臺灣博碩士論文加值系統

線性回歸中，在共線性問題發生的情況下，常見的最小平方估計量表現得並不理想。這類問題發生在混合實驗 ( mixture experiments ) 中，原因是此種實驗對自變數 ( regressors ) 的數學限制式所導致的。Hoerl和Kennard在1970年提出了ridge 回歸方法的概念，解決了最小平方估計量在共線性問題下的缺陷。近來，ridge回歸方法也成功地應用在混合實驗中。然而，混合實驗通常會將截距項合併至自變數的係數之中，若我們在線性模型中保有截距項並對自變數做標準化的話，ridge回歸方法的應用將變得比較複雜。此篇論文考慮了一種特殊的劉氏估計量( Liu-type estimators )，並保有模型中的截距項。我們推導新估計量的均方差( mean squared error )函式並且透過統計模擬來比較新估計量和既有的ridge估計量。最後，我們用兩組實驗的資料來說明新估計量的表現。

In the linear regression, the least square estimator does not perform well in terms of mean squared error when multicollinearity exists. The problem of multicollinearity occurs in industrial mixture experiments, where regressors are constrained.Hoerl and Kennard (1970) proposed the ordinary ridge estimator to overcome the problem of the least squared estimator under multicollinearity. Recently, the ridge regression is successfully applied to mixture experiments. However, the application of ridge becomes difficult if the linear model has the intercept term and the regressors are standardized as occurring in mixture experiments. This paper considers a special class of Liu-type estimators (Liu, 2003) with intercept. We derive the theoretical formula of the mean squared error for the proposed method. We perform simulations to compare the proposed estimator with the ridge estimator in terms of mean squared error. We demonstrate this special class using the dataset on Portland cement with mixture experiment (Woods et al., 1932).

摘要 …….………………………………………………………………………………………...I

Abstract ...………………………………………………………………………………………..II

致謝辭 ...…………………………………………………………………………………………III

Contents ...………………………………………………………………………………………IV

List of Tables …………………………………………………………………………………….V

List of Figures …………………………………………………………………………………..VI

1. Introduction ………………………………………………………...………………………....1

2. Background …………………………………………………………...……………………….2

3. Proposed Method ……………………………………………………………………............ 10

4. Theory………………………………………………………………..……………….……….14

5. Simulation ………………………………………………………………………...…………..22

6. Data analysis ………………………………………………………………………...……….32

7. Conclusion ………………………………………………………………………...………….41

8. Reference ………………………………………………………………………...…………...42

Allen, D. M. (1974). The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16(1), 125-127.
Brown, P. J. (1977). Centering and scaling in ridge regression. Technometrics 19(1), 35-36.
Chuang, S. C. and Hung, Y. C. (2010). Uniform design over general input domains with applications to target region estimation in computer experiments. Computational Statistics & Data Analysis 54(1), 219-232.
Cornell, J. A. (2011). A primer on experiments with mixture. Hoboken, N. J.: Wiley.
Dempster, A. P., Schatzoff, M. and Wermuth, N. (1977). A simulation study of alternatives to ordinary least squares. Journal of the American Statistical Association 72, 77-91.
Emura, T., Chen Y. H. and Chen H. Y. (2012). Survival prediction based on compound covariate under cox proportional hazard models. PLoS ONE 7, DOI: 10.1371/journal.pone.0047627.
Emura, T. and Chen, Y. H. (2014). Gene selection for survival data under dependent censoring: a copula-based approach. Statistical Methods in Medical Research, DOI: 10.1177/0962280214533378.
Gibbons, D. G. (1981). A simulation study of some ridge estimators. Journal of the American Statistical Association 76, 131-139.
Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12, 55-67.
Hoerl, A. E., Kennard, R. W. and Baldwin K. F. (1975). Ridge regression: some simulations, Communications in Statistics 4, 105-123.
Hsu, H.-L. (2003). Robust D-optimal designs for mixture experiments in Scheffé models. Master thesis, Department of Applied Mathematics, National Sun Yat-sen University.
Jang, D.-H. and Anderson-Cook C. M. (2010). Fraction of design space plots for evaluating ridge estimators in mixture experiments. Quality and Reliability Engineering International 27, 27-34.
Jang, D.-H. and Anderson-Cook C. M. (2014). Visualization approaches for evaluating ridge regression estimators in mixture and mixture-process experiments. Quality and Reliability Engineering International. DOI: 10.1002/qre.1683.
Jimichi, M. and Inagaki, N. (1993). Centering and scaling in ridge regression. Statistical Science and Data Analysis 3, 77-86.
Jimichi, M. (2005). Improvement of regression estimators by shrinkage under multicollinearity and its feasibility. Ph.D. Thesis. Osaka University: Japan.
Li, Y. and Yang, H. (2010). A new Liu-type estimator in linear regression model. Statistical Papers 53, 427-437.
Liu, K. (1993). A new class of biased estimate in linear regression. Communications in Statistics – Theory and Methods 22, 393-402.
Liu, K. (2003). Using Liu-type estimator to combat collinearity. Communications in Statistics – Theory and Methods 32, 1009-1020.
Mazerolle, M. J. (2014). AICcmodavg: Model selection and multimodel inference based on (Q)AIC(c). R package version 2.0-1.http://CRAN.R-project.org/package=AICcmodavg.
McLean, R. A. and Anderson, V. L. (1966). Extreme Vertices Design of Mixture Experiments. Technometrics 8(3), 447-454.
Montgomery, D. C., Peck, E. A. and Vining G. G. (2012). Introduction to Linear Regression Analysis. New Jersey: Wiley, 2012. Print.
Sakallıoğlu, S. and Kaçıranlar, S. (2006). A new biased estimator on ridge estimation. Statistical Papers 49, 669-689.
Theobald, C. M. (1973). Generalizations of mean square error applied to ridge regression. Journal of the Royal Statistical Society. Series B (Methodological) 36, 103-106.
Tukey, J. W. (1993). Tightening the clinical trial. Controlled Clinical Trials 14, 266-285.
Wong, K. Y. and Chiu, S. N. (2015). An iterative approach to minimize the mean squared error in ridge regression. Computational Statistics. DOI: 10.1007/s00180-015-0557-y.
Woods, H., Steinour, H. H. and Starke, H. R. (1932) Effect of composition of Portland cement on heat evolved during hardening. Industrial Engineering and Chemistry 24, 1207-1214.
Yang, S. P. (2014). A class of generalized ridge estimator for high-dimensional linear regression. Master thesis, National Central University Electronic Theses & Dissertations.