Probabilidades imprecisas: intervalar, fuzzy e fuzzy intuicionista

The idea of considering imprecision in probabilities is old, beginning with the Booles George work, who in 1854 wanted to reconcile the classical logic, which allows the modeling of complete ignorance, with probabilities. In 1921, John Maynard Keynes in his book made explicit use of intervals to represent the imprecision in probabilities. But only from the work ofWalley in 1991 that were established principles that should be respected by a probability theory that deals with inaccuracies. With the emergence of the theory of fuzzy sets by Lotfi Zadeh in 1965, there is another way of dealing with uncertainty and imprecision of concepts. Quickly, they began to propose several ways to consider the ideas of Zadeh in probabilities, to deal with inaccuracies, either in the events associated with the probabilities or in the values of probabilities. In particular, James Buckley, from 2003 begins to develop a probability theory in which the fuzzy values of the probabilities are fuzzy numbers. This fuzzy probability, follows analogous principles to Walley imprecise probabilities. On the other hand, the uses of real numbers between 0 and 1 as truth degrees, as originally proposed by Zadeh, has the drawback to use very precise values for dealing with uncertainties (as one can distinguish a fairly element satisfies a property with a 0.423 level of something that meets with grade 0.424?). This motivated the development of several extensions of fuzzy set theory which includes some kind of inaccuracy. This work consider the Krassimir Atanassov extension proposed in 1983, which add an extra degree of uncertainty to model the moment of hesitation to assign the membership degree, and therefore a value indicate the degree to which the object belongs to the set while the other, the degree to which it not belongs to the set. In the Zadeh fuzzy set theory, this non membership degree is, by default, the complement of the membership degree. Thus, in this approach the non-membership degree is somehow independent of the membership degree, and this difference between the non-membership degree and the complement of the membership degree reveals the hesitation at the moment to assign a membership degree. This new extension today is called of Atanassov s intuitionistic fuzzy sets theory. It is worth noting that the term intuitionistic here has no relation to the term intuitionistic as known in the context of intuitionistic logic. In this work, will be developed two proposals for interval probability: the restricted interval probability and the unrestricted interval probability, are also introduced two notions of fuzzy probability: the constrained fuzzy probability and the unconstrained fuzzy probability and will eventually be introduced two notions of intuitionistic fuzzy probability: the restricted intuitionistic fuzzy probability and the unrestricted intuitionistic fuzzy probability

Algoritmos genéticos: uso de lógica nebulosa e análise de convergência por cadeia de Markov

In this work, the Markov chain will be the tool used in the modeling and analysis of convergence of the genetic algorithm, both the standard version as for the other versions that allows the genetic algorithm. In addition, we intend to compare the performance of the standard version with the fuzzy version, believing that this version gives the genetic algorithm a great ability to find a global optimum, own the global optimization algorithms. The choice of this algorithm is due to the fact that it has become, over the past thirty yares, one of the more importan tool used to find a solution of de optimization problem. This choice is due to its effectiveness in finding a good quality solution to the problem, considering that the knowledge of a good quality solution becomes acceptable given that there may not be another algorithm able to get the optimal solution for many of these problems. However, this algorithm can be set, taking into account, that it is not only dependent on how the problem is represented as but also some of the operators are defined, to the standard version of this, when the parameters are kept fixed, to their versions with variables parameters. Therefore to achieve good performance with the aforementioned algorithm is necessary that it has an adequate criterion in the choice of its parameters, especially the rate of mutation and crossover rate or even the size of the population. It is important to remember that those implementations in which parameters are kept fixed throughout the execution, the modeling algorithm by Markov chain results in a homogeneous chain and when it allows the variation of parameters during the execution, the Markov chain that models becomes be non - homogeneous. Therefore, in an attempt to improve the algorithm performance, few studies have tried to make the setting of the parameters through strategies that capture the intrinsic characteristics of the problem. These characteristics are extracted from the present state of execution, in order to identify and preserve a pattern related to a solution of good quality and at the same time that standard discarding of low quality. Strategies for feature extraction can either use precise techniques as fuzzy techniques, in the latter case being made through a fuzzy controller. A Markov chain is used for modeling and convergence analysis of the algorithm, both in its standard version as for the other. In order to evaluate the performance of a non-homogeneous algorithm tests will be applied to compare the standard fuzzy algorithm with the genetic algorithm, and the rate of change adjusted by a fuzzy controller. To do so, pick up optimization problems whose number of solutions varies exponentially with the number of variables

Modelagem de usinas eólicas através de um processo de Markov e técnicas de confiabilidade para a estimativa anual da energia produzida

