異質變異與高維數據在統計是一個非常實用的議題。本論文將會回顧Ing與Lai (2011)和Lin (2018)對高維迴歸與高維異質變異迴歸的議題提出的選模方法與理論性質，並套用他們的想法，以正交貪婪演算法和柴比雪夫貪婪演算法兩種貪婪法的搭配為主要選模工具，針對變異為線性增長的高維異質變異迴歸提出了一非線性模型選擇程序稱為SOS。此外，在不知道真實的變異函數是線性或指數結構時，本文提出一程序，能依據數據推薦使用者較為適合的配適模型。模擬結果與晶圓實際資料分析的應用說明了我們的方法具有實用性。

We consider the problem of high-dimensional regression under non-constant error variances. First of all, we review the model selection methods and theoretical properties of Ing and Lai (2011) and Lin (2018) on the topics of high-dimensional regression and high-dimensional heterogeneous regression (HHR) respectively. After that, we borrow their ideas and propose a non-linear model selection method called Suggestion and One Stage (SOS) which use orthogonal greedy algorithm (OGA) and Chebyshev greedy algorithm (CGA) as the main tool to select the relevant variables. In addition, we propose a procedure that can recommend a suitable variation function of the HHR, linear or exponential, based on the data. Simulation results and applications to wafer data are provided to shed light on the performance and usefulness of our approach.

摘要
目錄
1 緒論 ----------------------- 1
2 文獻回歸 ---------------------- 4
3 SOS與推薦程序 ---------------------- 15
4 模擬結果 ---------------------- 23
5 實際資料分析 ---------------------- 33
6 結論與後續研究 ---------------------- 39
7 參考文獻 ---------------------- 41

1. Chen, Y.-L, Dai, C.-S and Ing, C.-K (2018). Model selection for high-dimensional sparse nonlinear models using Chebyshev greedy algorithms. Working paper.

2. Friedman, J., Hastie, T. and Tibshirani, R. (2010). glmant: Lasso and elastic-net regularized generalized linear models.

3. Ing, C.-K and Lai, T. L. (2011). A stepwise regression method and consistent model selection for high-dimensional sparse linear models. Statistica Sinica,21,1473–1513.

4. Lin, S.-C (2018). High-dimensional location-dispersion models with applications to root cause analysis in wafer fabrication processes. Master thesis, National Tsing Hua University, Hsinchu, Taiwan.

5. Zou, H. and Gu, Y.-W (2016). High-dimensional generalizations of asymmetric least squares regression and their applications. Ann. Statistic,44, 2661-2694.