A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

Prediction of free-stall occupancy rate in dairycattle barns through fuzzy sets

The goal of this study was to develop a fuzzy model to predict the occupancy rate of free-stalls facilities of dairy cattle, aiding to optimize the design of projects. The following input variables were defined for the development of the fuzzy system: dry bulb temperature (Tdb, °C), wet bulb temperature (Twb, °C) and black globe temperature (Tbg, °C). Based on the input variables, the fuzzy system predicts the occupancy rate (OR, %) of dairy cattle in free-stall barns. For the model validation, data collecting were conducted on the facilities of the Intensive System of Milk Production (SIPL), in the Dairy Cattle National Research Center (CNPGL) of Embrapa. The OR values, estimated by the fuzzy system, presented values of average standard deviation of 3.93%, indicating low rate of errors in the simulation. Simulated and measured results were statistically equal (P>0.05, t Test). After validating the proposed model, the average percentage of correct answers for the simulated data was 89.7%. Therefore, the fuzzy system developed for the occupancy rate prediction of free-stalls facilities for dairy cattle allowed a realistic prediction of stalls occupancy rate, allowing the planning and design of free-stall barns.

Clinical signs of pneumonia in children: association with and prediction of diagnosis by fuzzy sets theory

The present study compares the performance of stochastic and fuzzy models for the analysis of the relationship between clinical signs and diagnosis. Data obtained for 153 children concerning diagnosis (pneumonia, other non-pneumonia diseases, absence of disease) and seven clinical signs were divided into two samples, one for analysis and other for validation. The former was used to derive relations by multi-discriminant analysis (MDA) and by fuzzy max-min compositions (fuzzy), and the latter was used to assess the predictions drawn from each type of relation. MDA and fuzzy were closely similar in terms of prediction, with correct allocation of 75.7 to 78.3% of patients in the validation sample, and displaying only a single instance of disagreement: a patient with low level of toxemia was mistaken as not diseased by MDA and correctly taken as somehow ill by fuzzy. Concerning relations, each method provided different information, each revealing different aspects of the relations between clinical signs and diagnoses. Both methods agreed on pointing X-ray, dyspnea, and auscultation as better related with pneumonia, but only fuzzy was able to detect relations of heart rate, body temperature, toxemia and respiratory rate with pneumonia. Moreover, only fuzzy was able to detect a relationship between heart rate and absence of disease, which allowed the detection of six malnourished children whose diagnoses as healthy are, indeed, disputable. The conclusion is that even though fuzzy sets theory might not improve prediction, it certainly does enhance clinical knowledge since it detects relationships not visible to stochastic models.

Selection of patients for myocardial perfusion scintigraphy based on fuzzy sets theory applied to clinical-epidemiological data and treadmill test results

Coronary artery disease (CAD) is a worldwide leading cause of death. The standard method for evaluating critical partial occlusions is coronary arteriography, a catheterization technique which is invasive, time consuming, and costly. There are noninvasive approaches for the early detection of CAD. The basis for the noninvasive diagnosis of CAD has been laid in a sequential analysis of the risk factors, and the results of the treadmill test and myocardial perfusion scintigraphy (MPS). Many investigators have demonstrated that the diagnostic applications of MPS are appropriate for patients who have an intermediate likelihood of disease. Although this information is useful, it is only partially utilized in clinical practice due to the difficulty to properly classify the patients. Since the seminal work of Lotfi Zadeh, fuzzy logic has been applied in numerous areas. In the present study, we proposed and tested a model to select patients for MPS based on fuzzy sets theory. A group of 1053 patients was used to develop the model and another group of 1045 patients was used to test it. Receiver operating characteristic curves were used to compare the performance of the fuzzy model against expert physician opinions, and showed that the performance of the fuzzy model was equal or superior to that of the physicians. Therefore, we conclude that the fuzzy model could be a useful tool to assist the general practitioner in the selection of patients for MPS.

Fuzzy sets in distributed traffic control

The task of controlling urban traffic requires flexibility, adaptability and handling uncertain information spread through the intersection network. The use of fuzzy sets concepts convey these characteristics to improve system performance. This paper reviews a distributed traffic control system built upon a fuzzy distributed architecture previously developed by the authors. The emphasis of the paper is on the application of the system to control part of Campinas downtown area. Simulation experiments considering several traffic scenarios were performed to verify the capabilities of the system in controlling a set of coupled intersections. The performance of the proposed system is compared with conventional traffic control strategies under the same scenarios. The results obtained show that the distributed traffic control system outperforms conventional systems as far as average queues, average delay and maximum delay measures are concerned.

