Fuzzy Set Membership Research Articles

A study of parameter values for a Mahalanobis Distance fuzzy classifier

A supervised Mahalanobis Distance fuzzy classifier (and the related fuzzy c-means clustering algorithm) requires the a priori selection of a weighting parameter called the fuzzy exponent. Guidance in the existing literature on an appropriate value is not definitive. This paper attempts to rigorously justify previous experimental findings on suitable values for this fuzzy exponent, using the criterion that fuzzy set memberships reflect class proportions in the mixed pixels of a remotely sensed image.

PARTIAL CORRELATION COEFFICIENTS OF INTUITIONISTIC FUZZY SETS

In many applications, partial correlation for three or more intuitionistic fuzzy sets are very important, but Hung [12] do not discuss this problem. In this paper, we propose a method to calculate the partial correlation of intuitionistic fuzzy sets by means of multivariate correlation model. In order to fit into normal framework, we use empirical logit transform (see Agresti [1] and Johnson and Wichern [13]) for the degree of membership of intuitionistic fuzzy set to achive this.

Development of Highway Congestion Index with Fuzzy Set Models

Development of a highway congestion index on the basis of expert and user opinions with fuzzy set theory was attempted. Forty 1-min videotape segments showing different highway traffic flow conditions were displayed to a group of highway experts and users. They were asked to complete the relevant questionnaires on the existence and acceptability of congestion for each video segment display. Five congestion-descriptive variables of travel speed, travel rate, delay rate, travel rate ratio, and delay ratio were determined for each video segment. Information about the values of congestion-descriptive variables for the displayed videos, questionnaire responses, and fuzzy set theory were used to develop congestion fuzzy set membership functions. The membership functions characterized the congestion intensity fuzzy values of “congested” and “unacceptably congested.” The membership functions were suggested as congestion indexes measured on a scale of 0 to 1. In the questionnaire, the surveyed group also provided opinions regarding the relative importance of the descriptive variables in explaining congestion. This information was used to determine average weighting values for the developed indexes. Using the average weighting values, overall indexes for “congested” and “unacceptably congested” were developed. The overall congestion indexes were also suggested for congestion intensity assessment and evaluation. The relationships between the Highway Capacity Manual levels of service and the developed congestion indexes were studied. The results were used to address some issues regarding normative index development. Although the study findings are based on a rather limited database and are location specific, the applied methodology can be used to capture congestion intensity perception in any future index development. The study index development was based on fuzzy set theory and is suggested as an alternative or complementary approach to past normative index development methods.

MPEG VBR video traffic modeling and classification using fuzzy technique

We present an approach for MPEG variable bit rate (VBR) video modeling and classification using fuzzy techniques. We demonstrate that a type-2 fuzzy membership function, i.e., a Gaussian MF with uncertain variance, is most appropriate to model the log-value of I/P/B frame sizes in MPEG VBR video. The fuzzy c-means (FCM) method is used to obtain the mean and standard deviation (std) of T/P/B frame sizes when the frame category is unknown. We propose to use type-2 fuzzy logic classifiers (FLCs) to classify video traffic using compressed data. Five fuzzy classifiers and a Bayesian classifier are designed for video traffic classification, and the fuzzy classifiers are compared against the Bayesian classifier. Simulation results show that a type-2 fuzzy classifier in which the input is modeled as a type-2 fuzzy set and antecedent membership functions are modeled as type-2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters.

Evolutionary learning of rule premises for fuzzy modelling

The task of fuzzy modelling involves specification of rule antecedents and determination of their consequent counterparts. Rule premises appear here a critical issue since they determine the structure of a rule base. This paper proposes a new approach to extracting fuzzy rules from training examples by means of genetic-based premise learning. In order to construct a 'parsimonious' fuzzy model with high generalization ability, general premise structure allowing incomplete compositions of input variables as well as OR connectives of linguistic terms is considered. A genetic algorithm is utilized to optimize both the premise structure of rules and fuzzy set membership functions at the same time. Determination of rule conclusions is nested in the premise learning, where consequences of individual rules are determined under fixed preconditions. The proposed method was applied to the well-known gas furnace data of Box and Jenkins to show its validity and to compare its performance with those of other works.

