Cubic B-spline插值與迴歸方法比較

Abstract

插值方法被廣泛的應用在工程學上，其應用有各種波型的還原及使影像放大不失真等等。而一般的插值方法其概念是由一個連續形函數通過已知的有限觀測資料，還原原始的函數值，其模型由觀測資料與interpolation kernel兩大部份所組成，本研究選用cubic B-spline曲線做為插值函數的interpolation kernel, 探討在具有平滑連續特性的函數資料下，其插值還原方法的效果，並在觀測資料具有大誤差時，提供先平均再插值的修正方式。隨後以統計迴歸分析的角度去看此插值問題，選取適當基底下估計迴歸函數，以進行插值並且比較其與一般插值方法和先平均再使用插值方法的還原效果，可以得到以下結論：(1)在函數較平滑時，不論資料的誤差大小，我們都建議使用迴歸的方法來得到較佳的還原效果，但在資料誤差大時且必須使用插值方法的情況下，可以使用本研究建議的先平均再進行插值方法來改善傳統的插值方法。(2)當函數資料不那麼平滑時，將不建議先平均再插值的處理方法，而建議在資料誤差小時使用一般插值方法，反之若資料誤差大時，則使用迴歸方法還原較佳。(3)若資料函數極不平滑時，不論誤差大小，應使用傳統的插值方法會有較佳的還原能力。

In this thesis, three interpolation approaches are studied: interpolation based on original data, interpolation based on averaged data, and B-spline regression. For the two interpolation approaches, a cubic B-spline kernel is used. The three approaches are compared based on simulation results, where the data are generated according to a regression model with various regression functions and error variances. The findings based on the simulation experiments are given below. (1) When the data are generated using smooth regression functions, B-spline regression restores the regression functions best. In such case, if the data are generated with large errors, interpolation based on averaged data performs better than interpolation based on original data. (2) When the data are generated using less smooth regression functions, interpolation based on averaged data does not perform well and is not recommanded. In such case, if the data are generated with small errors, interpolation based on original data performs better than B-spline regression; if the data are generated with large errors, B-spline regression outperforms interpolation based on original data. (3) When the data are generated using extremely unsmooth regression functions, interpolation based on original data performs better than the other two approaches.