This study aims to use a computational model that considers the statistical characteristics of the wind and the reliability characteristics of a wind turbine, such as failure rates and repair, representing the wind farm by a Markov process to determine the estimated annual energy generated, and compare it with a real case. This model can also be used in reliability studies, and provides some performance indicators that will help in analyzing the feasibility of setting up a wind farm, once the power curve is known and the availability of wind speed measurements. To validate this model, simulations were done using the database of the wind farm of Macau PETROBRAS. The results were very close to the real, thereby confirming that the model successfully reproduced the behavior of all components involved. Finally, a comparison was made of the results presented by this model, with the result of estimated annual energy considering the modeling of the distribution wind by a statistical distribution of Weibull

Métodos estatisticos em cadeias de Markov

Este trabalho tem como objetivo o estudo do comportamento assintótico da estatística de Pearson (1900), que é o aparato teórico do conhecido teste qui-quadrado ou teste x2 como também é usualmente denotado. Inicialmente estudamos o comportamento da distribuição da estatística qui-quadrado de Pearson (1900) numa amostra {X1, X2,...,Xn} quando n → ∞ e pi = pi0 , 8n. Em seguida detalhamos os argumentos usados em Billingley (1960), os quais demonstram a convergência em distribuição de uma estatística, semelhante a de Pearson, baseada em uma amostra de uma cadeia de Markov, estacionária, ergódica e com espaço de estados finitos S

Estimação paramétrica e não-paramétrica em modelos de markov ocultos

In this work we study the Hidden Markov Models with finite as well as general state space. In the finite case, the forward and backward algorithms are considered and the probability of a given observed sequence is computed. Next, we use the EM algorithm to estimate the model parameters. In the general case, the kernel estimators are used and to built a sequence of estimators that converge in L1-norm to the density function of the observable process

Estimador do Tipo Núcleo para Densidades Limites de Cadeias de Markov com Espaço de Estados Geral

In this work we studied the consistency for a class of kernel estimates of f f (.) in the Markov chains with general state space E C Rd case. This study is divided into two parts: In the first one f (.) is a stationary density of the chain, and in the second one f (x) v (dx) is the limit distribution of a geometrically ergodic chain

Modelagem dos algorítmos simples e Simulated Annealing por cadeias de Markov

Os Algoritmos Genético (AG) e o Simulated Annealing (SA) são algoritmos construídos para encontrar máximo ou mínimo de uma função que representa alguma característica do processo que está sendo modelado. Esses algoritmos possuem mecanismos que os fazem escapar de ótimos locais, entretanto, a evolução desses algoritmos no tempo se dá de forma completamente diferente. O SA no seu processo de busca trabalha com apenas um ponto, gerando a partir deste sempre um nova solução que é testada e que pode ser aceita ou não, já o AG trabalha com um conjunto de pontos, chamado população, da qual gera outra população que sempre é aceita. Em comum com esses dois algoritmos temos que a forma como o próximo ponto ou a próxima população é gerada obedece propriedades estocásticas. Nesse trabalho mostramos que a teoria matemática que descreve a evolução destes algoritmos é a teoria das cadeias de Markov. O AG é descrito por uma cadeia de Markov homogênea enquanto que o SA é descrito por uma cadeia de Markov não-homogênea, por fim serão feitos alguns exemplos computacionais comparando o desempenho desses dois algoritmos

Teoria da Ruína em um Modelo de Markov com dois Estados

In this work, we present a risk theory application in the following scenario: In each period of time we have a change in the capital of the ensurance company and the outcome of a two-state Markov chain stabilishs if the company pays a benece it heat to one of its policyholders or it receives a Hightimes c > 0 paid by someone buying a new policy. At the end we will determine once again by the recursive equation for expectation the time ruin for this company

Estimadores do tipo núcleo para a densidade de transição de uma cadeia de markov com espaço de estados geral

In this work, we studied the strong consistency for a class of estimates for a transition density of a Markov chain with general state space E ⊂ Rd. The strong ergodicity of the estimates for the density transition is obtained from the strong consistency of the kernel estimates for both the marginal density p(:) of the chain and the joint density q(., .). In this work the Markov chain is supposed to be homogeneous, uniformly ergodic and possessing a stationary density p(.,.)

Ergodicidade em cadeias de Markov não-homogêneas e cadeias de Markov com transições raras

The central objective of a study Non-Homogeneous Markov Chains is the concept of weak and strong ergodicity. A chain is weak ergodic if the dependence on the initial distribution vanishes with time, and it is strong ergodic if it is weak ergodic and converges in distribution. Most theoretical results on strong ergodicity assume some knowledge of the limit behavior of the stationary distributions. In this work, we collect some general results on weak and strong ergodicity for chains with space enumerable states, and also study the asymptotic behavior of the stationary distributions of a particular type of Markov Chains with finite state space, called Markov Chains with Rare Transitions