Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.

科學研究中，我們常關注變數間是否存在某種相關，及其相關的程度與方向。但傳統的相關分析方法，並不適用於更能表達真實情況的模糊資料。 在統計學中，討論資料之相關性的統計量有許多，本研究旨在針對討論兩變數間之線性關係的皮爾森相關係數 (Pearson Product-Moment Correlation Coefficient)，以模糊統計方法的角度，提出合理的模糊直線相關係數定義，以協助處理區間模糊資料，瞭解模糊資料間的線性關係。 / In the scientific research, we often pay attention to whether there are some relations between two variables, and the strength and direction of a linear relationship. But the traditional statistics method is not suitable for the fuzzy data. There are a lot of statistics of discussing the relevance between two variables. In this study, a modified method, combining Pearson Product-Moment Correlation Coefficient and fuzzy theory, was applied to deal with the fuzzy data, and find the linear relation among them.

台灣加權股價指數(TAIEX)，可以說是台灣最重要的經濟指數之一。在預測的方法中，時間序列分析一直都是熱門的課題，也是最常被使用來研究股價預測的方法。近年來，模糊理論在生醫、財務、社會、電機等各領域都有不錯的應用與發展 。本研究欲透過模糊區間的預測，主要是以時間序列預測台灣加權股價指數，來作為模糊區間精確度的探討，並針對區間時間序列進行模式的建構診斷和預測。最後我們將以2012年第一季(Q1)，每日交易股價指數的最高價與最低價作為實際研究的例子，同時也比較不同預測方法所得的結果。結果顯示模糊區間預測提供不同於傳統預測方法所得的資訊，希望能提供投資者另一種投資的參考。 關鍵字 : 台灣加權股價指數(TAIEX) 、模糊理論、模糊區間、區間預測 / Taiwan Weighted Stock Index (TAIEX) is one of Taiwan's most important economic indicators. Among the forecasting methods of time series analysis is always a hot issue on the forecasting methods and is also the most commonly used to make the stock price predictions. In recent years , fuzzy theory makes a great of application and development in various fields , such as , biomedical , financial and social …etc.. For this study, through the fuzzy interval forecasting is mainly based on time series forecasting TAIEX as fuzzy interval accuracy of the construction of diagnosis and prediction of the mode and interval time series. Finally, we will take the daily highest / lowest stock index prices data in the first quarter of 2012 (Q1) for actual research example , and will compare different forecasting methods of the results. The results show that the fuzzy interval forecasting differented from the traditional one on the basis of these information. We hope to offer investors an alternative investment advice. Keyword : Taiwan Capitalization Weighted Stock Index (TAIEX) 、 Fuzzy theory 、 Fuzzy interval、Interval forecasting.

