Nous aspectes de la teoria dels subconjunts borrosos i estudi d'algunes aplicacions a models econòmics

Fonaments de la Matemàtica per al tractament de la Incertesa. Noves aportacions a l’estudi de les Equacions Borroses i de les Equacions Diferencials Borroses. Aplicacions de la Matemàtica de la Incertesa al comportament de models de la teoria econòmica.

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

Past, present and future mathematical models for buildings (i)

This is the first of two articles presenting a detailed review of the historical evolution of mathematical models applied in the development of building technology, including conventional buildings and intelligent buildings. After presenting the technical differences between conventional and intelligent buildings, this article reviews the existing mathematical models, the abstract levels of these models, and their links to the literature for intelligent buildings. The advantages and limitations of the applied mathematical models are identified and the models are classified in terms of their application range and goal. We then describe how the early mathematical models, mainly physical models applied to conventional buildings, have faced new challenges for the design and management of intelligent buildings and led to the use of models which offer more flexibility to better cope with various uncertainties. In contrast with the early modelling techniques, model approaches adopted in neural networks, expert systems, fuzzy logic and genetic models provide a promising method to accommodate these complications as intelligent buildings now need integrated technologies which involve solving complex, multi-objective and integrated decision problems.

Past, present and future mathematical models for buildings (ii)

This article is the second part of a review of the historical evolution of mathematical models applied in the development of building technology. The first part described the current state of the art and contrasted various models with regard to the applications to conventional buildings and intelligent buildings. It concluded that mathematical techniques adopted in neural networks, expert systems, fuzzy logic and genetic models, that can be used to address model uncertainty, are well suited for modelling intelligent buildings. Despite the progress, the possible future development of intelligent buildings based on the current trends implies some potential limitations of these models. This paper attempts to uncover the fundamental limitations inherent in these models and provides some insights into future modelling directions, with special focus on the techniques of semiotics and chaos. Finally, by demonstrating an example of an intelligent building system with the mathematical models that have been developed for such a system, this review addresses the influences of mathematical models as a potential aid in developing intelligent buildings and perhaps even more advanced buildings for the future.

Construction of neurofuzzy models for imbalanced data classification

We propose a new class of neurofuzzy construction algorithms with the aim of maximizing generalization capability specifically for imbalanced data classification problems based on leave-one-out (LOO) cross validation. The algorithms are in two stages, first an initial rule base is constructed based on estimating the Gaussian mixture model with analysis of variance decomposition from input data; the second stage carries out the joint weighted least squares parameter estimation and rule selection using orthogonal forward subspace selection (OFSS)procedure. We show how different LOO based rule selection criteria can be incorporated with OFSS, and advocate either maximizing the leave-one-out area under curve of the receiver operating characteristics, or maximizing the leave-one-out Fmeasure if the data sets exhibit imbalanced class distribution. Extensive comparative simulations illustrate the effectiveness of the proposed algorithms.

ANFIS BASED MODELS FOR ACCESSING QUALITY OF WIKIPEDIA ARTICLES

Wikipedia is a free, web-based, collaborative, multilingual encyclopedia project supported by the non-profit Wikimedia Foundation. Due to the free nature of Wikipedia and allowing open access to everyone to edit articles the quality of articles may be affected. As all people don’t have equal level of knowledge and also different people have different opinions about a topic so there may be difference between the contributions made by different authors. To overcome this situation it is very important to classify the articles so that the articles of good quality can be separated from the poor quality articles and should be removed from the database. The aim of this study is to classify the articles of Wikipedia into two classes class 0 (poor quality) and class 1(good quality) using the Adaptive Neuro Fuzzy Inference System (ANFIS) and data mining techniques. Two ANFIS are built using the Fuzzy Logic Toolbox [1] available in Matlab. The first ANFIS is based on the rules obtained from J48 classifier in WEKA while the other one was built by using the expert’s knowledge. The data used for this research work contains 226 article’s records taken from the German version of Wikipedia. The dataset consists of 19 inputs and one output. The data was preprocessed to remove any similar attributes. The input variables are related to the editors, contributors, length of articles and the lifecycle of articles. In the end analysis of different methods implemented in this research is made to analyze the performance of each classification method used.

Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and Mart´ınez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale.We find that the key of computational techniques
based on linguistic 2-tuples is to set suitable numerical scale with
the purpose of making transformations between linguistic 2-tuples
and numerical values. By defining the concept of the transitive
calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.

Short term load forecasting using interval type-2 fuzzy logic systems

Accurate Short Term Load Forecasting (STLF) is essential for a variety of decision making processes. However, forecasting accuracy may drop due to presence of uncertainty in the operation of energy systems or unexpected behavior of exogenous variables. This paper proposes the application of Interval Type-2 Fuzzy Logic Systems (IT2 FLSs) for the problem of STLF. IT2 FLSs, with extra degrees of freedom, are an excellent tool for handling prevailing uncertainties and improving the prediction accuracy. Experiments conducted with real datasets show that IT2 FLS models appropriately approximate future load demands with an acceptable accuracy. Furthermore, they demonstrate an encouraging degree of accuracy superior to feedforward neural networks used in this study.

