近幾年來，非線性時間數列轉折點的研究愈來愈受到重視，學者們也提出許多關於轉折點的偵測及檢定方法。若考慮實際資料走勢轉變的情形，“轉折區間”的概念更可以解釋結構改變的現象。但文獻中對於如何找尋時間數列結構改變之轉折區間的研究並不多。本文擬以時間數列統計模式及模糊學理論的角度來研究，並結合遺傳演算的規則而提出主導模式的概念，來架構出時間數列遺傳模式，再藉由轉折區間決策法則來找出數列的轉折區間。其中，我們以統計模式為遺傳演化過程中的染色體，而以候選模式之隸屬度函數為衡量染色體適應能力的指標。最後，我們舉出臺灣股價收盤指數之實例，分別以我們所提出的方法及其他方法找出數列的轉折區間及轉折點，並做比較。 / For recent years, the research of change point in nonlinear time series has been considered to be more and more important. Scholars have proposed a lot of detecting and testing methods about change points.If considering the trend of real situation, the concept of change period will show the phenomena of structure change.But there are not many researches about how to find change period in time series.My paper is based on the points of time series models and fuzzy theory.Besides,it combines the rules of genetic algorithm and provides the concepts of leading model to construct time seriep genetic model and to find out change period by decision rule.ln this paper, we use time series statistical models as chromosome in procedure of genetic evolution, and we also use membership function of selected models as pereformance: index of chromosome.Finally, the empirical application about change periods and change points detecting by our method and other's for Taiwan stock closing prices is demonstrated and make a comparision with these results.

模糊理論開始於 1960 年代中期,關於這方面的研究與發展均已獲得相當不錯的成果.其中尤以在群落分析應用上的專題研究更是廣泛.Bezdek 提出的模糊分類演算法,乃根據 Dunn 的C平均法所作的一改良方法.但仍有其缺點,例如,未考慮權重且以靜態資料為主. 有鑑於此,本研究對 Bezdek 之方法加以改進推廣,提出加權模糊分類法.對於評價因素為多變量時,應加入模糊權重的考量.此外更結合時間因素,使準則函數成為動態的模式,將傳統的模糊分類法由靜態資料轉為動態資料形式,以反映真實 的情況. / Research on the theory of fuzzy sets has been growing steadily since itsinception during the mid-1960s. The literature especially dealing with fuzzycluster analysis is quite extensive. But the research on FCM still has somedisadvantages. For instance, the

本文嘗試以模糊理論的觀念，應用到時間數列分析上。研究重點包括模糊關係、模糊規則庫和模糊時間數列模式建構與預測等。我們首先給定模糊時間數列模式的概念與一些重要性質。接著提出模糊規則庫的定義，以及模式建構的流程，並以模糊關係方程式的推導，提出模糊時間數列模式建構方法。最後，利用提出的方法，對台灣地區加權股票指數建立模糊時間數列模式，並對未來進行預測，且考慮以平均預測準確度來做預測效果之比較。這對於財務金融的未來走勢分析將深具意義。 / The paper has attempted to apply the concept of fuzzy method on the analysis of time series. This reserch is also to include fuzzy relation, fuzzy rule base, fuzzy time series model constructed and forecasting. First, we'll define the concept of fuzzy time series model and some important properties. Next, the definition of fuzzy rule base will also be put forward, along with procedure of model constructed, the formation of fuzzy relation polynomial, and the methods to construct fuzzy time series model. At last, with the above methods, we'll build up fuzzy time series model on Taiwan Weighted Index and predict future trend while examine the predictive results with average forecasting accuracy. This shall carry profund signifigornce on the analysis of future trend in terms of financialism.

The result of my dissertation justifies the use of fuzzy-sets theory to make a prediction of cost risk of a PPP project, when there is not enough information available to clearly describe the project, and, when the probability distributions of the variables that characterize the project are unknown. I showed that fuzzy-sets theory and linguistic variables may be effectively used in such a case. In this thesis were classified different types of uncertainty and investigated traditional methods for estimating efficiency of a investment project in conditions of uncertainty. On the basis of the analysis were offered new ways of conducting risk analysis for PPP projects with use of fuzzy sets theory. The main goal was to create an application model for risk assessment of the PPP project which, with a high degree of reliability, suggests a general assessment of situation. The goal set in my work was met. Model of risk assessment of the project proposed by me gives more stable results in comparison with the probabilistic model. For comparison were used different types of probability distribution functions and membership functions. The following conclusions and statements describe the novelty of the work on fuzzy logic and economic theory: develops a method of cash-flow (future expenditure connected with the appearance of risk) modeling of investment project in fuzzy environment, demonstrates the use of fuzzy sets theory in projects analyses and describes how to calculate and interpret this value, demonstrates example of the use of results applied to the analysis of infrastructure development project in Moscow, Russia. The possibility of using this method is not only in the analysis of infrastructure development projects, but also in realization of non-commercial projects by social institutes and government agencies.

