Knot selection for least-squares and penalized splines

Journal of Statistical Computation and Simulation

Volumn 83, Issue 6, 2013, Pages 1020-1036

  Spiriti, Steven ^a   Eubank, Randall ^a   Smith, Philip W ^b   Young, Dennis ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

^b TEXAS TECH UNIVERSITY   (United States)

Author keywords

blind random search; generalized cross validation; genetic algorithm; golden section; parallel computing

Indexed keywords

EID: 84878966411     PISSN: 00949655     EISSN: 15635163     Source Type: Journal    
DOI: 10.1080/00949655.2011.647317     Document Type: Article

References (40)

1. Wold, S. 1974. Spline functions in data analysis. Technometrics, 16: 1 - 11. (doi:10.1080/00401706.1974.10489142)
2. Smith, P. 1979. Splines as a useful and convenient statistical tool. Am. Statist., 33: 57 - 62.
3. Jupp, D. 1975. The 'Lethargy' Theorem-A property of approximation by γ-polynomials. J. Approx. Theory, 14: 204 - 217. (doi:10.1016/0021-9045(75)90056-8)
4. Jupp, D. 1978. Approximation to data by splines with free knots. SIAM J. Numer. Anal., 15: 328 - 343. (doi:10.1137/0715022)
5. de Boor, C. 2001. A Practical Guide to Splines, Berlin: Springer. Revised ed.
6. Hu, Y. 1993. An algorithm for data reduction using splines with free knots. IMA J. Numer. Anal., 13: 365 - 381. (doi:10.1093/imanum/13.3.365)
7. Dierckx, P. 1993. Curve and Surface Fitting with Splines, Oxford: Oxford University Press.
8. Lindstrom, M. 1999. Penalized estimation of free knot splines. J. Comput. Graph. Statist., 8: 333 - 352.
9. Ruppert, D. 2002. Selecting the number of knots for penalized splines. J. Comput. Graph. Statist., 11: 735 - 757. (doi:10.1198/106186002853)
10. Spall, J. 2003. Introduction to Stochastic Search and Optimization, New York: Wiley.
11. Pitman, J. and Murthy, C. A. 2000. Fitting optimal piecewise linear functions using genetic algorithms. IEEE Trans. Pattern Anal. Mach. Intell., 22: 701 - 718. (doi:10.1109/34.865188)
12. Pitman, J. 2002. Adaptive splines and genetic algorithms. J. Comput. Graph. Statist., 11: 615 - 638. (doi:10.1198/106186002448)
13. Lee, T. 2002a. Automatic smoothing for discontinuous regression functions. Statistica Sinica, 12: 823 - 842.
14. Lee, T. 2002b. On algorithms for ordinary least squares regression spline fitting: A comparative study. J. Statist. Comput. Simul., 72: 647 - 663. (doi:10.1080/00949650213743)
15. Miyata, S. and Shen, X. 2005. Free knot splines and adaptive knot selection. J. Japanese Statist. Soc., 35: 303 - 324.
16. Smith, M. and Kohn, R. 1996. Nonparametric regression using Bayesian variable selection. J. Econometrics, 75: 317 - 343. (doi:10.1016/0304-4076(95)01763-1)
17. Denison, D., Mallick, B. and Smith, A. 1998. Automatic Bayesian curve fitting. J. Roy. Statist. Soc. Ser. B, 60: 333 - 350. (doi:10.1111/1467-9868.00128)
18. DiMatteo, I., Genovese, C. and Kass, R. 2001. Bayesian curve fitting with free-knot splines. Biometrika, 88: 1055 - 1071. (doi:10.1093/biomet/88.4.1055)
19. Lindstrom, M. 2002. Bayesian estimation of free knot splines using reversible jumps. Comput. Statist. Data Anal., 41: 255 - 269. (doi:10.1016/S0167-9473(02)00066-X)
20. Eilers, P. and Marx, B. 1996. Flexible smoothing with B-splines and penalties. Statist. Sci., 11: 89 - 121. (doi:10.1214/ss/1038425655)
21. Ruppert, D., Wand, M. and Carroll, R. 2003. Semiparametric Regression, Cambridge: Cambridge University Press.
22. Imoto, S. and Konishi, S. 2003. Selection of smoothing parameters in B-spline nonparametric regression models using information criteria. Ann. Inst. Statist. Math., 55: 671 - 687. (doi:10.1007/BF02523388)
23. Kauermann, G. 2005. A note on smoothing parameter selection for penalized splines. J. Statist. Plann. Inference, 127: 53 - 69. (doi:10.1016/j.jspi.2003.09.023)
24. Schwetlick, H. and Schöutze, T. 1995. Least squares approximation by splines with free knots. BIT Numer. Math., 35: 361 - 384. (doi:10.1007/BF01732610)
25. Yao, F. and Lee, C. M. 2008. On knot placement for penalized spline regression. J. Korean Statist. Soc., 2: 259 - 267. (doi:10.1016/j.jkss.2008.01.003)
26. Agarwal, G. and Studden, W. 1980. Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist., 8: 1307 - 1325. (doi:10.1214/aos/1176345203)
27. Claeskens, G., Krivobokova, T. and Opsomer, J. 2009. Asymptotic properties of penalized spline estimators. Biometrika, 96: 529 - 544. (doi:10.1093/biomet/asp035)
28. Golub, G. and Pereyra, V. 1973. The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal., 10: 413 - 432. (doi:10.1137/0710036)
29. Smith, P. 1982. Hypothesis testing in B-spline regression. Commun. Statist. B: Simul. Comput., 11: 143 - 157. (doi:10.1080/03610918208812251)
30. Hutchinson, M. and de Hoog, F. 1985. Smoothing noisy data with spline functions. Numer. Math., 47: 99 - 106. (doi:10.1007/BF01389878)
31. Solis, F. and Wets, R. 1981. Minimization by random search techniques. Math. Oper. Res., 6: 19 - 30. (doi:10.1287/moor.6.1.19)
32. Haupt, R. and Haupt, S. 2004. Practical Genetic Algorithms, 2, New York: Wiley.
33. Wahba, G. 1990. Spline Models for Observational Data, Philadelphia, PA: SIAM.
34. Silverman, B. 1984. Spline smoothing: The equivalent variable kernel method. Ann. Statist., 12: 898 - 916. (doi:10.1214/aos/1176346710)
35. L'Ecuyer, P., Simard, R., Chen, J. and Kelton, W. 2002. An object-oriented random-number package with many long streams and substreams. Oper. Res., 50: 1073 - 1075. (doi:10.1287/opre.50.6.1073.358)
36. Eubank, R. 1999. Nonparametric Regression and Spline Smoothing, New York: Marcel-Dekker.
37. Friedman, J. 1991. Multivariate adaptive regression splines. Ann. Statist., 19: 1 - 141. (doi:10.1214/aos/1176347963)
38. Eubank, R. 2004. A simple smoothing spline, III. Comput. Statist., 19: 227 - 241. (doi:10.1007/BF02892058)
39. Li, X. and Ruppert, D. 2008. On the asymptotics of penalized splines. Biometrika, 95: 415 - 436. (doi:10.1093/biomet/asn010)
40. Welham, S., Cullis, B., Kenward, M. and Thompson, R. 2007. A comparison of mixed model splines for curve fitting. Austral. New Zealand J. Statist., 49: 1 - 23.