線性迴歸模型具巢狀結構下透過廣義自由度選取最適模型之探討

Study on adaptive model selection through generalized degrees of freedom in nested linear regression models

湯泉發(Chiuan-Fa Tang)

指導教授 ： 陳宏

國立臺灣大學/理學院/數學研究所/碩士(2010年)

https://doi.org/10.6342/NTU.2010.03487

摘要

選擇模型來解釋資料的方法有很多種, 像是AIC (Akaike
1974), BIC (Schwarz 1978), 以及Mallows’ Cp. 當考慮線性
迴歸模型選取時, 可將上述的模型選取法則寫成廣義最終
誤差的選擇方法, 而各個方法之間的差異僅在於選擇方法
的懲罰項λ. Shen and Ye (2002) 提出從所有可能的廣義最
終誤差選擇方法中, 透過決定懲罰項λ 來找出來最適模型
的選取方法.
在本文裡我們將會介紹由Shen and Ye (2002) 提出所謂
透過廣義自由度選取最適模型. 並且誤差為常態分配, 線
性迴歸模型具巢狀結構以及某些正規條件下來評量這個
方法. 我們將會透過模擬的方式呈現如果這個方法如果不
估計廣義自由度而是帶入真實值. 那麼將不會是完全地選
到最適當的模型. 並且將給這樣的結果解釋.

關鍵字

最是模型選取方法 ； 廣義最終誤差選擇方法 ； 廣義自由度

並列摘要

Various model selection criteria have been proposed to fit models to data, such as
AIC (Akaike 1974), BIC (Schwarz 1978), and Mallows’ Cp (1973). For linear regression
with suitable regularity conditions, we can combine those criterion into general
final prediction error criterion with different lambda. If we consider all possible general
final prediction error criterion over an interval including λ = 2 and λ = log n.
Shen and Ye (2002) proposed the adaptive model selection by determining proper
lambda through general final prediction error.
In this thsis, we will introduce the adaptive models selection criterion through
generalized degrees of freedom which proposed by Shen and Ye (2002) and evaluate
the performance of this criterion in a most widely used linear regression model with
normal error and some further bias and sample size assumption. We will demonstrate
that the adaptive model selection criterion is not fully adaptive. As a remedy, we
suggest that the interval should be restricted. We will provide some simulation results
to show the performance of adaptive model selection through generalized degrees of
freedom in nested linear regression models and the conclusions. We will provide
some simulation results to motivate the procedure of solving problems and support
our conclusions.

並列關鍵字

adaptive model selection ； final prediction error ； generalized degrees of freedom

參考文獻

Akaike, H. (1973). Information Theory and an Extension of the Maximum Likelihood

Mallows, C.L. (1973). Some Comments on Cp. Technometrics, 15(4), pp. 661-675.

Schwarz, G. (1978). Estimating the Dimension of a Model. The Annals of Statistics,

Shen, G. and Ye (2002). Adaptive Model Selection, Journal of American Statistical

Spitzer, F.(1959). A combinatorial Lemma and Its Application to Probability Theory,