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.

在統計學上，我們常使用皮爾森相關係數(Pearson’s Correlation Coefficient)來表達兩變數間線性關係的強度，同時也表達出關係之方向。傳統之相關係數所處理的資料都是明確的實數值，但是當資料是模糊數時，並不適合使用傳統的方法來計算模糊相關係數。而本研究探討區間模糊樣本資料值求得模糊相關係數，首先將區間型模糊資料分為離散型和連續型，提出區間模糊相關係數定義，並提出廣義誤差公式，將相關係數作合理的調整，使所求的出相關係數更加精確。在第三章我們以影響數學成就評量的因素，作實證研究分析，得出合理的分析。而此相關係數定義和廣義誤差公式也能應用在兩資料值為實數或其中一筆資料值為實數的情況，可以解釋更多在實務上所發生的相關現象。 / In the statistic research, we usually express the magnitude of linear relation between two variables by means of Pearson’s Correlation Coefficient, which is also used to convey the direction of such relation. Traditionally, correlation coefficient deals with data which consist of specific real numbers. But when the data are composed of fuzzy numbers, it is not feasible to use this traditional approach to figure out the fuzzy correlation coefficient. The present study investigates the fuzzy samples of interval data to find out the fuzzy correlation coefficient. First, we categorize the fuzzy interval data into two types: discrete and continuous. Second, we define fuzzy correlation with interval data and propose broad formulas of error in order to adjust the coefficient more reasonably and deal with it more accurately. In Chapter Three, we conduct empirical research by the factor which affects the evaluation of mathematical achievement to acquire reasonable analysis. By doing so, broad definition of coefficient and formulas of error can also be applied to the conditions of either both values of the data are real number or one value of the data is real number, and can explain more related practical phenomenon.

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.

近年來隨著科技的進步與工商業的發展，預測技術的創新與改進愈來愈受到重視。相對地，對於預測準確度的要求也愈來愈高。尤其在經濟建設、經營規畫、管理控制等問題上，預測更是決策過程中不可或缺的重要資訊。然而僅用單一數值形式收集來的資料，其建立的模式是不足以描述每日或每月的發展趨勢。因為有太多模糊且不完整訊息，以致於無法用傳統以點資料建構的系統來進行預測。基於點預測的不確定性，因此嘗試以區間資料來建構模式並進行預測。本論文探討區間時間序列之動態走勢及預測結果之效率性，共三部份，分別為區間時間序列之分析與預測、區間預測準確度之探討和計算區間資料的相關係數。 第一部份，利用區間具有糢糊數的特質，將其分解成區間平均數及區間長度，提出區間時間數列建構過程及預測方法，如區間移動平均、區間加權移動平均、ARIMA區間預測等方法。並藉由模擬方式設計出數組穩定及非穩定之區間時間數列，再利用本文所提出的區間預測方法進行預測。根據這些計算預測結果效率性的方法，發現ARIMA區間預測，提供了較傳統的預測方法更為準確及具有彈性的預測結果。 第二部份，我們特別針對區間預測結果的準確度提出效率性的分析，如平均區間預測誤差平方和、平均相對區間誤差及平均XOR比率。而在預測效率性的實證分析上，平均XOR比率能給與決策者更正確的資訊，做出更客觀的判斷。 第三部份，在探討如何將區間資料應用在計算相關係數。利用單一數值資料的收集 ，並以傳統的相關係數r來說明兩變數之間是否相關? 是較為便利且易懂的統計方法。但資料是否足以代表母體特性？這樣求出來的相關係數值會不會太主觀?有鑑於此，以區間就是模糊數的概念，建構模糊相關係數。最後舉出應用實例，比較模糊相關係數與傳統的相關係數的差異性，在說明兩變數關係的強弱程度，模糊相關係數提供了一個較有彈性的統計分析方法。 / Point forecasting provides important information during decision-making processes, especially in economic developments, population policies, management planning or financial controls. Nevertheless, the forecasting model constructed only by single values may not demonstrate the whole trend of a daily or monthly process. Since there are so many unpredictable and continuous fluctuations on the process to be predicted, the observed values are discrete instantaneous values which are insufficient to represent the true process. Therefore, the collected information is generally vague and incomplete so that the real number system is not sufficient to express the forecasting model. In additional, due to the business marketing is full of uncertainty and the continuous fluctuations, intervals are used to express and establish the forecasting model to estimate the prediction values. This dissertation investigates the dynamic trend of interval time series and the performance evaluation of interval forecasting. It consists of three parts: the analysis and forecasting of interval time series, the evaluation of forecasting performance for interval data, and the calculation of the fuzzy correlation coefficient. First of all, we propose the conception of fuzzy for interval and propose interval forecasting approaches, such as the interval moving average, the weighted interval moving average, and ARIMA interval forecasting. The soft computing technique as well as the model simulation is used to carry out the interval forecasting. The forecast results are compared by the mean squared interval error and the mean relative interval error. Finally, we take two practical cases study. By the comparison of forecasting performance, it is found that ARIMA interval forecasting provides more efficiency and flexibility than the traditional ones do. Secondly, we concentrated on the forecasting performance evaluation for interval data. The evaluation techniques are developed to determine the validity of the forecast results. The forecast results are compared by three criteria which are the mean squared error of interval, mean relative interval error, and the mean ratio of exclusive-or. It is found that the empirical studies show that the mean ratio of exclusive-or can provide a more objective suggestion in interval forecasting for policymakers. The third part considers the evaluation of the correlation coefficient interval by collecting sample data whose types are real and interval. When an interval is considered as a fuzzy number, the aspect of fuzzy can be utilized to construct the fuzzy correlation coefficient for interval data. As compared with the traditional correlation coefficient, the fuzzy correlation coefficient can demonstrate conservative correlation coefficient and provide an objective statistical method for discovering the correlation between two variables.