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| Title: | 一種基於BIC的B-Spline節點估計方式 |
| Authors: | 何昕燁
Ho, Hsin Yeh |
| Contributors: | 黃子銘
Huang, Tzee Ming
何昕燁
Ho, Hsin Yeh |
| Keywords: | B-樣條
節點
馬可夫鏈蒙地卡羅
B-Spline
knot
reversible-jump Morkov chain Monte Carlo
Bayesian information criterion |
| Date: | 2012 |
| Issue Date: | 2013-07-22 11:10:51 (UTC+8) |
| Abstract: | 在迴歸分析中，若變數間具有非線性的關係時，B-Spline線性迴歸是以無母數的方式建立模型。B-Spline函數為具有節點(knots)的分段多項式，選取合適節點的位置對B-Spline的估計有重要的影響，在近年來許多的文獻中已提出一些尋找節點位置的估計方法，而本文中我們提出了一種基於Bayesian information criterion(BIC)的節點估計方式。
我們想要深入瞭解在不同類型的迴歸函數間，各種選取節點方法的配適效果與模擬時間，並且加以比較，在使用B-Spline函數估計時，能夠使用合適的方法尋找節點。
In regression analysis, when the relation between the response variable and the explanatory variable is nonlinear, one can use nonparametric methods to estimate the regression function.
B-Spline regression is one of the popular nonparametric regression methods. B-Splines are piecewise polynomial joint at knots, and the choice of knot locations is crucial.
Zhou and Shen (2001) proposed to use spatially adaptive regression splines (SARS), where the knots are estimated using a selection scheme. Dimatteo, Genovese, and Kass (2001) proposed to use Bayesian adaptive regression splines (BARS), where certain priors for knot locations are considered. In this thesis, a knot estimation method based on the Bayesian information criterion (BIC) is proposed, and simulation studies are carried out to compare BARS, SARS and the proposed BIC-based method. |
| Reference: | [1] C. Biller. Adaptive Bayesian regression splines in semiparametric generalized linear models. Journal of Computational and Graphical Statistics,9:122-40,2000.
[2] C.G. Broyden. The convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications,(6):222-231,1970.
[3] C. de Boor. On calculating with B-Splines. Journal of approximation theory, (6):50-62, 1972.
[4] D.G.T. Denison, B.K. Mallick, and A.F.M. Smith. Automatic Bayesian curve fitting. Journal of Royal Statistical Society: Series B, (60):333-350, 1998.
[5] I. Dimatteo, C.R. Genovese, and R.E. Kass. Bayesian curve-fitting with free-knot splines. Biometrika,88(4):1055-1071, 2001.
[6] R. Fletcher. A new approach to variable metric algorithms. Computer Journal, 13(3):317-322, 1970.
[7] J.H. Friedman. Multivariate adaptive regression splines. The Annals of
Statistics, 19:1{141, 1991.
[8] D. Goldfarb. A family of variable metric updates derived by variational means. Mathematics of Computation, 24(109):23-26, 1970.
[9] P.J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82:711-32, 1995.
[10] E.F. Halpern. Bayesian spline regression when the number of knots is unknown. Journal of Royal Statistical Society: Series B, 35:347-60,1973.
[11] R.E. Kass and L. Wasserman. A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of American Statistical Association, 90:928-34, 1995.
[12] M.J. Lindstrom. Penalized estimation of free-knot splines. Journal of Computational and Graphical Statistics, 8:333-52, 1999.
[13] David Ruppert. Selecting the number of knots for penalized splines.Journal of Computational and Graphical Statistics, 11(4):735-757, 2002.
[14] D.F. Shanno. Conditioning of quasi-Newton methods for function min-imization. Mathematics of Computation, 24(111):647-656, 1970.
[15] C.M. Stein. Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, 9(6):1135-1151, 1981.
[16] S.ZHOU and X.SHEN. Spatially adaptive regression splines and accurate knot selection schemes. Journal of American Statistical Association,96(453):247-259, 2001. |
| Description: | 碩士
國立政治大學
統計研究所
100354006
101 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G1003540062 |
| Data Type: | thesis |
| Appears in Collections: | [統計學系] 學位論文
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