always partitioned into "grids". As the number of variables increases, the number of partitioned grids grows combinatorially. To avoid this problem in some complex systems, FALCON-ART partitions the I/O spaces flexibly based on data distribution. It can create and train FALCON in a highly autonomous way. In its initial form, there is no membership function, fuzzy partition, and fuzzy logic rule. They are created and begin to grow as the first training pattern arrives. Thus, the users need not give it any a priori knowledge or initial information. FALCON-ART can online partition the I/O spaces, tune membership functions, find proper fuzzy logic rules, and annihilate redundant rules dynamically upon receiving online data.
This paper proposes a genetic fuzzy predictor ensemble (GFPE) for the accurate prediction of the future in the chaotic or nonstationary time series. Each fuzzy predictor in the GFPE is built from two design stages, where each stage is performed by different genetic algorithms (GA). The first stage generates a fuzzy rule base that covers as many of training examples as possible. The second stage builds fine-tuned membership functions that make the prediction error as small as possible. These two design stages are repeated independently upon the different partition combinations of input-output variables. The prediction error will be reduced further by invoking the GFPE that combines multiple fuzzy predictors by an equal prediction error weighting method. Applications to both the Mackey-Glass chaotic time series and the nonstationary foreign currency exchange rate prediction problem are presented. The prediction accuracy of the proposed method is compared with that of other fuzzy and neural network predictors in terms of the root mean squared error (RMSE).
In this paper, we discuss a fuzzy classifier with ellipsoidal regions which has a learning capability. First, we divide the training data for each class into several clusters. Then, for each cluster, we define a fuzzy rule with an ellipsoidal region around a cluster center. Using the training data for each cluster, we calculate the center and the covariance matrix of the ellipsoidal region for the cluster. Then we tune the fuzzy rules, i.e., the slopes of the membership functions, successively until there is no improvement in the recognition rate of the training data. We evaluate our method using the Fisher iris data, numeral data of vehicle license plates, thyroid data, and blood cell data. The recognition rates (except for the thyroid data) of our classifier are comparable to the maximum recognition rates of the multilayered neural network classifier and the training times (except for the iris data) are two to three orders of magnitude shorter.
Jobshop scheduling problems are NP-hard problems. The durations in the reality of manufacturing are often imprecise and the imprecision in data is very critical for the scheduling procedures. Therefore, the fuzzy approach, in the framework of the Dempster-Shafer theory, commands attention. The fuzzy numbers are considered as sets of possible probabilistic distributions. After a review of some issues concerning fuzzy numbers, we discuss the determination of a unique optimal solution of the problem and then we cast a meta-heuristic (simulated annealing-SA) to this particular framework for optimization. It should be stressed that the obtained schedule remains feasible for all realizations of the operations durations.
Defuzzification is used to transform fuzzy inference results into crisp output. The standard defuzzification methods fail in some applications. It is, therefore, important to select appropriate defuzzification methods depending on the application. This paper presents some of the most important defuzzification methods and investigates their properties. With three application examples, it illustrates how to select appropriate defuzzification methods using application specific properties.
This paper presents different approaches to the problem of fuzzy rules extraction by using fuzzy clustering as the main tool. Within these approaches we describe six methods that represent different alternatives in the fuzzy modeling process and how they can be integrated with a genetic algorithms. These approaches attempt to obtain a first approximation to the fuzzy rules without any assumption about the structure of the data. Because the main objective is to obtain an approximation, the methods we propose must be as simple as possible, but also, they must have a great approximative capacity and in that way we work directly with fuzzy sets induced in the variables input space. The methods are applied to four examples and the errors obtained are specified in the different cases.
Derives an interpretation for a family of competitive learning algorithms and investigates their relationship to fuzzy c-means and fuzzy learning vector quantization. These algorithms map a set of feature vectors into a set of prototypes associated with a competitive network that performs unsupervised learning. Derivation of the new algorithms is accomplished by minimizing an average generalized distance between the feature vectors and prototypes using gradient descent. A close relationship between the resulting algorithms and fuzzy c-means is revealed by investigating the functionals involved. It is also shown that the fuzzy c-means and fuzzy learning vector quantization algorithms are related to the proposed algorithms if the learning rate at each iteration is selected to satisfy a certain condition.
In this paper, we present a formal derivation of general nonsingleton fuzzy logic systems (NSFLSs) and show how they can be efficiently computed. We give examples for special cases of membership functions and inference and we show how an NSFLS can be expressed as a "nonsingleton fuzzy basis function" expansion and present an analytical comparison of the nonsingleton and singleton fuzzy logic systems formulations. We prove that an NSFLS can uniformly approximate any given continuous function on a compact set and show that our NSFLS does a much better job of predicting a noisy chaotic time series than does a singleton fuzzy logic system (FLS).
Checking the coherence of a set of rules is an important step in knowledge base validation. Coherence is also needed in the field of fuzzy systems. Indeed, rules are often used regardless of their semantics, and it sometimes leads to sets of rules that make no sense. Avoiding redundancy is also of interest in real-time systems for which the inference engine is time consuming. A knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base. We more particularly consider knowledge bases composed of parallel fuzzy rules. Then, coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted (i.e., any value for these variables is possible) or, at least, not more restrictive than integrity constraints. Fuzzy rule representations can be implication-based or conjunction-based; we show that only implication-based models may lead to coherence problems. However, unlike conjunction-based models, they allow to design coherence checking processes. Some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules. The problem of redundancy, which is also of interest for fuzzy knowledge bases validation, is addressed for these two kinds of rules.
