本論文旨在研究一種利用灰色系統 (Grey System)、傅立葉級數 (Fourier Series)、馬可夫鏈 (Markov chain) 與模糊系統 (Fuzzy System) 等理論為基礎的方法，發展出一個可精準預測時間序列的預測模型。首先，利用灰色系統理論中的灰關聯度 (Grey Relational Grade, GRG) 找出與台灣加權股價指數 (TAIEX) 關聯度最高的價的移動平均線 (Moving Average Price, MAP) 以供灰預測模型建模之用，在該研究中使用了距離灰關聯度分析二者的相關性。其次本論文利用灰色理論中的灰預測模型與新陳代謝檢驗法所組合而成的灰新陳代謝預測模型 (Grey Metabolizing Model, GMM) 來找出最佳的建模點數並進行預測。雖然 GMM 在大盤轉折點的預測結果不錯但仍有改善的空間，為了降低預測誤差以提高預測的精確度，我們利用傅立葉級數來修正經 GMM 產生的預測誤差，這種結合傅立葉級數與灰預測模型的方法稱之為傅立葉灰預測模型 (Fourier Grey Model, FGM)。經實驗證明，傅立葉灰預測模型的確可改善預測的精確度。接著，為了加強預測的效能，本論文提出一種結合灰預測模型、傅立葉級數與馬可夫模型的預測模型, 即所謂的馬可夫-傅立葉灰預測模型 (Markov-Fourier Grey Model, MFGM) 來提昇 FGM 的預測結果。在本論文中 MFGM 除了應用於 TAIEX 時間序列的預測外，也應用於阿拉巴馬大學的新生註冊率以及台灣的國民生產毛額之預測，由於預測的精確度頗佳，因此可證實其在預測上的貢獻。另外經實驗結果得知，在利用馬可夫轉移矩陣處理經傅立葉灰預測模型產生的預測值時，選取涵蓋整體的資料所得到的預測結果將優於僅使用最近幾點的資料。最後，基於沒有最好的，只有更好的理念，我們利用模糊理論中的模糊規則來選擇不同預測模型的預測值，以期能獲得更好的預測結果，實驗結果證明這種結合模糊理論與 MFGM 的方法，稱之為 Fuzzy-Markov-Fourier Grey Model (FMFGM) 是可行的。

The main purpose of this dissertation is to develop an effective prediction model that forecasts time series data based on grey theory, Fourier series theory, Markov chain theory, and Fuzzy theory. First, the grey relational grade (GRG) by relative distance is employed to analyze the relationship between the moving average of price (MAP) and the Taiwan weighted stock index (TAIEX) for the purpose of constructing the grey prediction model. Secondly, based on both the grey prediction model GM(1,1) and metabolizing checking, the grey metabolizing model is used to forecast the next value of the TAIEX and to generate the optimal number of modeling data. In spite of the fact that the grey prediction model has shown a respectable performance for the TAIEX time series prediction, in order to lower the residual error for enhancing the prediction accuracy, this study utilizes the Fourier series to correct the residual errors generated by the grey metabolizing prediction model. This method of using the Fourier correction approach [42] to modify the residual error of GMM is called the Fourier Grey Model (FGM). As indicated by the simulation results, it is evident that the FGM method can improve forecasting accuracy. Next, in order to improve prediction capability, an effective method that is based on the grey model, the Fourier series and the Markov state transition matrices (termed the Markov-Fourier Grey Model, MFGM), is proposed in this dissertation. The proposed method is not only applied to forecast the TAIEX time series, but also used to predict enrollment at the University of Alabama as well as the Gross Domestic Product (GDP) of Taiwan, in order to demonstrate its superiority, robustness and general suitability. The simulation results reveal that the proposed schemes outperform the previously proposed methods found in the literature. From the simulation results, it was determined that in utilizing the Markov state transition matrices to deal with the predictive value produced by the Fourier Grey model (FGM), the prediction results of choosing the entire body of data will be superior to only using the most recent data. Finally, based on the idea that ‘some are better and none is best’, this research applied the rules of fuzzy theory to select the forecasting results of various prediction models used, to obtain a high level of prediction accuracy. The simulation results prove that this method that integrates fuzzy rules with MFGM, entitled FMFGM, is feasible.

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