在灰色系統理論的GM(1,1)模型，是屬於預測模型，主要是構建在一階微分的模型，而是採取兩個相鄰點的平均值(背景值)所產生的。根據過去的研究，發現背景值是影響預測誤差最重要的因素，因此許多論文都集中在背景值的調整，包括背景值中alpha值的數位跳躍變動，或者背景值中alpha值的連續變動以低預測的誤差，但是在此種形式下背景值，有時仍無法充分顯現離散點所隱含之規律，因此形成GM(1,1)模型的誤差無法大幅降低。因此本論文提出了一種新的方法，內容包含三大要項，首先將背景值以分數的型態表示，並且結合組合數學的方法獲得實際的數值，最為降低誤差的基礎。接著整合GM(1,1)模型，實際分析降低預測誤差的內容，做出分析結果之數學模式。最後本文還自行研發體工具箱，做為減少複雜的計算和驗證最終結果的準確性，不僅擺脫傳統的背景值方法，也打開了GM(1,1)模型降低預測誤差的新方向。

bstract

According to the basic of empirical mathematical operation of GM(1,1) model in grey system theory, the background value is used to establish in the first order differential model, and the traditional method is taken the average of the two neighbor points. According to the past research in GM(1,1) model, it can fine that the background value not only is the main influence factor, but also is the most important influence factor in prediction error. Therefore, the paper based on the past research, focused on the adjustment of the background values, present a new approach, which is called the fractional order accumulation method, and combine with the combination mathematics method to get the mathematics result. Besides, the paper also develops computer toolbox to reduce the complex calculation and to verify the accuracy of final results. From the mentioned above, the paper it not only get rid of the default of traditional background value method, but also to open a new approach for GM(1,1) model.

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