在許多科學研究領域，我們經常會遇到高維度資料模型選擇的問題。Ing and Lai(2011)提出一個擁有選模一致性的三階段方法。但三階段選模法在變數高相關、低樣本與模型訊噪比小時會有較差的表現。為了提升三階段選模法的表現，我們將提出兩個方法分別使用弱正交貪婪演算法與多步正交貪婪演算法。除此之外，我們將介紹一個新的變數觀念名為"中心變數"去幫助我們克服低訊噪比的問題。在結合新概念與兩種選模方法後，我們的方法比三階段選模法擁有更好的表現並且證明這兩種選模方法也擁有選模一致性的特性。

In many scientific fields, there have feature selection problems for linear regression model with high-dimensional data. To select relevant variables, Ing and Lai (2011) introduce a three stage procedure which has model selection consistency property. But we find out that the procedure does not work well when covariates are highly correlated, sample size is less than 100 and signal to noise ratio is small. To improve the three stage procedure, we propose two methods using weak orthogonal greedy algorithm (WOGA) and multi-step orthogonal greedy algorithm (MOGA). Moreover, there introduce a new concept of variable, named ”central variable”, which can explain all covariates around itself. By combining two methods and new concept, there have better performance comparing with the three stage method. We also show that two methods preserve the model selection consistency property.

Abstract
1 Introduction 4
2 Weak Orthogonal Greedy Algorithm and Multi-step Orthogonal
Greedy Algorithm 7
2.1 Central variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Uniform convergence rates of WOGA and MOGA 10
3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Uniform Convergence Rates . . . . . . . . . . . . . . . . . . . . . . 11
4 Consistency of variable selection 14
4.1 Consistency of central variable selection . . . . . . . . . . . . . . . 14
4.2 Sure screening property for WOGA and MOGA . . . . . . . . . . . 17
4.3 Sure screening property for HDBIC . . . . . . . . . . . . . . . . . . 18
4.4 Model selection consistency property . . . . . . . . . . . . . . . . . 19
5 Simulation Study 19
5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Conclusion and future work 28
A Lemma 30
B Theorem 3.1 MOGA part 34

Fan, J. and Lv, J. (2008). Sure independence screening for ultrahigh dimensional
feature space. Journal of the Royal Statistical Society. Series B: Statistical
Methodology, 70(5):849–911.
Gao, F., Ing, C. K., and Yang, Y. (2013). Metric entropy and sparse linear
approximation of ff q-hulls for 0 < q ff 1. Journal of Approximation Theory,
166(1):42–55.
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.
Ing, C.-K. and Lai, T. L. (2018). An Efficient Pathwise Variable Selection Criterion
In Weakly Sparse Regression Models. submitted to Annals of Statistics.
29
Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable
selection with the Lasso. Annals of Statistics, 34(3):1436–1462.
Temlyakov, V. N. (2000). Weak greedy algorithms. Advances in Computational
Mathematics, 12:213.
Tibshirani, R. (1996). Rregression shrinkage and selection via the lasso. Journal
of the Royal Statistical Society. Series B (Methodological), 58:267–288.
Wu, W. B. and Wu, Y. N. (2016). Performance bounds for parameter estimates
of high-dimensional linear models with correlated errors. Electronic Journal of
Statistics, 10(1):352–379.
Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the
American Statistical Association, 101(476):1418–1429.