This research realises a type-2 fuzzy probabilistic system for proactive monitoring of uncertain data-intensive seasonal time series in both theoretical and practical implications. In this thesis, a new form of representation, J˜-plane, is proposed for concave and unnormalized type-2 fuzzy events as well as convex and normalized ones, which facilitates bridging the gaps between higher order fuzzy probability realizations and real world problems. Since J˜-plane representation, the investigation of type-2 fuzzy probability theory and the proposal of a type-2 fuzzy probabilistic system become possible. Based on J˜-plane representation, a new fuzzy systemmodel - a type-2 fuzzy probabilistic system is proposed incorporating probabilistic inference with type-2 fuzzy sets. A special case study, a type-2 fuzzy SARIMA system is proposed and experimented in forecasting singleton and uncertain non-singleton bench mark data - Mackey-Glass time series. The results show that the type-2 fuzzy SARIMA system has achieved significant improvements beyond its predecessors - the classical statistical model - SARIMA, type-1 and general type-2 fuzzy logic systems, no matter whether in the singleton or the non-singleton experiments, whereas a SARIMA model cannot forecast non-singleton data at all. The type-2 fuzzy SARIMA system is applied in a real world scenario - WSS CAPS proactive monitoring, and compared with the results of the statistical model SARIMA, type-1 and general type-2 fuzzy logic systems to show that, the type-2 fuzzy SARIMA system can monitor practical uncertain data-intensive seasonal time series proactively and accurately, whereas its predecessors - the statistical model SARIMA, type-1 and general type-2 fuzzy logic systems - cannot deal with this at all. As a series of concepts, algorithms, experiments, practical implements and comparisons prove that, a type-2 fuzzy probabilistic system is viable in practice which realises that type-2 fuzzy systems evolve from rule-based fuzzy systems to the systems incorporating probabilistic inference with type-2 fuzzy sets.

Computing with words (CW) provides symbolic and semantic methodology to deal with imprecise information associated with natural language. The CW paradigm rooted in fuzzy logic, when coupled with an expert system, offers a general methodology for computation with fuzzy variables and a fusion of natural language propositions for this purpose. Fuzzy variables encode the semantic knowledge, and hence, the system can understand the meaning of the symbols. The use of words not only simplifies the knowledge acquisition process, but can also eliminate the need of a human knowledge engineer. CW encapsulates various fuzzy logic techniques developed in past decades and formalizes them. Z-number is an emerging paradigm that has been utilized in computing with words among other constructs. The concept of Z-number is intended to provide a basis for computation with numbers that deals with reliability and likelihood. Z-numbers are confluence of the two most prominent approaches to uncertainty, probability and possibility, that allow computations on complex statements. Certain computations related to Z-numbers are ambiguous and complicated leading to their slow adaptation into areas such as computing with words. Moreover, as acknowledged by Zadeh, there does not exist a unique solution to these problems. The biggest contributing factor to the complexity is the use of probability distributions in the computations. This dissertation seeks to provide an applied model of Z-number based on certain realistic assumptions regarding the probability distributions. Algorithms are presented to implement this model and integrate it into an expert system shell for computing with words called CWShell. CWShell is a software tool that abstracts the underlying computation required for computing with words and provides a convenient way to represent and reason on a unstructured natural language.

Decision support systems have emerged over five decades ago to serve decision makers in uncertain conditions and usually rapidly changing and unstructured problems. Most decision support approaches, such as Bayesian decision theory and computing with words, compare and analyze the consequences of different decision alternatives. Bayesian decision methods use probabilities to handle uncertainty and have been widely used in different areas for estimating, predicting, and offering decision supports. On the other hand, computing with words (CW) and approximate reasoning apply fuzzy set theory to deal with imprecise measurements and inexact information and are most concerned with propositions stated in natural language. The concept of a Z-number [69] has been recently introduced to represent propositions and their reliability in natural language. This work proposes a methodology that integrates Z-numbers and Bayesian decision theory to provide decision support when precise measurements and exact values of parameters and probabilities are not available. The relationships and computing methods required for such integration are derived and mathematically proved. The proposed hybrid methodology benefits from both approaches and combines them to model the expert knowledge and its certainty (reliability) in natural language and apply such model to provide decision support. To the best of our knowledge, so far there has been no other decision support methodology capable of using the reliability of propositions in natural language. In order to demonstrate the proof of concept, the proposed methodology has been applied to a realistic case study on breast cancer diagnosis and a daily life example of choosing means of transportation.