In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1 1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2 1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 11 2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 3 Optimization problem with fuzzy objective 19 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25 4 Linear optimization with fuzzy objective 27 4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 5 Optimality conditions 47 5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48 5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49 5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51 5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52 5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Fuzzy linear optimization problem over fuzzy polytope 59 6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65 6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Bilevel optimization with fuzzy objectives 73 7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78 7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Conclusions 95 Bibliography 97

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在許多實際情形下，傳統的統計檢定方法是不足以應付的。故本論文提出模糊檢定方法，我們定義出模糊樣本期望值與模糊樣本變異數的計算方法，再針對不同的模糊資料，分別提出不同的檢定方法，去解決最實際需要解決的問題，其中包括推廣古典的統計檢定方法與自創的檢定方法。 關鍵字：隸屬度函數，模糊樣本取樣，模糊樣本期望值，模糊樣本變異數，人性思考，t檢定，F檢定，模糊常態分配。 / In many expositions of fuzzy methods, fuzzy techniques are described as an alternative to a more traditional statistical approach. In this paper, we present a class of fuzzy statistical decision process in which testing hypothesis can be naturally reformulated in terms of interval-valued statistics. We provide the definitions of fuzzy mean, fuzzy distance as well as investigation of their related properties. We also give some empirical examples to illustrate the techniques and to analyze fuzzy data. Empirical studies show that fuzzy hypothesis testing with soft computing for interval data are more realistic and reasonable in the social science research. Finally certain comments are suggested for the further studies. We hope that this reformation will make the corresponding fuzzy techniques more acceptable to researchers whose only experience is in using traditional statistical methods. Key words: Membership function, fuzzy sampling survey, fuzzy mean, human thought, t-test, F-test, normally distributed.

使用傳統迴歸的方式對未知事物做預測，往往不能夠精準的做出結論，縱使在相同的條件下實際去操作，也很難得到相同的結果，因此模糊數概念的建立，並運用在迴歸分析上更能有效描述預測結果的不確定性。然而模糊線性迴歸(Fuzzy Linear Regression)在利用最小平方法處理問題時，往往過於著重在模糊區間的中心與分展度上，而忽略了描述資料的模糊性，使得隸屬度函數(membership function)的功能受到相當大的限制。本文在D'Urso和Gastaldi(2000)所提出的雙重模糊線性迴歸(doubly fuzzy linear regression)模型架構下，利用Yang和Ko(1996)在LR空間下所定義模糊數間的距離公式，導出能反映隸屬度函數的最小平方估計，並引進一些傳統迴歸中常用來偵測離群值(outlier)與具影響力觀察值(influence observation)的概念與技巧，應用在模糊線性迴歸資料的偵測上。

本文將模糊理論的觀念，應用到時間數列分析上。研究重點包括模糊自相似度的定義與度量，模糊自迴歸係數的分析，模糊相似度辨識與自迴歸階次認定、模糊時間數列模式建構與預測等。我們首先給定模糊時間數列模式的概念與一些重要性質。接著提出模糊相似度的定義與度量，以及模式建構的流程。經由系統性的模擬與分析，我們建立階次認定的演算法則與認定程序。藉著詳細的演算比較這些類型的模糊時間數列。並以模糊關係方程式推導，提出合適的模糊時間數列模式建構方法。並利用提出的方法對台灣的景氣對策信號，及台灣結婚率建立模糊時間數列模式。最後，使用所建構的模糊時間數列模式對未來進行預測，以驗證所建構模糊時間數列模式的效率性與實用性。 / In modeling a time series the accuracy of various model constructions and forecasting techniques, certain rules and models are adhered to. Traditional methods on the model construction for a time series are based on the researchers' experience by choosing a "good" model, which will satisfactorily explain its dynamic behavior, from a model-base. But a fundamental question that often arises is: does the data exhibit the real case honestly? In this research we show how fuzzy time series construction be applied for this purpose. An order detection process for fuzzy time series is presented. Simulation has been used extensively to explore general properties of statistical procedures, and the approach is particularly useful in fuzzy time series construction. Statistical strategies typically consist of sequences of rules used repeatedly on the same data set. This paper is organized as follows: In Chapter 2 we will discuss about the definition of fuzzy time series as well as certain important properties. In Chapter 3, We use the similarity comparison process to decide the order of a fuzzy time series. Simulations and analysis with the results about various types of autocorrelation are experienced in Chapter 4. Finally, we apply our methods to three empirical examples, Taiwan business cycle index, marriage rate and numbers of students enrollment in Chapter 5. Chapter 6 is the conclusion and the discussion of future researches.