An empirical analysis of colour image segmentation using fuzzy c-means clustering

In this paper, an empirical analysis to examine the effects of image segmentation with different colour models using the fuzzy c-means (FCM) clustering algorithm is conducted. A qualitative evaluation method based on human perceptual judgement is used. Two sets of complex images, i.e., outdoor scenes and satellite imagery, are used for demonstration. These images are employed to examine the characteristics of image segmentation using FCM with eight different colour models. The results obtained from the experimental study are compared and analysed. It is found that the CIELAB colour model yields the best outcomes in colour image segmentation with FCM.

Enhancing the failure mode and effect analysis methodology with fuzzy inference techniques

Traditional Failure Mode and Effect Analysis (FMEA) adopts the Risk Priority Number (RPN) ranking model to evaluate failure risks, to rank failures, as well as to prioritize actions. Although this approach is simple, it suffers from several shortcomings. In this paper, we investigate a number of fuzzy inference techniques for determining the RPN scores, in an attempt to overcome the weaknesses associated with the traditional RPN model. The main objective is to examine the possibility of using fuzzy rule interpolation and reduction techniques to design new fuzzy RPN models. The performance of the fuzzy RPN models is evaluated using a real-world case study pertaining to the test handler process in a semiconductor manufacturing plant. The FMEA procedure for the test handler is performed, and a fuzzy RPN model is developed. In addition, improvement to the fuzzy RPN model is proposed by refining the weights of the fuzzy production rules, hence a new weighted fuzzy RPN model. The ability of the weighted fuzzy RPN model in failure risk evaluation with a reduced rule base is also demonstrated.

Fuzzy ARTMAP and hybrid evolutionary programming for pattern classification

In this paper, an Evolutionary Artificial Neural Network (EANN) that combines the Fuzzy ARTMAP (FAM) network and a Hybrid Evolutionary Programming (HEP) model is introduced. The proposed FAM-HEP model, which combines the strengths of FAM and HEP, is able to construct its network structure autonomously as well as to perform learning and evolutionary search and adaptation concurrently. The effectiveness of the proposed FAM-HEP network is assessed empirically using several benchmark data sets and a real medical diagnosis problem. The performance of FAM-HEP is analyzed, and the results are compared with those of FAM-EP, FAM, and other classification models. In general, the results of FAM-HEP are better than those of FAM-EP and FAM, and are comparable with those from other classification models. The study also reveals the potential of FAM-HEP as an innovative EANN model for undertaking pattern classification problems in general, and a promising computerized decision support tool for tackling medical diagnosis tasks in particular.

Authenticating the identity of computer users with typing biometrics and the fuzzy min-max neural network

In this paper, typing biometrics is applied as an additional security measure to the password-based or Personal Identification Number (PIN)-based systems to authenticate the identity of computer users. In particular, keystroke pressure and latency signals are analyzed using the Fuzzy Min-Max (FMM) neural network for authentication purposes. A special pressure-sensitive keyboard is designed to collect keystroke pressure signals, in addition to the latency signals, from computer users when they type their passwords. Based on the keystroke pressure and latency signals, the FMM network is employed to classify the computer users into two categories, i.e., genuine users or impostors. To assess the effectiveness of the proposed approach, two sets of experiments are conducted, and the results are compared with those from statistical methods and neural network models. The experimental outcomes positively demonstrate the potentials of using typing biometrics and the FMM network to provide an additional security layer for the current password-based or PIN-based methods in authenticating the identity of computer users.

An evolutionary-based similarity reasoning scheme for monotonic multi-input fuzzy inference systems

In this paper, an Evolutionary-based Similarity Reasoning (ESR) scheme for preserving the monotonicity property of the multi-input Fuzzy Inference System (FIS) is proposed. Similarity reasoning (SR) is a useful solution for undertaking the incomplete rule base problem in FIS modeling. However, SR may not be a direct solution to designing monotonic multi-input FIS models, owing to the difficulty in getting a set of monotonically-ordered conclusions. The proposed ESR scheme, which is a synthesis of evolutionary computing, sufficient conditions, and SR, provides a useful solution to modeling and preserving the monotonicity property of multi-input FIS models. A case study on Failure Mode and Effect Analysis (FMEA) is used to demonstrate the effectiveness of the proposed ESR scheme in undertaking real world problems that require the monotonicity property of FIS models.

Modeling of a stereo vision system using a genetic algorithm based fuzzy linear regression

In this paper a fuzzy linear regression (FLR) model integrated with a genetic algorithm (GA) is proposed. The proposed GA-FLR model is applied to modeling of a stereo vision system. A set of empirical data from stereo vision object measurement is collected based on the full factorial design technique. Three regression models, namely ordinary least-squares regression (OLS), FLR, and GA-FLR, are developed, and with their performances compared. The results show that the proposed GA-FLR model performs better than OLS and FLR in modeling of a stereo vision system.

A new neural network-based type reduction algorithm for interval type-2 fuzzy logic systems

This paper introduces a new type reduction (TR) algorithm for interval type-2 fuzzy logic systems (IT2 FLSs). Flexibility and adaptiveness are the key features of the proposed non-parametric algorithm. Lower and upper firing strengths of rules as well as their consequent coefficients are fed into a neural network (NN). NN output is a crisp value that corresponds to the defuzzified output of IT2 FLSs. The NN type reducer is trained through minimization of an error-based cost function with the purpose of improving modelling and forecasting performance of IT2 FLS models. Simulation results indicate that application of the proposed TR algorithm greatly enhances modelling and forecasting performance of IT2 FLS models. This benefit is achieved in no cost, as the computational requirement of the proposed algorithm is less than or at most equivalent to traditional TR algorithms.