Application of fuzzy measures in multi-criteria evaluation in GIS

Multi-criteria evaluation (MCE) is perhaps the most fundamental of decision support operations in geographical information systems (GIS). This paper reviews two main MCE approaches employed in GIS, namely Boolean and Weighted Linear Combination (WLC), and discusses issues and problems associated with both. To resolve the conceptual differences between the two approaches, this paper proposes the application of fuzzy measures, a concept that is broader but that includes fuzzy set membership, and argues that the standardized factors of MCE belong to a general class of fuzzy measures and the more specific instance of fuzzy set membership. This perspective provides a strong theoretical basis for the standardization of factors and their subsequent aggregation. In this context, a new aggregation operator that accommodates and extends the Boolean and WLC approaches is discussed: the Ordered Weighted Average. A case study of industrial allocation in Nakuru, Kenya is employed to illustrate the different approaches.

A ROUGH SET/FUZZY LOGIC BASED DECISION MAKING SYSTEM FOR MEDICAL APPLICATIONS

A method of performing prognostic modeling of disease states is proposed. The technique uses rough sets to extract rules from a database. The data is then reformatted into a fuzzy logic template, and a learning algorithm is used to adjust the fuzzy set membership functions. The method is applied to the POSCH problem, which looks at risk factors associated with the progression of coronary artery disease. The POSCH data has several shortcomings, including a limited number of cases, correlated inputs, as well as noise on both the inputs and outcome. The problem was to predict progression of atherosclerosis in the LAD three years after baseline based on physiologic data available at baseline. The proposed rough/fuzzy set method correctly predicted progression of atherosclerotic disease in 69% of the patients, which is statistically better than neural network, rough set and logistic models performed.

Multicriteria-multistages linguistic evaluation and ranking of machine tools

A model for linguistic evaluation of machine tools parameters, as well as procedure for alternatives ranking is shown in the paper. Linguistic quantification of machine tools' parameters values and their significance for given machining conditions provides the generation of machine tools suitability measure. It enables ranking of alternatives which is done in two stages. First-stage ranking is related to selection of linguistic value from pre-defined dictionary according to generated suitability measure. Second-stage ranking includes evaluation of some additional parameters and experts' subjective belief about their importance for given conditions. For this purpose Choquet integral and Sugeno's γ-measure have been chosen. Fuzzy sets membership functions of values of appropriate linguistic variables are formed according to literature and empirical information and data and results of an experiment conducted in laboratory and real industrial environment.

A Bayesian Evaluation of Fuzzy Logic in a Classification Problem

Models based on fuzzy sets are increasingly being used in academic and industrial research and applications, primarily for data analysis and systems control. The possible consequences of their popular use necessitate their evaluation from a probabilistic viewpoint. In this paper, a simple fuzzy logic model is translated into a Bayesian setting and used to address a discrimination and attribution (or classification) problem. Probabilistic tools and concepts are then employed to evaluate the fuzzy-set paradigm. The translation suggests that applications of fuzzy-set theory may suffer from many serious problems, primarily due to the unobservability of the fuzzy-set membership vector and to the poor structure of the fuzzy-set model. These difficulties are illustrated using the well-known Fisher iris data. For example, Table 11 suggests that fuzzy-set models may find probabilistic mixtures only when the evidence is thoroughly overwhelming in Bayesian models

An improved fuzzy logic controller using genetic algorithmic techniques for building management systems

Abstract This paper presents the design and control performance of a fuzzy logic controller (FLC) based on the system’s output errors for a building management system (BMS). The system comprises a heating, ventilating and air conditioning (HVAC) plant which regulates the air temperature and relative humidity in a full-scale office room. The plant has 3 control inputs, namely the heater, cooler and humidifier. It has been shown that good set point regulation can be achieved by employing 5 fuzzy input set membership functions for each output error and 3 fuzzy output set membership functions for each incremental control input as well as having good relational rules to determine the control strategy. It is also shown that the performance can be further improved by employing genetic algorithms to search for the 9 optimal fuzzy set output membership functions while maintaining the 10 fuzzy input sets constant.

