兩變數之間是否相關，以及相關的程度與方向是統計研究學者所關注的一項課題。傳統上使用皮爾森相關係數(Pearson’s Correlation Coefficient)來表達兩實數變數間線性關係的強度與方向。然而，對於反映人類思維不確定性的模糊資料而言，傳統的相關分析方法卻有不足與不適用之缺失。 　　本論文的主要目的在於尋求一個合理、適用的區間模糊資料相關係數，提供研究者簡單且容易計算的模糊相關係數求法，用以了解區間模糊資料間的相關程度。接著利用轉換離散型模糊數成為區間模糊數的方式，處理離散型模糊資料間的相關係數。最後，以國中數學教學現場所調查的資料做實例應用。 / In statistical studies, the correlation between two variables and its strength and direction are always concerned. Traditionally, the Pearson’s Correlation Coefficient is used to convey the linear relationship between two variables. However, the traditional correlation analysis is not applicable to the fuzzy data which are able to reflect more appropriately the uncertainty of human thinking. 　　The main purpose of the study is to find a reasonable and usable correlation coefficient of interval fuzzy data which provides researchers a simple and easy way to calculate and find the fuzzy correlation coefficient. Meanwhile, it can help us understand the correlation of interval fuzzy data. Moreover, we use the process of transforming discrete fuzzy number into the interval fuzzy number to deal with the correlation coefficient of discrete fuzzy data. Finally, we utilize the data from mathematics teaching in junior high school for application.

To achieve a shared blueprint for peace and prosperity for people and the planet, the United Nations has defined the 2030 Agenda for Sustainable Development, including economic, social, and environmental sustainability as three pillars. This thesis work is aimed at building a sustainability evaluation framework to investigate the product, which in this study is transformer insulating liquid. What’s more, this study focuses on the sustainability evaluation of the extraction step of the raw material which is crude oil and vegetable oil for insulating liquids. The evaluation method is based on the triangular fuzzy number variables, which represent the ambiguity of linguistic variables. Meanwhile, the sustainable development goals are discussed and developed into feasible evaluation criteria. According to the defined criteria, the performance of each indicator is taken from published literature, reports, and official organization. After that, the sustainability index of each oil is calculated and compared, thus a general sustainable result is given. The result shows that mineral oil is more sustainable than vegetable oil only in economic sustainability, while in other aspects including the overall aspect vegetable oil is more sustainable than mineral oil. Among that, mineral oil performs most weakly in environmental sustainability, while vegetable oil is in economic sustainability. Furthermore, the framework designed is expected to be used to evaluate the entire life cycle of a product and can be referred to in future supply chain improvement, since more indexes can be developed based on the mathematics fuzzy number calculation principle. / För att uppnå en gemensam plan för fred och välstånd för människor och planeten har FN definierat 2030-agendan för hållbar utveckling, inklusive ekonomisk, social och miljömässig hållbarhet, som tre pelare. Detta examensarbete syftar till att bygga ett ramverk för hållbarhetsutvärdering för att undersöka produkten, som i denna studie är transformatorisoleringsvätska. Dessutom fokuserar denna studie på hållbarhetsutvärderingen av utvinningssteget av råmaterialet som är råolja och vegetabilisk olja för isolerande vätskor. Utvärderingsmetoden är baserad på de triangulära luddiga talvariablerna, som representerar tvetydigheten hos språkliga variabler. Samtidigt diskuteras målen för hållbar utveckling och utvecklas till genomförbara utvärderingskriterier. Enligt de definierade kriterierna är prestandan för varje indikator hämtad från publicerad litteratur, rapporter och officiell organisation. Därefter beräknas och jämförs hållbarhetsindexet för varje olja, vilket ger ett generellt hållbart resultat. Resultatet visar att mineralolja är mer hållbar än vegetabilisk olja endast i ekonomisk hållbarhet, medan i andra aspekter inklusive den övergripande aspekten är vegetabilisk olja mer hållbar än mineralolja. Bland annat presterar mineralolja sämst i miljömässig hållbarhet, medan vegetabilisk olja har ekonomisk hållbarhet. Dessutom förväntas det utformade ramverket användas för att utvärdera hela livscykeln för en produkt och kan hänvisas till i framtida förbättringar av försörjningskedjan, eftersom fler index kan utvecklas baserat på matematikens luddiga talberäkningsprincip.

