A Framework of Fuzzy Diagnosis

Fault diagnosis has become an important component in intelligent systems, such as intelligent control systems and intelligent eLearning systems. Reiter's diagnosis theory, described by first-order sentences, has been attracting much attention in this field. However, descriptions and observations of most real-world situations are related to fuzziness because of the incompleteness and the uncertainty of knowledge, e. g., the fault diagnosis of student behaviors in the eLearning processes. In this paper, an extension of Reiter's consistency-based diagnosis methodology, Fuzzy Diagnosis, has been proposed, which is able to deal with incomplete or fuzzy knowledge. A number of important properties of the Fuzzy diagnoses schemes have also been established. The computing of fuzzy diagnoses is mapped to solving a system of inequalities. Some special cases, abstracted from real-world situations, have been discussed. In particular, the fuzzy diagnosis problem, in which fuzzy observations are represented by clause-style fuzzy theories, has been presented and its solving method has also been given. A student fault diagnostic problem abstracted from a simplified real-world eLearning case is described to demonstrate the application of our diagnostic framework.

A fuzzy Lyapunov function approach for stabilization and control of switched TS fuzzy systems

In this work, sufficient conditions for the existence of switching laws for stabilizing switched TS fuzzy systems via a fuzzy Lyapunov function are proposed. The conditions are found by exploring properties of the membership functions and are formulated in terms of linear matrix inequalities (LMIs). Stabilizing switching conditions with bounds on the decay rate solution and H1 performance are also obtained. Numerical examples illustrate the effectiveness of the proposed design methods.

Hybrid fuzzy Monte Carlo and logic programming model for distribution network reconfiguration in the presence of outages

This paper presents a methodology for distribution networks reconfiguration in outage presence in order to choose the reconfiguration that presents the lower power losses. The methodology is based on statistical failure and repair data of the distribution power system components and uses fuzzy-probabilistic modelling for system component outage parameters. Fuzzy membership functions of system component outage parameters are obtained by statistical records. A hybrid method of fuzzy set and Monte Carlo simulation based on the fuzzy-probabilistic models allows catching both randomness and fuzziness of component outage parameters. Once obtained the system states by Monte Carlo simulation, a logical programming algorithm is applied to get all possible reconfigurations for every system state. In order to evaluate the line flows and bus voltages and to identify if there is any overloading, and/or voltage violation a distribution power flow has been applied to select the feasible reconfiguration with lower power losses. To illustrate the application of the proposed methodology to a practical case, the paper includes a case study that considers a real distribution network.

An Automatic Approach for Learning and Tuning Gaussian Interval Type-2 Fuzzy Membership Functions Applied to Lung CAD Classification System

The potential of type-2 fuzzy sets for managing high levels of uncertainty in the subjective knowledge of experts or of numerical information has focused on control and pattern classification systems in recent years. One of the main challenges in designing a type-2 fuzzy logic system is how to estimate the parameters of type-2 fuzzy membership function (T2MF) and the Footprint of Uncertainty (FOU) from imperfect and noisy datasets. This paper presents an automatic approach for learning and tuning Gaussian interval type-2 membership functions (IT2MFs) with application to multi-dimensional pattern classification problems. T2MFs and their FOUs are tuned according to the uncertainties in the training dataset by a combination of genetic algorithm (GA) and crossvalidation techniques. In our GA-based approach, the structure of the chromosome has fewer genes than other GA methods and chromosome initialization is more precise. The proposed approach addresses the application of the interval type-2 fuzzy logic system (IT2FLS) for the problem of nodule classification in a lung Computer Aided Detection (CAD) system. The designed IT2FLS is compared with its type-1 fuzzy logic system (T1FLS) counterpart. The results demonstrate that the IT2FLS outperforms the T1FLS by more than 30% in terms of classification accuracy.

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.