Universal Pavement Distress Evaluator Based on Fuzzy Sets

Setting priorities for pavement maintenance and rehabilitation depends on the availability of a universal scale for assessing the condition of every element in the network. The condition of a pavement section has traditionally been assessed by several condition indexes. The present serviceability index (PSI) is one common evaluator used to describe the functional condition with respect to ride quality. Pavement condition index is another index commonly used to describe the extent of distress on a pavement section. During the decision-making process, both classes of indexes are needed to evaluate the overall status of a pavement section in comparison to other sections in the network. Traditionally, regression techniques were used for the development of functions that relate condition indexes to the information recorded in the pavement management database. This approach produces mathematical functions that are limited to a particular database. The functions so developed may also suffer from inaccuracies due to errors in data collection and recording. There is a need for a more generalized approach for the evaluation of pavement conditions to enable efficient management of large transportation networks. The development of a universal measure capable of formally assessing the condition of a pavement section within the universe of pavement conditions is described. This is accomplished by the fusion of a set of fuzzy membership functions that describe different parameters in the database with the perception of each parameter’s significance. The model output is the fuzzy distress index (FDI), which combines the extent of structural distress with traditional performance parameters such as roughness to describe the overall status of the pavement section. The behavior of FDI over time is examined for a random sample of pavement sections and is compared with trends in the corresponding PSI values (PSI was used only because it was readily available in the database). The results indicate that the flexible, universal FDI is a consistent and accurate measure of the overall pavement condition. The set of generated membership functions describing the different extents of every distress type can be easily standardized over the 50 states, allowing the model to be implemented on any pavement at any location. Also, the parameter weights used in the assessment may be easily adjusted (increased or decreased) to reflect changes in maintenance policies or budget availability at the local, state, or national decision-making level. Moreover, the concept allows for the omission of any number of parameters that might not be available in a particular pavement management database.

Evolving fuzzy rule based controllers using genetic algorithms

The synthesis of genetics-based machine learning and fuzzy logic is beginning to show promise as a potent tool in solving complex control problems in multi-variate non-linear systems. In this paper an overview of current research applying the genetic algorithm to fuzzy rule based control is presented. A novel approach to genetics-based machine learning of fuzzy controllers, called a Pittsburgh Fuzzy Classifier System # 1 (P-FCS1) is proposed. P-FCS1 is based on the Pittsburgh model of learning classifier systems and employs variable length rule-sets and simultaneously evolves fuzzy set membership functions and relations. A new crossover operator which respects the functional linkage between fuzzy rules with overlapping input fuzzy set membership functions is introduced. Experimental results using P-FCS 1 are reported and compared with other published results. Application of P-FCS1 to a distributed control problem (dynamic routing in computer networks) is also described and experimental results are presented.

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 new method for implementing fuzzy retrieval from a spatial database

An algorithm has been defined to implement a new type of fuzzy query (i.e., iterative, ranked retrieval) from an exclusive, partitional, spatial database. First, a working definition of the type of fuzzy query is developed. This type of query does not depend on the traditional definition of a fuzzy set membership function. This membership function allows only a predetermined number ( N) of the highest ranking items to belong to the set. Membership in the set is determined not just by an item's facets, but also by the distribution of other items which are retrieved as a result of the requirements established by the query. Next, a solution technique for implementing this type of fuzzy query is developed. The algorithm works by first converting a fuzzy query to a range query, and then retrieving and ranking all items which fall within the range query's boundaries. The algorithm proceeds by iteratively lowering the desired target rank until the N “best” items are found. When the target rank is lowered, range query limits are implicitly calculated from the lowered target rank. A prototype system implementing this technique for retrieval from a PLOP hashing database is described. A simple example is presented. Finally, conclusions are drawn and future research directions discussed.

Sigmoidal theory

Sigmoidal theory is a new theory that defines and analyzes a new family of functions called sigmoidal functions. A number of theorems proved show that there is an infinite number of classes of sigmoidal functions and provide us with a method of constructing those classes. Subsequently, multi-sigmoidal functions can be constructed using classes of sigmoidal functions. The applications of sigmoidal functions range from fuzzy sets to neural networks and pattern recognition. In particular, sigmoidal functions are shown to be very appropriate functions to model fuzzy set membership. The formalization of sigmoidal functions here is expected to increase the interest in them and result in a number of new applications.