Epidemiological Models of Directly Transmitted Diseases: An Approach via Fuzzy Sets Theory

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Modeling and measuring the spatial relation "along": Regions, contours and fuzzy sets

The analysis of spatial relations among objects in an image is an important vision problem that involves both shape analysis and structural pattern recognition. In this paper, we propose a new approach to characterize the spatial relation along, an important feature of spatial configurations in space that has been overlooked in the literature up to now. We propose a mathematical definition of the degree to which an object A is along an object B, based on the region between A and B and a degree of elongatedness of this region. In order to better fit the perceptual meaning of the relation, distance information is included as well. In order to cover a more wide range of potential applications, both the crisp and fuzzy cases are considered. In the crisp case, the objects are represented in terms of 2D regions or ID contours, and the definition of the alongness between them is derived from a visibility notion and from the region between the objects. However, the computational complexity of this approach leads us to the proposition of a new model to calculate the between region using the convex hull of the contours. On the fuzzy side, the region-based approach is extended. Experimental results obtained using synthetic shapes and brain structures in medical imaging corroborate the proposed model and the derived measures of alongness, thus showing that they agree with the common sense. (C) 2011 Elsevier Ltd. All rights reserved.

Von Fuzzy-Sets zu Computing-with-Words

Dieser Artikel bietet einen Überblick über die Entwicklung und Zusammenhänge der einzelnen Elemente der Fuzzy-Logik, wovon Fuzzy-Set-Theorie die Grundlage bildet. Die Grundproblematik besteht in der Handhabung von linguistischen Informationen, die häufig durch Ungenauigkeit gekennzeichnet sind. Die verschiedenen technischen Anwendungen von Fuzzy-Logik bieten eine Möglichkeit, intelligentere Computersysteme zu konstruieren, die mit unpräzisen Informationen umgehen können. Solche Systeme sind Indizien für die Entstehung einer neuen Ära des Cognitive-Computing, di in diesemArtikel ebenfalls zur Sprache kommt. Für das bessere Verständnis wird der Artikel mit einem Beispiel aus der Meteorologie (d. h. Schnee in Adelboden) begleitet.

Some geometrical methods for constructing contradiction measures on Atanassov's intuitionistic fuzzy sets

Trillas et al. (1999, Soft computing, 3 (4), 197–199) and Trillas and Cubillo (1999, On non-contradictory input/output couples in Zadeh's CRI proceeding, 28–32) introduced the study of contradiction in the framework of fuzzy logic because of the significance of avoiding contradictory outputs in inference processes. Later, the study of contradiction in the framework of Atanassov's intuitionistic fuzzy sets (A-IFSs) was initiated by Cubillo and Castiñeira (2004, Contradiction in intuitionistic fuzzy sets proceeding, 2180–2186). The axiomatic definition of contradiction measure was stated in Castiñeira and Cubillo (2009, International journal of intelligent systems, 24, 863–888). Likewise, the concept of continuity of these measures was formalized through several axioms. To be precise, they defined continuity when the sets ‘are increasing’, denominated continuity from below, and continuity when the sets ‘are decreasing’, or continuity from above. The aim of this paper is to provide some geometrical construction methods for obtaining contradiction measures in the framework of A-IFSs and to study what continuity properties these measures satisfy. Furthermore, we show the geometrical interpretations motivating the measures.

Supplementarity measures on fuzzy sets.

In this paper, we commence the study of the so called supplementarity measures. They are introduced axiomatically and are then related to incompatibility measures by antonyms. To do this, we have to establish what we mean by antonymous measure. We then prove that, under certain conditions, supplementarity and incompatibility measuresare antonymous. Besides, with the aim of constructing antonymous measures, we introduce the concept of involution on the set made up of all the ordered pairs of fuzzy sets. Finally, we obtain some antonymous supplementarity measures from incompatibility measures by means of involutions.

Obtaining contradiction measure on intuitionistic fuzzy sets from fuzzy connectives

In a previous paper, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open. The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

About t-norms on type-2 fuzzy sets.

Walker et al. defined two families of binary operations on M (set of functions of [0,1] in [0,1]), and they determined that, under certain conditions, those operations are t-norms (triangular norm) or t-conorms on L (all the normal and convex functions of M). We define binary operations on M, more general than those given by Walker et al., and we study many properties of these general operations that allow us to deduce new t-norms and t-conorms on both L, and M.

Multi-argument fuzzy measures on lattices of fuzzy sets

In this paper, we axiomatically introduce fuzzy multi-measures on bounded lattices. In particular, we make a distinction between four different types of fuzzy set multi-measures on a universe X, considering both the usual or inverse real number ordering of this lattice and increasing or decreasing monotonicity with respect to the number of arguments. We provide results from which we can derive families of measures that hold for the applicable conditions in each case.