Neurofuzzy network model construction using Bézier-Bernstein polynomial functions

Neurofuzzy modelling systems combine fuzzy logic with quantitative artificial neural networks via a concept of fuzzification by using a fuzzy membership function usually based on B-splines and algebraic operators for inference, etc. The paper introduces a neurofuzzy model construction algorithm using Bezier-Bernstein polynomial functions as basis functions. The new network maintains most of the properties of the B-spline expansion based neurofuzzy system, such as the non-negativity of the basis functions, and unity of support but with the additional advantages of structural parsimony and Delaunay input space partitioning, avoiding the inherent computational problems of lattice networks. This new modelling network is based on the idea that an input vector can be mapped into barycentric co-ordinates with respect to a set of predetermined knots as vertices of a polygon (a set of tiled Delaunay triangles) over the input space. The network is expressed as the Bezier-Bernstein polynomial function of barycentric co-ordinates of the input vector. An inverse de Casteljau procedure using backpropagation is developed to obtain the input vector's barycentric co-ordinates that form the basis functions. Extension of the Bezier-Bernstein neurofuzzy algorithm to n-dimensional inputs is discussed followed by numerical examples to demonstrate the effectiveness of this new data based modelling approach.

Improving the stability conditions of TS fuzzy systems with fuzzy Lyapunov functions

In this paper, the fuzzy Lyapunov function approach is considered for stabilizing continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing a slack LMI variable into the problem formulation. The stability results are thus used in the state feedback design which is also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilizing conditions presented. © 2011 IFAC.

Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions

In this article, the fuzzy Lyapunov function approach is considered for stabilising continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing slack LMI variables into the problem formulation. The relaxation conditions given can also be used with a class of fuzzy Lyapunov functions which also depends on the membership function first-order time-derivative. The stability results are thus extended to systems with large number of rules under membership function order relations and used to design parallel-distributed compensation (PDC) fuzzy controllers which are also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilising conditions presented. © 2013 Copyright Taylor and Francis Group, LLC.

Autonomous classification models in ubiquitous environments

Stream-mining approach is defined as a set of cutting-edge techniques designed to process streams of data in real time, in order to extract knowledge. In the particular case of classification, stream-mining has to adapt its behaviour to the volatile underlying data distributions, what has been called concept drift. Moreover, it is important to note that concept drift may lead to situations where predictive models become invalid and have therefore to be updated to represent the actual concepts that data poses. In this context, there is a specific type of concept drift, known as recurrent concept drift, where the concepts represented by data have already appeared in the past. In those cases the learning process could be saved or at least minimized by applying a previously trained model. This could be extremely useful in ubiquitous environments that are characterized by the existence of resource constrained devices. To deal with the aforementioned scenario, meta-models can be used in the process of enhancing the drift detection mechanisms used by data stream algorithms, by representing and predicting when the change will occur. There are some real-world situations where a concept reappears, as in the case of intrusion detection systems (IDS), where the same incidents or an adaptation of them usually reappear over time. In these environments the early prediction of drift by means of a better knowledge of past models can help to anticipate to the change, thus improving efficiency of the model regarding the training instances needed. By means of using meta-models as a recurrent drift detection mechanism, the ability to share concepts representations among different data mining processes is open. That kind of exchanges could improve the accuracy of the resultant local model as such model may benefit from patterns similar to the local concept that were observed in other scenarios, but not yet locally. This would also improve the efficiency of training instances used during the classification process, as long as the exchange of models would aid in the application of already trained recurrent models, that have been previously seen by any of the collaborative devices. Which it is to say that the scope of recurrence detection and representation is broaden. In fact the detection, representation and exchange of concept drift patterns would be extremely useful for the law enforcement activities fighting against cyber crime. Being the information exchange one of the main pillars of cooperation, national units would benefit from the experience and knowledge gained by third parties. Moreover, in the specific scope of critical infrastructures protection it is crucial to count with information exchange mechanisms, both from a strategical and technical scope. The exchange of concept drift detection schemes in cyber security environments would aid in the process of preventing, detecting and effectively responding to threads in cyber space. Furthermore, as a complement of meta-models, a mechanism to assess the similarity between classification models is also needed when dealing with recurrent concepts. In this context, when reusing a previously trained model a rough comparison between concepts is usually made, applying boolean logic. The introduction of fuzzy logic comparisons between models could lead to a better efficient reuse of previously seen concepts, by applying not just equal models, but also similar ones. This work faces the aforementioned open issues by means of: the MMPRec system, that integrates a meta-model mechanism and a fuzzy similarity function; a collaborative environment to share meta-models between different devices; a recurrent drift generator that allows to test the usefulness of recurrent drift systems, as it is the case of MMPRec. Moreover, this thesis presents an experimental validation of the proposed contributions using synthetic and real datasets.