動機與目的：傳統的統計迴歸模式假設觀測值的不確定性來自於隨機現象，而模糊迴歸則考慮不確定性來自於多重隸屬現象。不同的模型建構所得到的估計值也不一致。如何衡量模型的優劣程度，至今仍沒有一套嚴謹的標準。 研究方法：本研究以區間模糊數建構模糊迴歸模式，如此一來對樣本的解釋方式將更為貼近現實，並提出一套區間模糊數距離測度，以衡量估計值與實際值之間的差距。實證分析中（懸浮微粒PM_10濃度預測、台灣加權股價指數預測），我們藉由此距離測度衡量二維模糊迴歸與傳統二項最小平方法對於樣本的配適性。 創新與推廣：提出區間模糊數距離衡量估計值與原樣本之差異程度。在符合傳統統計迴歸精神之下，當距離最小就是差異最小的估計，最能符合所抽取的樣本，也是最佳估計。 重要發現：利用本區間模糊數距離測度，我們發現二維模糊迴歸方法比起傳統二項最小平方法更有效率且廣義殘差（generalized residual）將更小。 結論：過去以來，我們對於模糊迴歸架構一直都沒有完整的衡量標準。文中我們定義區間模糊數區間距離與平均距離，並推導賦距空間等性質。結合實例分析及應用，建構一合適模糊迴歸模式，以利統計決策分析參考。 / Objective: This study concerns how to develop effective fuzzy regression models. In the literature, little is addressed on how to evaluate the effectiveness of fuzzy regression models developed with different regression methods. We consider this issue in this work and present a framework for such evaluation. Method: We consider fuzzy regression models developed with different regression approaches. A method to evaluate the developed models is proposed. We then show that the proposed method possesses desirable mathematical properties and it is applied to compare the two-dimensional regression method and the traditional least square based regression method in our case studies: predicating the concentration of and the volatility of the weighted price index of the Taiwanese stock exchange. Innovation: We propose a new metric to define a distance between two fuzzy numbers. This metric can be used to evaluate the performance of different fuzzy regression models. When a prediction from one model is closest to the sample data measured in terms of the proposed metric, it can be recognized as the optimal predication. Results: Based on the proposed metric, it can be obtained that the two-dimensional fuzzy regression method is better than the traditional least square based regression method. Especially, its resulting generalized residual is smaller. Conclusion: In the literature, no unified framework has been previously proposed in evaluating the effectiveness of developed fuzzy regression models. In this work, we present a metric to achieve this goal. It facilitates the work to determine whether a fuzzy regression model suitably fits obtained samples and whether the model has potential to provide sufficient accuracy for follow-up analysis in a considered problem.

The thesis deals with investment decision making. It starts from the decision making situation about the making of an investment portfolio of the open unit trusts. The whole decision making process is described, namely the methodical approaches used in terms of the portfolio making procedure. Thus, the main part of this paper focuses on a detailed description of an algorithm of the fuzzy weight estimation method, fuzzy multiple criteria evaluation method and fuzzy multiple objective programming method. The methods are proposed on the basis of current concepts; they conclude the new ideas as well. The fuzzy weight estimation method is able to calculate the weights of criteria according to their linguistically expressed importance. The fuzzy multiple criteria evaluation method accepts uncertain input data in the form of the fuzzy numbers. The alternatives are evaluated by the concept of the preference relations. This method provides a division of the alternatives into the effective and ineffective. The fuzzy multiple objective programming method also works with the uncertain elements as fuzzy numbers. To solve the particular mathematical models, the Bellman's optimality approach is applied. The method is proposed in the interactive form. The decision maker can change a current solution by his/her additional (vague) preferences. The proposed concepts form the two-phase decision making procedure that is applied in the practical situation of the portfolio making in the field of the capital market with open unit trusts offered by the Česká spořitelna investment company. Two types of investors are defined, the investment situations are described, and the results are analyzed in detail. The decision making theory, the fuzzy sets theory and the capital market of collective investment, or with unit trusts, are introduced in a necessary scope. The instigation of my dissertation is to solve the real decision making situation. The investment decision making process is described and the methodical approaches are proposed in order to make the portfolio of unit trusts.

