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題名 比較使用Kernel和Spline法的傘型迴歸估計
Compare the Estimation on Umbrella Function by Using Kernel and Spline Regression Method
作者 賴品霖
Lai, Pin Lin
貢獻者 黃子銘
賴品霖
Lai, Pin Lin
關鍵詞 核迴歸
樣條迴歸
無母數迴歸
傘型函數
Kernel regression
Spline regression
Nonparametric regression
Umbrella function
日期 2016
上傳時間 11-Jul-2016 16:55:04 (UTC+8)
摘要 本研究探討常用的兩個無母數迴歸方法，核迴歸與樣條迴歸，在具有傘型限制式下，對於傘型函數的估計與不具限制式下的傘型函數估計比較，同時也探討不同誤差變異對估計結果的影響，並進一步探討受限制下兩方法的估計比較。本研究採用「估計頂點位置與實際頂點位置差」及「誤差平方和」作為衡量估計結果的指標。在帶寬及節點的選取上，本研究採用逐一剔除交互驗證法來篩選。模擬結果顯示，受限制的核函數在誤差變異較大的頂點位置估計較佳，誤差變異縮小時反而頂點位置估計較差，受限制的B-樣條函數也有類似的狀況。而在兩方法的比較上，對於較小的誤差變異，核函數的頂點位置估計能力不如樣條函數，但在整體的誤差平方和上卻沒有太大劣勢，當誤差變異較大時，核函數的頂點位置估計能力有所提升，整體誤差平方和仍舊維持還不錯的結果。
In this study, we give an umbrella order constraint on kernel and spline regression model. We compare their estimation in two measurements, one is the difference of estimate peak and true peak, the other one is the sum of square difference on predict and the true value. We use leave-one-out cross validation to select bandwidth for kernel function and also to decide the number of knots for spline function. The effect of different error size is also considered. Some of R packages are used when doing simulation. The result shows that when the error size is bigger, the prediction of peak location is better in both constrained kernel and spline estimation. The constrained spline regression tends to provide better peak location estimation compared to constrained kernel regression.
參考文獻 1. Boor, C. D. (1972) “On calculating with B-splines.”, Journal of Approximation theorey, 6, 50-62.
2. Cressie, N. A. C. and Read, T. R. C. (1984) “Multinomial goodness-of-fit tests.”, J. Roy. Statist. Soc. Ser. B, 46, 440-464
3. Du, P., Parmeter, C. F. and Racine, J. S. (2013) “Nonparametricc kernel regression with multiple predictors and multiple shape constraints.”, Statistica Sinica, 23, 1347-1371.
4. Fan, J. (1992) “Design-adaptive nonparametric regression.”, J. Amer. Statist. Assoc., 87, 998-1004.
5. Gasser, T. and Müller, H.-G. (1979) “Kernel estimation of regression functions.”, In Smoothing Techniques for Curve Estimation, 23(68), Springer-Verlag, New York.
6. Hall, P. and Haung, L.-S. (2001) “Noparametric kernel regresson subject to monotonicity constraints.”, Ann. Statist, 29(3), 624-647.
7. He, X., and Shi, P.(1998) “Monotone B-spline smoothing.”, J. Amer. Statist. Assoc., 93(442), 643-650.
8. Mammen, E. and Thomas-Agnan, C. (1998) “Smoothing splines and shape restrictions.”, Scandinavian Journal of Statistics, 26, 239-252.
9. Nadaraya, E. A. (1965) “On nonparametric estimates of density functions and regression curves”, Theory Probab. Appl., 10, 186-190.
10. Priestley, M. B. and Chao, M. T. (1972) “Nonparametric function fitting.”, J. Roy. Statist. Soc. Ser. B, 34, 385-392
11. Racine, J. and Li, Q. (2004) “Nonparametric estimation of regression functions with both categorical and continuous data.”, J. Econometrics, 119, 99-130.
12. Schumaker, L. L. (1981) Spline functions, Wiley, New York.
13. Stone, M. (1974) “Cross-validatory choice and assessment of statistical predictions.”, Roy. Statist. Soc. Ser. B, 36(2), 111-147
14. Stout, F. (2008) “Unimodal regression via prefix isotonic regression.”, Computational Statistics and Data Analysis, 53, 289-297.