Studies the global asymptotic stability of a class of fuzzy systems. It demonstrates the equivalence of stability properties of fuzzy systems and linear time invariant (LTI) switching systems. A necessary and sufficient condition for the stability of such systems are given, and it is shown that under the sufficient condition, a common Lyapunov function exists for the LTI subsystems. A particular case when the system matrices can be simultaneously transformed to normal matrices is shown to correspond to the existence of a common quadratic Lyapunov function. A constructive procedure to check the possibility of simultaneous transformation to normal matrices is provided.
A procedure is presented for designing fuzzy controllers based upon variable structures techniques. Three such controllers are presented: the fuzzy equivalents of sliding-mode controllers, saturating controllers, and tanh controllers. By using an approach based upon variables structures (VSS) techniques, the stability of each of these controllers is assured. By using a sliding surface, the order of the rule base is reduced to size r/spl times/m, where r is the number of inputs and m is the number of fuzzification levels. This combination makes the proposed design procedure able to generate simple controllers with guaranteed stability properties. To illustrate the proposed design procedure, fuzzy controllers are designed for a ball-and-beam system. It is demonstrated that in spite of this system being a fourth-order unstable system, the proposed design procedure results in simple stable fuzzy controllers.
Advances in nonlinear control theory have provided the mathematical foundations necessary to establish conditions for stability of several types of adaptive fuzzy controllers. However, very few, if any, of these techniques have been compared to conventional adaptive or nonadaptive nonlinear controllers or tested beyond simulation; therefore, many of them remain as purely theoretical developments whose practical value is difficult to ascertain. In this paper we develop three case studies where we perform a comparative analysis between the adaptive fuzzy techniques in Spooner and Passino (1995,1996) and some conventional adaptive and nonadaptive nonlinear control techniques. In each case, the analysis is performed both in simulation and in implementation, in order to show practical examples of how the performance of these controllers compares to conventional controllers in real systems.
Objective function-based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors. Although there are special algorithms to discover lines, the detection of rectangles needs further treatment. A simple line-detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle, additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary. In this paper, a special fuzzy shell-clustering algorithm for rectangular contours is developed. The principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles, ellipses, and lines.
We investigate dynamic versions of fuzzy logic systems (FLSs) and, specifically, their non-Singleton generalizations (NSFLSs), and derive a dynamic learning algorithm to train the system parameters. The history-sensitive output of the dynamic systems gives them a significant advantage over static systems in modeling processes of unknown order. This is illustrated through an example in nonlinear dynamic system identification. Since dynamic NSFLS's can be considered to belong to the family of general nonlinear autoregressive moving average (NARMA) models, they are capable of parsimoniously modeling NARMA processes. We study the performance of both dynamic and static FLSs in the predictive modeling of a NARMA process.
This paper presents a new method for learning a fuzzy logic controller automatically. A reinforcement learning technique is applied to a multilayer neural network model of a fuzzy logic controller. The proposed self-learning fuzzy logic control that uses the genetic algorithm through reinforcement learning architecture, called a genetic reinforcement fuzzy logic controller, can also learn fuzzy logic control rules even when only weak information such as a binary target of "success" or "failure" signal is available. In this paper, the adaptive heuristic critic algorithm of Barto et al. (1987) is extended to include a priori control knowledge of human operators. It is shown that the system can solve more concretely a fairly difficult control learning problem. Also demonstrated is the feasibility of the method when applied to a cart-pole balancing problem via digital simulations.
This paper presents a methodology for subjective safety analysis of safety requirements specifications of software for safety-critical systems. The methodology incorporates fuzzy set modeling and evidential reasoning to assess the safety associated with safety requirements specifications. Three basic parameters-failure likelihood, consequence severity, and failure consequence probability are used to analyze a safety rule in terms of membership functions. The subjective safety description associated with the safety rule is then mapped back to the defined safety expressions which are also characterized in terms of membership functions. Such a mapping results in the production of the safety evaluation associated with the safety rule. The information produced for all safety rules can then be synthesised using an evidential reasoning approach to obtain the safety evaluation associated with the safety requirements specifications. The developed methodology is capable of dealing with multiple safety analysts who make judgements on each safety rule. A case study based on a train-set crossing is used to demonstrate the methodology.
Fuzzy system identification was applied to a biomedical system for classification purposes. Gait achieved through functional electrical stimulation (FES) of paraplegics was divided using sensor measurements of kinematic variables as inputs to five discrete events. Two identification algorithms were used to estimate the system model. Both max-min and max-product composition were used. Membership functions were either trapezoidal or triangular and all membership functions in a particular universe of discourse had the same shape and size. The universe of discourse was varied by altering the overlap between membership functions. The classification performance was assessed quantitatively, by measuring the percentage of time steps in which the correct event was found, and qualitatively, by observing types of errors. The identification algorithm affected system performance. No difference in classification was found between max-min and max-product composition. The performance was dependent on membership function overlap. A comparison of the classification found using the fuzzy rule base versus that found using a traditional look-up table demonstrated that the fuzzy approach was superior. It is speculated that the use of fuzzy logic decreased errors stemming from sensor noise and/or small variations in the input signals. The performance of this approach was compared to that of a feedforward neural network and the fuzzy system is found superior.
This paper discusses architectural and circuit-level aspects related to hardware realizations of fuzzy controllers. A brief overview on fuzzy inference methods is given focusing on chip implementation. The singleton or zero-order Sugeno's method is chosen since it offers a good tradeoff between hardware simplicity and control efficiency. The CMOS microcontroller described herein processes information in the current-domain, but input-output signals are represented as voltage to ease communications with conventional control circuitry. Programming functionalities are added by combining analog and digital techniques, giving rise to a versatile microcontroller, capable of solving different control problems. After identifying the basic component blocks, the circuits used for their implementation are discussed and compared with other alternatives. This study is illustrated with the experimental results of prototypes integrated in different CMOS technologies.