指數加權複迴歸及相對重要性指標之研究

Abstract

本研究在Gauss,K.F.(1821,1823,1826)指數加權遞迴之最小平方法(exponentially weighted recursive least square)的基礎之下，研究指數加權對複迴歸係數的影響，並針對指數加權導致之估計不穩特性，提出有別於針對每筆新收集到的資料加權，而是隨著時間以區塊(block)為單位對資料加權，發展出區塊加權遞迴之最小平方法(block-weighted recursive least square)，減低複迴歸係數的變異程度。比較指數加權遞迴之最小平方法與區塊加權遞迴之最小平方法時，選出在不同資料量、權重、區塊大小時各自有最佳表現的組合，並觀察區塊加權遞迴之最小平方法對於原本演算法的改進程度。經實驗發現在變數個數少的時候，區塊加權遞迴之最小平方法才會有改善的空間。經多個案例研究發現，以區塊(block)做加權並不會改善指數加權遞迴之最小平方法之穩定性。 複迴歸分析在變數間不存在共線性關係時，迴歸係數之t統計量及檢定可代表各變數的重要性程度，但存在共線性關係時，文獻使用Johnson[1]的Dominance index先將原始變數轉換成彼此正交(orthogonal)且無相依性(uncorrelated)的最佳估計，再進行迴歸分析以及變數重要性的估算。然而製程可能隨著時間變動，因此本研究將最新收集到的資料給予較大的權重，權重依時間呈指數遞減，發展出指數加權之變數相對重要性方法。而加權的方法又可分成兩種，第一種是利用指數加權移動相關係數(EWMC)，第二種是利用指數加權迴歸法(exponentially weighted regression)，兩種方法各有其優劣。 同時考慮到資料變化的趨勢，利用指數加權移動相關係數(EWMC)所估計的指數加權之相對重要性在我們研究的案例中表現相對較好。研究方法上，將會以模擬案例及實際半導體製程資料驗證本研究所提出的兩種方法的偵測能力。

In this research, we first focus on the effect of applying exponential weight to the regression analysis data based on the exponentially weighted recursive least square method. Gauss,K.F.(1821,1823,1826) . Because of the unstability of regreesion coefficients resulting from the weighting scheme, we attempt to propose a new method which is called the block-weighted recursive least square method. The proposed method applies weights to a block of observations in order to reduce the variation of regreesion coefficients. We use simulations to compare the results of the exponentially weighted recursive least square method and the block-weighted recursive least square method. It is found that applying weights to the block does not significantly imporove the variation of the regreesion coefficients through adding an additional parameter on the block size.. When the predictor variables are uncorrelated, we can simply use the t statistic as the test statistic of regression coefficient to find out the relative importance of each predictor variable. When the predictor variables are correlated, previous articles have shown that the sets of variables can be approximated by a set of uncorrelated variables that can be used for estimation of relative importance. (Johnson,2000). Since the manufacturing factors may undergo structural changes from time to time, we propose in this research applying the exponential weights to the data to capture the dynamic changes of causal effects. We propose two weighting schemes : exponentially weighted moving correlation (EWMC), and exponentially weighted regression. In our study , the exponentially weighted relative importance with EWMC is found better than the exponentially weighted regression when the response variable appears to be nonstationary. Simulated and actual semi-conductor engineering data are used to illustrate and validate the proposed methods.