15. Watson, G. S. (1964) “Smooth regression analysis.”, Sankhya ̅, 26(15), 175-184.
描述 碩士
國立政治大學
統計學系
102354008
資料來源 http://thesis.lib.nccu.edu.tw/record/#G1023540081
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.author (Authors) 賴品霖zh_TW
dc.contributor.author (Authors) Lai, Pin Linen_US
dc.creator (作者) 賴品霖zh_TW
dc.creator (作者) Lai, Pin Linen_US
dc.date (日期) 2016en_US
dc.date.accessioned 11-Jul-2016 16:55:04 (UTC+8)-
dc.date.available 11-Jul-2016 16:55:04 (UTC+8)-
dc.date.issued (上傳時間) 11-Jul-2016 16:55:04 (UTC+8)-
dc.identifier (Other Identifiers) G1023540081en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/98847-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 102354008zh_TW
dc.description.abstract (摘要) 本研究探討常用的兩個無母數迴歸方法，核迴歸與樣條迴歸，在具有傘型限制式下，對於傘型函數的估計與不具限制式下的傘型函數估計比較，同時也探討不同誤差變異對估計結果的影響，並進一步探討受限制下兩方法的估計比較。本研究採用「估計頂點位置與實際頂點位置差」及「誤差平方和」作為衡量估計結果的指標。在帶寬及節點的選取上，本研究採用逐一剔除交互驗證法來篩選。模擬結果顯示，受限制的核函數在誤差變異較大的頂點位置估計較佳，誤差變異縮小時反而頂點位置估計較差，受限制的B-樣條函數也有類似的狀況。而在兩方法的比較上，對於較小的誤差變異，核函數的頂點位置估計能力不如樣條函數，但在整體的誤差平方和上卻沒有太大劣勢，當誤差變異較大時，核函數的頂點位置估計能力有所提升，整體誤差平方和仍舊維持還不錯的結果。zh_TW
dc.description.abstract (摘要) In this study, we give an umbrella order constraint on kernel and spline regression model. We compare their estimation in two measurements, one is the difference of estimate peak and true peak, the other one is the sum of square difference on predict and the true value. We use leave-one-out cross validation to select bandwidth for kernel function and also to decide the number of knots for spline function. The effect of different error size is also considered. Some of R packages are used when doing simulation. The result shows that when the error size is bigger, the prediction of peak location is better in both constrained kernel and spline estimation. The constrained spline regression tends to provide better peak location estimation compared to constrained kernel regression.en_US
dc.description.tableofcontents 第壹章 緒論 1
第一節 研究動機與背景 1
第二節 研究問題與目的 2
第三節 研究架構與流程 3
第貳章 文獻探討 4
第一節 傘型函數相關文獻探討 4
第二節 核函數相關文獻回顧 4
第三節 樣條函數文獻探究 7
第參章 研究設計 8
第一節 核函數迴歸模型 8
第二節 核函數估計流程 9
第三節 B-樣條函數迴歸模型 10
第四節 B-樣條函數估計流程 11
第肆章 模擬實驗 12
第一節 資料產生 12
第二節 實證結果 13
第伍章 結論與建議 21
第一節 研究結論 21
第二節 研究建議 22
附錄一 模擬之傘型函數 23
附錄二 平滑後之傘型函數 27
參考文獻 31		zh_TW
dc.format.extent 801246 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G1023540081en_US
dc.subject (關鍵詞) 核迴歸zh_TW
dc.subject (關鍵詞) 樣條迴歸zh_TW
dc.subject (關鍵詞) 無母數迴歸zh_TW
dc.subject (關鍵詞) 傘型函數zh_TW
dc.subject (關鍵詞) Kernel regressionen_US
dc.subject (關鍵詞) Spline regressionen_US
dc.subject (關鍵詞) Nonparametric regressionen_US
dc.subject (關鍵詞) Umbrella functionen_US
dc.title (題名) 比較使用Kernel和Spline法的傘型迴歸估計zh_TW
dc.title (題名) Compare the Estimation on Umbrella Function by Using Kernel and Spline Regression Methoden_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 1. Boor, C. D. (1972) “On calculating with B-splines.”, Journal of Approximation theorey, 6, 50-62.
2. Cressie, N. A. C. and Read, T. R. C. (1984) “Multinomial goodness-of-fit tests.”, J. Roy. Statist. Soc. Ser. B, 46, 440-464
3. Du, P., Parmeter, C. F. and Racine, J. S. (2013) “Nonparametricc kernel regression with multiple predictors and multiple shape constraints.”, Statistica Sinica, 23, 1347-1371.
4. Fan, J. (1992) “Design-adaptive nonparametric regression.”, J. Amer. Statist. Assoc., 87, 998-1004.
5. Gasser, T. and Müller, H.-G. (1979) “Kernel estimation of regression functions.”, In Smoothing Techniques for Curve Estimation, 23(68), Springer-Verlag, New York.
6. Hall, P. and Haung, L.-S. (2001) “Noparametric kernel regresson subject to monotonicity constraints.”, Ann. Statist, 29(3), 624-647.
7. He, X., and Shi, P.(1998) “Monotone B-spline smoothing.”, J. Amer. Statist. Assoc., 93(442), 643-650.
8. Mammen, E. and Thomas-Agnan, C. (1998) “Smoothing splines and shape restrictions.”, Scandinavian Journal of Statistics, 26, 239-252.
9. Nadaraya, E. A. (1965) “On nonparametric estimates of density functions and regression curves”, Theory Probab. Appl., 10, 186-190.
10. Priestley, M. B. and Chao, M. T. (1972) “Nonparametric function fitting.”, J. Roy. Statist. Soc. Ser. B, 34, 385-392
11. Racine, J. and Li, Q. (2004) “Nonparametric estimation of regression functions with both categorical and continuous data.”, J. Econometrics, 119, 99-130.
12. Schumaker, L. L. (1981) Spline functions, Wiley, New York.
13. Stone, M. (1974) “Cross-validatory choice and assessment of statistical predictions.”, Roy. Statist. Soc. Ser. B, 36(2), 111-147
14. Stout, F. (2008) “Unimodal regression via prefix isotonic regression.”, Computational Statistics and Data Analysis, 53, 289-297.
15. Watson, G. S. (1964) “Smooth regression analysis.”, Sankhya ̅, 26(15), 175-184.		zh_TW