<p>U ovoj tezi dokazane su teoreme o nepokretnoj tački koje predstavljaju jednoznačna i vi&scaron;eznačna uop&scaron;tenja Banahovog prin&shy;cipa kontrakcije u verovatnosnim metričkim i fazi metričkim pros&shy;torima. Dokazana je teorema koja predstavlja uop&scaron;tenje teoreme o nepokretnoj tački za verovatnosnu ^-kontrakciju / :&nbsp; S &mdash;* S,gde je&nbsp; ( S ,&nbsp; J7, T ) kompletan Mengerov prostor. Uveden je pojam jake (6n)-kontrakcije i dokazana je teorema koja predstavlja uop&scaron;tenje teoreme Sehgala i Bharuche-Reid kada je preslikavanje / :&nbsp; S &mdash;&gt; S jaka (6n)-kontrakcija. Teorema Caristija, koja predstavlja jedan od najvažnijih rezultata za teoriju nepokretne tačke i nelinearnu analizu uop&scaron;tena je u kompletnom Mengerovom prostoru&nbsp; (S ,&nbsp; F , T ), gde je t-norma&nbsp; T H -tipa. Kako Mengerovi prostori pripadaju klasi kvazi-uniformnih prostora dokazana je teorema o nepokretnoj tački tri preslikavanja u jednoj specijalnoj klasi kvazi-uniformnih prostora. Dokazana je teorema o nepokretnoj tački koja predstavlja verovatnosno uop&scaron;tenje Nadlerove g-kontrakcije za tri preslikavanja kao i uop&scaron;tenje Hiksovog principa kontrakcije za tri preslikavanja. Teorija<br />kontraktora, koju je uveo M. Altman, odnosi se na re&scaron;avanje nelin&shy;earnih operatorskih jednačina u Banahovim prostorima. U tezi su dokazane teoreme koje obezbeđuju postojanje i jedinstvenost re&scaron;enja za nelinearne operatorske jednačine sa jednoznačnim i vi&scaron;eznačnim operatorom u nearhimedovskim Mengerovim verovatnosnim normi&shy; ranim prostorima.</p> / <p>In this thesis fixed point theorems which present singleval&shy;<br />ued and multivalued generalization of Banach contraction principle<br />in probabilistic metric and fuzzy metric spaces are proved. The<br />theorem which presents generalization of fixed point theorem for<br />probabilistic g-contraction / :&nbsp; S &mdash;* S is proved, where ( S ,&nbsp; J7, T ) is<br />complete Menger space. A notion of strong (&pound;&gt;n)-contraction is in&shy;<br />troduced and the theorem which presents a generalization of Sehgal<br />and Bharucha-Raid theorem when the mapping / :&nbsp; S&nbsp; &mdash;&raquo;&nbsp; S is strong<br />(6n)-contraction is proved. Caristi&rsquo;s theorem, which presents one of<br />the most imortant results for the fixed point theorem and nonlinear<br />analysis is generalized in complete Menger space&nbsp; (S, J-, T ), where<br />t-norm&nbsp; T is of&nbsp; H -type. As Menger&rsquo;s spaces belong to the class of<br />quasi-uniformizable spaces, the fixed point theorem for three map&shy;<br />pings in one special class of quasi-uniformizable spaces is proved.<br />The fixed point theorem which presents a probabilistic generaliza&shy;<br />tion of Nadler g-contraction for three mappings is proved as well<br />as the generalization of Hicks&rsquo;s contraction principle for three map&shy;<br />pings. The theory of contractor, which was introduced by M. A lt&shy;<br />man refers to solving nonlinear operator equations in Banach spaces.<br />This thesis proves the theorems which provide the existence and<br />uniqueness of the solutions for nonlinear operator equations with<br />singlevalued and multivalued operators in nonarhimedian Menger&rsquo;s<br />probabilistic normed spaces</p>