未知改變點下之連續分段迴歸分析 - 修正後目標規劃法的應用

Abstract

改變點位置偵測在逐段多項式函數是一個重要問題﹐一般均需假設改變點 分配才能找到其位置﹐之後﹐求出迴歸方程。由於這是一個棘手問題﹐所 以通常假設改變點位置為已知﹐如此便可利用最小平方法或spline method。 本論文提出以修正後目標規劃法求解在未知改變點下之連續 逐段多項式﹐首先先 介紹本法逐段多項式表示法､特性;其次﹐藉由其特 性利用零壹變數控制改變點個數與修正後目標規劃法﹐以互動的方式﹐完 成能同時偵測改變點位置､求解迴歸方程﹐進而決定滿意的改變點個數﹔ 修正後目標規劃法的應用在於能提高求解的速度﹔最後﹐以兩個範例說明 本法的使用﹐如何找到改變點､並與Poirier's方法與最小平方法做比較 ﹐以利凸顯其優點。 An essence problem in estimating a piecewise polynomial function is the positions of change-points. Suppose the positions of the change-points are known(fixed constants), the polynomial function can then be estimated straight forward by least squares methods or spline method. This paper proposes a Least Absolute Deviations( LAD, L1-norm ) method to estimate a piecewise polynomial function with unknown change-points. We first express a piecewise polynomial function by a series of absolute terms. Utilizing the properties of this function, a goal programming model is formulated to minimize the estimation errors within a given number of change-points. The model is solved by a modified goal programming technique which is more computational efficiency than conventional goal programing methods. We show two examples in Chapter 4 to describe how the proposed method does.