Uma fundamentação matemática para processamento digital de sinais intervalares

This work deals with a mathematical fundament for digital signal processing under point view of interval mathematics. Intend treat the open problem of precision and repesention of data in digital systems, with a intertval version of signals representation. Signals processing is a rich and complex area, therefore, this work makes a cutting with focus in systems linear invariant in the time. A vast literature in the area exists, but, some concepts in interval mathematics need to be redefined or to be elaborated for the construction of a solid theory of interval signal processing. We will construct a basic fundaments for signal processing in the interval version, such as basic properties linearity, stability, causality, a version to intervalar of linear systems e its properties. They will be presented interval versions of the convolution and the Z-transform. Will be made analysis of convergences of systems using interval Z-transform , a essentially interval distance, interval complex numbers , application in a interval filter.

Uma fundamentação para sinais e sistemas intervalares

In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given

Generalizações do conceito de distância, i-distâncias, distâncias intervalares e topologia

In this dissertation we present some generalizations for the concept of distance by using more general value spaces, such as: fuzzy metrics, probabilistic metrics and generalized metrics. We show how such generalizations may be useful due to the possibility that the distance between two objects could carry more information about the objects than in the case where the distance is represented just by a real number. Also in this thesis we propose another generalization of distance which encompasses the notion of interval metric and generates a topology in a natural way. Several properties of this generalization are investigated, and its links with other existing generalizations

Algoritmos de agrupamentos fuzzy intervalares e índice de validação para agrupamento de dados simbólicos do tipo intervalo

Symbolic Data Analysis (SDA) main aims to provide tools for reducing large databases to extract knowledge and provide techniques to describe the unit of such data in complex units, as such, interval or histogram. The objective of this work is to extend classical clustering methods for symbolic interval data based on interval-based distance. The main advantage of using an interval-based distance for interval-based data lies on the fact that it preserves the underlying imprecision on intervals which is usually lost when real-valued distances are applied. This work includes an approach allow existing indices to be adapted to interval context. The proposed methods with interval-based distances are compared with distances punctual existing literature through experiments with simulated data and real data interval

Integrais de linha intervalares: fundamentos e aplicações

A necessidade de uma precisão e de uma aproximação dos resultados numéricos zeram com que diversas teorias surgissem: dentre elas, destacamos a Matemática Intervalar. A Matemática Intervalar surgiu na década de 60 com os trabalhos de pesquisa de Moore (MOORE, 1959) , em que ele propôs trabalhar com uma Matemática baseada na noção de intervalo real e não mais com um número como aproximação. Com isso, surgiu a necessidade de revisitar e reformular os conceitos e resultados da Matemática Clássica utilizando como base a noção de intervalo de Moore. Uma das áreas da Matem ática Clássica que tem tido muitas aplicações em engenharias e ciências é a Análises Numérica, onde um dos seus pilares é o Cálculo Integral e em particular as integrais de linha. Assim, é muito desejável se ter um cálculo integral dentro da própria Matemática Intervalar. No presente trabalho apresenta-se uma noção de Integral de Linha Intervalar baseada na extensão de integração proposta por Bedregal em (BEDREGAL; BEDREGAL, 2010). Para a fundamentação apresenta-se incialmente uma introdução sobre a pespectiva em que o trabalho foi realizado, considerando alguns aspectos histórico-evolutivos da Matemática Clássica. Os conceitos de Integrais de Linha Clássica, bem como algumas das suas aplicações mais importantes. Alguns conceitos de Matemática Intervalar necessários para o entendimento do trabalho. Para nalizar propomos uma aplicação da integral de linha em um experimênto clássico da mecânica quântica (a difração de um elétron em uma fenda) que graças ao fato de ser a Matemática Intervalar utilizada, nos dá um foco mais detalhado e mais próximo da realidade

Uma Fundamentação Intervalar Aplicada à Morfologia Matemática

This work present a interval approach to deal with images with that contain uncertainties, as well, as treating these uncertainties through morphologic operations. Had been presented two intervals models. For the first, is introduced an algebraic space with three values, that was constructed based in the tri-valorada logic of Lukasiewiecz. With this algebraic structure, the theory of the interval binary images, that extends the classic binary model with the inclusion of the uncertainty information, was introduced. The same one can be applied to represent certain binary images with uncertainty in pixels, that it was originated, for example, during the process of the acquisition of the image. The lattice structure of these images, allow the definition of the morphologic operators, where the uncertainties are treated locally. The second model, extend the classic model to the images in gray levels, where the functions that represent these images are mapping in a finite set of interval values. The algebraic structure belong the complete lattices class, what also it allow the definition of the elementary operators of the mathematical morphology, dilation and erosion for this images. Thus, it is established a interval theory applied to the mathematical morphology to deal with problems of uncertainties in images

