因應指數股票型投資基金(Exchange Traded Funds, ETF)市場需求持續的擴大，促使指數追蹤 (index tracking)的課題受到廣大的重視；而線性模型(linear model)與最小誤差平方法(least squares method)是處理追蹤誤差(tracking error)最簡易且普遍的方式。然而，最小誤差平方法的凸目標函數(convex object function)雖可為線性模型解出唯一的最佳參數解，卻難以提供其它具有可行性的次佳解。因此，為能表列更具實務意義層面的最佳解，本研究以廣義奇異值分解法(Generalized Singular Value Decomposition，GSVD)來解構(decompose)凸目標函數，嘗試以封閉解(close-form solution)的形式詳列所有可行解，以利實務上權衡取捨之用。 後續，再進一步地將前述GSVD法所選定出的資產配置之權重參數引入約束型卡爾曼濾波器(constrained Kalman Filter)，作為即時預測指數的初始猜值(initial guess)，便於後續指數的預測。因推導、證明與實證的結果一致且符合期望，足資佐證廣義奇異值分解法、約束型卡爾曼濾波器適用在指數追蹤的課題上。

Since the expanding markets in the passive investment by Exchange Traded Funds, the financial index tracking has been received the widely attraction. For successfully tracking the financial index, a regression with least square method is effectively and inefficiently applied in the associated parameters estimations. However, the convex function is promised to find the unique best solution for estimating parameters, but it cannot offer other sufficient good candidates. Therefore, Generalized Singular Value Decomposition is proposed to decompose the mentioned convex function for finding the trade-off solutions. Furthermore, the deterministic parameters are employed in the regression and constrained Kalman filter for predicting financial index. In case of the prediction being closed to the actual value of index, it implies that the constrained GSVD and Kalman filter are the qualified methodology for index tracking and forecasting.

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