Made available in DSpace on 2014-12-17T14:55:08Z (GMT). No. of bitstreams: 1 ClaudileneGC_TESE.pdf: 853804 bytes, checksum: 011bfb4befb8b54fce2cd4b1b724efdb (MD5) Previous issue date: 2012-08-20 / 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 / A id?ia de considerar imprecis?o em probabilidades ? antiga, remontando aos trabalhos de George Booles, que em 1854 pretendia conciliar a l?gica cl?ssica, que permite modelar ignor?ncia completa, com probabilidades. Em 1921, John Maynard Keynes em seu livro fez uso expl?cito de intervalos para representar a imprecis?o nas probabilidades. Por?m, apenas a partir dos trabalhos de Walley em 1991 que foram estabelecidos princ?pios que deveriam ser respeitados por uma teoria de probabilidades que lide com imprecis?es. Com o surgimento da teoria dos conjuntos fuzzy em 1965 por Lotfi Zadeh, surge uma outra forma de lidar com incertezas e imprecis?es de conceitos. Rapidamente, come?aram a se propor diversas formas de considerar as id?ias de Zadeh em probabilidades, para lidar com imprecis?es, seja nos eventos associados ?s probabilidades como aos valores das probabilidades. Em particular, James Buckley, a partir de 2003 come?a a desenvolver uma teoria de probabilidade fuzzy em que os valores das probabilidades sejam n?meros fuzzy. Esta probabilidade fuzzy segue princ?pios an?logos ao das probabilidades imprecisas de Walley. Por outro lado, usar como graus de verdade n?meros reais entre 0 e 1, como proposto originalmente por Zadeh, tem o inconveniente de usar valores muito precisos para lidar com incertezas (como algu?m pode diferenciar de forma justa que um elemento satisfaz uma propriedade com um grau 0.423 de algo que satisfaz com grau 0.424?). Isto motivou o surgimento de diversas extens?es da teoria dos conjuntos fuzzy pelo fato de incorporar algum tipo de imprecis?o. Neste trabalho ? considerada a extens?o proposta por Krassimir Atanassov em 1983, que adicionou um grau extra de incerteza para modelar a hesita??o ao momento de se atribuir o grau de pertin?ncia, e portanto, um valor indicaria o grau com o qual o objeto pertence ao conjunto, enquanto o outro, o grau com o qual n?o pertence. Na teoria dos conjuntos fuzzy de Zadeh, esse grau de n?o-pertin?ncia por defeito ? o complemento do grau de pertin?ncia. Assim, nessa abordagem o grau de n?o-pertin?ncia ? de alguma forma independente do grau de pertin?ncia, e nessa diferencia entre essa n?o-pertin?ncia e o complemento do grau de pertin?ncia revela a hesita??o presente ao momento de se atribuir o grau de pertin?ncia. Esta nova extens?o hoje em dia ? chamada de teoria dos conjuntos fuzzy intuicionistas de Atanassov. Vale salientar, que o termo intuicionista aqui n?o tem rela??o com o termo intuicionista como conhecido no contexto de l?gica intuicionista. Neste trabalho ser? desenvolvida duas propostas de probabilidade intervalar: a probabilidade intervalar restrita e a probabilidade intervalar irrestrita; tamb?m ser?o introduzidas duas no??es de probabilidade fuzzy: a probabilidade fuzzy restrita e a probabilidade fuzzy irrestrita e por fim ser?o introduzidas duas no??es de probabilidade fuzzy intuicionista: a probabilidade fuzzy intuicionista restrita e a probabilidade fuzzy intuicionista irrestrita

[pt] Existem três tipos de incerteza: a de natureza aleatória, a gerada pelo conhecimento incompleto e a que ocorre em função do conhecimento vago ou impreciso. Há casos em que dois tipos de incerteza estão presentes, em especial nos experimentos aleatórios a partir de dados imprecisos. Para modelar a aleatoriedade quando a distribuição de probabilidade que rege o experimento não é conhecida, deve-se utilizar um método de estimação nãoparamétrico, tal como a janela de Parzen. Já a incerteza de medição, presente em qualquer medida de uma grandeza física, dá origem a dados imprecisos, tradicionalmente modelados por conceitos probabilísticos. Entretanto, como a probabilidade se aplica à análise de eventos aleatórios, mas não captura a imprecisão no evento, esta incerteza pode ser melhor representada por um número fuzzy segundo a transformação probabilidade-possibilidade superior. Neste trabalho é proposto um método de estimação não-paramétrico baseado em janela de Parzen para estimação da probabilidade fuzzy a partir de dados imprecisos. / [en] There are three kinds of uncertainty: one due to randomness, another due to incomplete knowledge and a third one due to vague or imprecise knowledge. Sometimes two kinds of uncertainty occur at the same time, especially in random experiments based on imprecise data. To model randomness when the probability distribution related to an experiment is unknown, a non-parametric estimation method must be used, such as the Parzen window. Uncertainty in measurement originates imprecise data, traditionally modelled through probability concepts. However, as probability applies to random events but does not capture their imprecision, this sort of uncertainty is better represented by a fuzzy number, through the superior probability-possibility transformation. This thesis proposes a non-parametric estimation method based on Parzen window to estimate fuzzy probability from imprecise data.