Máquina de vetores-suporte intervalar

The Support Vector Machines (SVM) has attracted increasing attention in machine learning area, particularly on classification and patterns recognition. However, in some cases it is not easy to determinate accurately the class which given pattern belongs. This thesis involves the construction of a intervalar pattern classifier using SVM in association with intervalar theory, in order to model the separation of a pattern set between distinct classes with precision, aiming to obtain an optimized separation capable to treat imprecisions contained in the initial data and generated during the computational processing. The SVM is a linear machine. In order to allow it to solve real-world problems (usually nonlinear problems), it is necessary to treat the pattern set, know as input set, transforming from nonlinear nature to linear problem. The kernel machines are responsible to do this mapping. To create the intervalar extension of SVM, both for linear and nonlinear problems, it was necessary define intervalar kernel and the Mercer s theorem (which caracterize a kernel function) to intervalar function

Uma nova forma de calcular os centros dos Clusters em algoritmos de agrupamento tipo fuzzy c-means

Clustering data is a very important task in data mining, image processing and pattern recognition problems. One of the most popular clustering algorithms is the Fuzzy C-Means (FCM). This thesis proposes to implement a new way of calculating the cluster centers in the procedure of FCM algorithm which are called ckMeans, and in some variants of FCM, in particular, here we apply it for those variants that use other distances. The goal of this change is to reduce the number of iterations and processing time of these algorithms without affecting the quality of the partition, or even to improve the number of correct classifications in some cases. Also, we developed an algorithm based on ckMeans to manipulate interval data considering interval membership degrees. This algorithm allows the representation of data without converting interval data into punctual ones, as it happens to other extensions of FCM that deal with interval data. In order to validate the proposed methodologies it was made a comparison between a clustering for ckMeans, K-Means and FCM algorithms (since the algorithm proposed in this paper to calculate the centers is similar to the K-Means) considering three different distances. We used several known databases. In this case, the results of Interval ckMeans were compared with the results of other clustering algorithms when applied to an interval database with minimum and maximum temperature of the month for a given year, referring to 37 cities distributed across continents

Em direção a uma representação para equações algébricas :uma lógica equacional local

The intervalar arithmetic well-known as arithmetic of Moore, doesn't possess the same properties of the real numbers, and for this reason, it is confronted with a problem of operative nature, when we want to solve intervalar equations as extension of real equations by the usual equality and of the intervalar arithmetic, for this not to possess the inverse addictive, as well as, the property of the distributivity of the multiplication for the sum doesn t be valid for any triplet of intervals. The lack of those properties disables the use of equacional logic, so much for the resolution of an intervalar equation using the same, as for a representation of a real equation, and still, for the algebraic verification of properties of a computational system, whose data are real numbers represented by intervals. However, with the notion of order of information and of approach on intervals, introduced by Acióly[6] in 1991, the idea of an intervalar equation appears to represent a real equation satisfactorily, since the terms of the intervalar equation carry the information about the solution of the real equation. In 1999, Santiago proposed the notion of simple equality and, later on, local equality for intervals [8] and [33]. Based on that idea, this dissertation extends Santiago's local groups for local algebras, following the idea of Σ-algebras according to (Hennessy[31], 1988) and (Santiago[7], 1995). One of the contributions of this dissertation, is the theorem 5.1.3.2 that it guarantees that, when deducing a local Σ-equation E t t in the proposed system SDedLoc(E), the interpretations of t and t' will be locally the same in any local Σ-algebra that satisfies the group of fixed equations local E, whenever t and t have meaning in A. This assures to a kind of safety between the local equacional logic and the local algebras

Java-XSC : módulo complexo e complexo intervalar

This work aims to develop modules that will increase the computational power of the Java-XSC library, and XSC an acronym for "Language Extensions for Scientific Computation . This library is actually an extension of the Java programming language that has standard functions and routines elementary mathematics useful interval. in this study two modules were added to the library, namely, the modulus of complex numbers and complex numbers of module interval which together with the modules original numerical applications that are designed to allow, for example in the engineering field, can be used in devices running Java programs

Especificação do tipo intervalar parametrizado em CASL

The interval datatype applications in several areas is important to construct a interval type reusable, i.e., a interval constructor can be applied to any datatype and get intervals this datatype. Since the interval is, of certain form, a set of elements limited for two bounds, left and right, with a order notions, then it s reasonable that interval constructor enclose datatypes with partial order. On the order hand, what we want is work with interval of any datatype like this we work with this datatype then. it s important to guarantee the properties of the datatype when maps to interval of this datatype. Thus, the interval constructor get a theory to parametrized interval type, i.e., a interval with generics parameters (for example rational, real, complex). Sometimes, the interval application in some algebras doesn t guarantee the mainutenance of their properties, for example, when we use interval of real, that satisfies the field properties, it doesn t guarantee the distributivity propertie. A form to surpass this problem Santiago introduced the local equality theory that weakened the notion of strong equality, and thus, allowing some properties are local keeped, what can be discard before. The interval arithmetic generalization aim to apply the interval constructor on ordered algebras weakened for local equality with the purpose of the keep their properties. How the intervals are important in applications with continuous data, it s interesting specify that theory using a specification language that supply a system development using intervals of form disciplined, trustworth and safe. Currently, the algebraic specification language, based in math models, have been use to that intention often. We choose CASL (Common Algebraic Specification Language) among others languages because CASL has several characteristics excellent to parametrized interval type, such as, provide parcialiy and parametrization