A stepwise fuzzy linear programming model with possibility and necessity relations

Linear programming (LP) is the most widely used optimization technique for solving real-life problems because of its simplicity and efficiency. Although conventional LP models require precise data, managers and decision makers dealing with real-world optimization problems often do not have access to exact values. Fuzzy sets have been used in the fuzzy LP (FLP) problems to deal with the imprecise data in the decision variables, objective function and/or the constraints. The imprecisions in the FLP problems could be related to (1) the decision variables; (2) the coefficients of the decision variables in the objective function; (3) the coefficients of the decision variables in the constraints; (4) the right-hand-side of the constraints; or (5) all of these parameters. In this paper, we develop a new stepwise FLP model where fuzzy numbers are considered for the coefficients of the decision variables in the objective function, the coefficients of the decision variables in the constraints and the right-hand-side of the constraints. In the first step, we use the possibility and necessity relations for fuzzy constraints without considering the fuzzy objective function. In the subsequent step, we extend our method to the fuzzy objective function. We use two numerical examples from the FLP literature for comparison purposes and to demonstrate the applicability of the proposed method and the computational efficiency of the procedures and algorithms. © 2013-IOS Press and the authors. All rights reserved.

Multi‐method global sensitivity analysis (MMGSA) for modelling floodplain hydrological processes

When studying hydrological processes with a numerical model, global sensitivity analysis (GSA) is essential if one is to understand the impact of model parameters and model formulation on results. However, different definitions of sensitivity can lead to a difference in the ranking of importance of the different model factors. Here we combine a fuzzy performance function with different methods of calculating global sensitivity to perform a multi-method global sensitivity analysis (MMGSA). We use an application of a finite element subsurface flow model (ESTEL-2D) on a flood inundation event on a floodplain of the River Severn to illustrate this new methodology. We demonstrate the utility of the method for model understanding and show how the prediction of state variables, such as Darcian velocity vectors, can be affected by such a MMGSA. This paper is a first attempt to use GSA with a numerically intensive hydrological model.

Multi-method global sensitivity analysis (MMGSA) for modelling floodplain hydrological processes

When studying hydrological processes with a numerical model, global sensitivity analysis (GSA) is essential if one is to understand the impact of model parameters and model formulation on results. However, different definitions of sensitivity can lead to a difference in the ranking of importance of the different model factors. Here we combine a fuzzy performance function with different methods of calculating global sensitivity to perform a multi-method global sensitivity analysis (MMGSA). We use an application of a finite element subsurface flow model (ESTEL-2D) on a flood inundation event on a floodplain of the River Severn to illustrate this new methodology. We demonstrate the utility of the method for model understanding and show how the prediction of state variables, such as Darcian velocity vectors, can be affected by such a MMGSA. This paper is a first attempt to use GSA with a numerically intensive hydrological model

The Interval Constructor on classes of ML-algebras

Monoidal logic, ML for short, which formalized the fuzzy logics of continuous t-norms and their residua, has arisen great interest, since it has been applied to fuzzy mathematics, artificial intelligence, and other areas. It is clear that fuzzy logics basically try to represent imperfect or fuzzy information aiming to model the natural human reasoning. On the other hand, in order to deal with imprecision in the computational representation of real numbers, the use of intervals have been proposed, as it can guarantee that the results of numerical computation are in a bounded interval, controlling, in this way, the numerical errors produced by successive roundings. There are several ways to connect both areas; the most usual one is to consider interval membership degrees. The algebraic counterpart of ML is ML-algebra, an interesting structure due to the fact that by adding some properties it is possible to reach different classes of residuated lattices. We propose to apply an interval constructor to ML-algebras and some of their subclasses, to verify some properties within these algebras, in addition to the analysis of the algebraic aspects of them

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.

Truth Values in t-norm based Systems Many-valued FUZZY Logic

In t-norm based systems many-valued logic, valuations of propositions form a non-countable set: interval [0,1]. In addition, we are given a set E of truth values p, subject to certain conditions, the valuation v is v=V(p), V reciprocal application of E on [0,1]. The general propositional algebra of t-norm based many-valued logic is then constructed from seven axioms. It contains classical logic (not many-valued) as a special case. It is first applied to the case where E=[0,1] and V is the identity. The result is a t-norm based many-valued logic in which contradiction can have a nonzero degree of truth but cannot be true; for this reason, this logic is called quasi-paraconsistent.