Systematic design and stability analysis of a fuzzy logic controller

Although the fuzzy logic controller has been adopted in many engineering applications, its performance is not guaranteed since there is no definite stability analysis. It may be the main factor that we hesitate to adopt the FLC in critical applications. In this paper, a guideline of systematic analysis and design of an FLC is presented for guaranteeing the stability and/or the performance. Based on the well-developed variable structure system theory, we account for the relationship between the tracking accuracy of a control system and the I/O fuzzy set membership shapes of the FLC. It is also possible to explain the performance or the stability of commonly used FLCs through the proposed guideline for the design and analysis.

The Tool Fuzzy-Detector for Evaluation of Dairy Farm Management with Fuzzy Sets

The method and tool FUZZY-DETECTOR has been developed to enable fast and easy building of knowledge-based systems (rule-based) for domains where data are qualitative and uncertain. The method is based on the fuzzy set theory and membership functions are used for modelling and calculation. FUZZY-DETECTOR is applied in two knowledge-based systems for the evaluation of dairy farm management.

A systematic approach to obtaining fuzzy sets for control systems

The behavior of a control practitioner, who may be acting only on subjective or intuitive knowledge could be represented by one or more fuzzy subsets of the set (assumed to be finite) of all states of a controlled system. In this paper a systematic approach is proposed for determining the grades of membership of such fuzzy sets. These sets and the plant are then optimized against a performance index. The results of applying this method to a particular second order system are presented and then compared with those obtained by using a continuous deterministic model of the same plant. Implications for the design of controllers are then drawn.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Classification of taste responses in brain stem: membership in fuzzy sets

1. Classification methods in sensory systems in general, and gustation in particular, tend to place each of the relevant objects, such as stimuli or neurons, into one class each. Some of these methods are based on the responsiveness of neurons to various stimuli; in these, each group must contain a variety of nonidentical members because of the individuality of each neuron or stimulus. 2. The "fuzzy" set method is appropriate for more accurate classification in such heterogeneous populations. In this method each member is given graded membership in several sets rather than membership in only one set. In the present paper we subjected previously published data on the responses of individual taste neurons to a variety of stimuli to fuzzy set analysis. 3. We found that the amounts of response of 46 neurons in the solitary nucleus of the rat to NaCl, HCl, sucrose, quinine HCl, and KCl could accurately be accounted for by giving each a grade of membership in three sets; the same held in the parabrachial nucleus of the rat for the responses of 41 neurons to the first four of these stimuli. The response was calculated as the sum of the products of the stimulus times neuron ratings in each set. 4. Temporal patterns of response have often been related, but with only moderate success, to the identity of the stimulus or neuron. These patterns could be accurately accounted for with the present method. Each of the products of designated parts of the stimulus ratings times the neuron ratings gave the basis for accurate description of the temporal course of the response of each neuron to each stimulus. 5. This method appears to account for the varieties of amount and temporal pattern of response of taste neurons with a simple mathematical process of few parameters. This suggests that within the known complexities of receptor mechanisms and mechanisms of neural processing, the neural message is reduced to a rather simple form. 6. The fuzzy set approach, which is based on disclosing underlying sets rather than placement of heterogeneous members into one of several essentialistic groups, may be useful in disclosure of the underlying mechanisms producing the neural responses in taste.

Interactive, multiobjective programming for forest resources management

Forest resources management is approached by a visual “free search” type of interactive procedure for multiple objective linear programming, which accounts for imprecision in the model by use of fuzziness. Specifically, a ponderosa pine forest resources management problem is examined. The problem is concerned with performing on given subwatersheds six different levels of vegetation treatment strategies ranging between 0 and 100% tree basal area cut in the forest. This problem is evaluated to “satisfice” six multiple and usually conflicting objective functions, namely, water yield, herbage production, sediment yield, recreational use, economicbenefit, and cost. The problem is subject to physical constraints, some of which, such as the total area that may be subjected to high percentage tree basal area removal, are not precisely defined and may be taken as fuzzy. Further, one objective, recreational use level, is defined as a fuzzy goal. The problem is formulated as a multiple objective linear programming model, in which fuzzy set membership value is also considered as an objective to be optimized. The effects of the fuzziness on the trade-offs between the conflicting objectives is examined using a visual interactive system called PARETO RACE. The system enables a decision-maker to search freely any part of the efficient frontier by controlling the speed and direction of motion. By using PARETO RACE, the decision-maker is also able to analyze relationships between the objective function values and visualize interactively how fuzziness in resources and objectives is reflected in objective functions. The model appears to lead to compromise policies acceptable to the decision-maker.