近年來，由於人類對自然現象、社會現象或經濟現象的認知意識逐漸產生多元化的研判與詮釋，也因此致使人類思維數據化的概念已逐漸廣泛的被應用，對數據分析已從傳統以單一數值或平均值的分析作法，演變為考量多元化數值的分析作為。有鑑於此，在數據資料具備「模糊性」特質的現今，藉由模糊區間的演算方法，進一步探討之間的關係。 傳統的統計分析，對於兩變數間線性關係的強度判斷，一般是藉由皮爾森相關係數（Pearson’s Correlation Coefficient）的方法予以衡量，同時也可以經由係數的正、負符號判斷變數間的關係方向。然而，在現實生活中無論是環境資料或社會經濟資料等，均可能以模糊的資料型態被蒐集，如果當資料型態係屬於模糊性質時，將無法透過皮爾森相關係數的方法計算。 因此，本研究欲研擬一個較簡而易懂的方法，計算模糊區間資料的相關係數，據以呈現兩組模糊區間資料的相互影響程度。此外，若時間性之模糊區間資料被蒐集之際，我們亦提出利用中心點與長度之模糊自相關係數（ACF with the Fuzzy Data of Center and Length；簡稱CLACF）及模糊區間資料之自相關函數(ACF with Fuzzy Interval Data；簡稱FIACF)的方法，探討時間性模糊資料的自相關係數予以衡量。 / The classical Pearson’s correlation coefficient has been widely adopted in various fields of application. However, when the data are composed of fuzzy interval values, it is not feasible to use such a traditional approach to evaluate the correlation coefficient. In this study, we propose the specific calculation of fuzzy interval correlation coefficient with fuzzy interval data to measure the relationship between various stocks. In addition, in time series analysis, the auto-correlation function (ACF) can evaluate the effect of stationary for time series data. However, as the fuzzy interval data could be occurred, then the classical time series analysis will be not applied. In this paper, we proposed two approaches, ACF with the fuzzy data of center and length (CLACF) and ACF with fuzzy interval data (FIACF), to calculate the auto-correlation coefficient for fuzzy interval data, and use the scheme of Mote Carlo simulation to illustrate the effect of evaluation methods. Finally, we offer empirical study to indentify the performance of CLACF and FIACF which may measure the effect of lagged period of fuzzy interval data for daily price (low, high) of the Centralized Securities Trading Market and the result show that the effect of evaluation lagged period via CLACF and FIACF may response the effect more easily than classical evaluation of ACF for the close price of Centralized Securities Trading Market.

Orientador: Edvaldo Assunção / Resumo: Neste trabalho estudamos problemas de controle ótimo intervalar e intervalar fuzzy. Em particular, propomos problemas de controle ótimo via teoria de incerteza generalizada e teoria dos conjuntos fuzzy. Dentre os vários tipos de incerteza generalizada utilizamos apenas a intervalar. Embora as abordagens do processo de solução dos problemas de controle ótimo intervalar e intervalar fuzzy sejam similares, as premissas iniciais para o uso e identificação de aplicação delas em problemas práticos são distintas assim como é distinto o processo de tomada de decisão. Assim, propomos inicialmente o problema de controle ótimo intervalar em tempo discreto. A primeira proposta de solução para o problema de controle ótimo intervalar em tempo discreto é construída usando a aritmética intervalar restrita de níveis simples juntamente com a técnica de programação dinâmica. As respostas do problema de controle ótimo intervalar contêm as possibilidades de soluções viáveis, e para implementar uma solução viável para o usuário final usamos a solução que minimiza o arrependimento máximo nos exemplos numéricos. A segunda proposta de solução para o problema de controle ótimo intervalar em tempo discreto é realizada com a aritmética intervalar restrita uma vez que essa aritmética intervalar é mais geral do que a aritmética intervalar restrita de níveis simples pois não considera os intervalos envolvidos nas operações variando de forma dependente. Exemplos numéricos também foram construídos e ilustram... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this work we study the interval optimal control problem and fuzzy interval optimal control problem. In particular, we propose optimal control problems via theory of generalized uncertainty and fuzzy set theory. Among the various types of generalized uncertainty we use only the interval uncertainty. Although the approaches to solve the interval optimal control problem and fuzzy interval optimal control problem are similar, the input data for problems with generalized uncertainty and flexibility are distinct as is distinct the decision-making process. Thus, we initially propose the discrete-time interval optimal control problem. The first solution method to solve the discrete-time interval optimal control problem is constructed using single-level constrained interval arithmetic coupled with a dynamic programming technique. The optimal interval solution contains the real-valued optimal solutions, and to implement a feasible solution to the user we use the minimax regret criterion in numerical examples. The second solution method to solve the discrete-time interval optimal control problem is done with the constrained interval arithmetic since this interval arithmetic is more general than the single-level constrained interval arithmetic because it does not have its intervals varying of dependent form in interval operations. Numerical examples have also been constructed and illustrate the method of solution. Finally, we study the discrete-time fuzzy interval optimal control prob... (Complete abstract click electronic access below) / Doutor