Thin Plate Regression Splines
Tóm tắt
I discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother. By making use of Lanczos iteration the basis change and truncation are computationally efficient. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems that are associated with ‘knot placement’ that usually complicate modelling with regression splines or penalized regression splines, provide a sensible way of modelling interaction terms in generalized additive models, provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, provide a means for incorporating smooth functions of more than one variable into non-linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. Given that the approach produces spline-like models with a sparse basis, it also provides a natural way of incorporating unpenalized spline-like terms in linear and generalized linear models, and these can be treated just like any other model terms from the point of view of model selection, inference and diagnostics.
Từ khóa
Tài liệu tham khảo
Akaike, 1973, Proc. 2nd Int. Symp. Information Theory, 267
Borchers, 1997, Improving the precision of the daily egg production method using generalized additive models, Can. J. Fish. Aq. Sci., 54, 2727, 10.1139/f97-134
Duchon, 1977, Construction Theory of Functions of Several Variables
Eilers, 1996, Flexible smoothing with B-splines and penalties, Statist. Sci., 11, 89, 10.1214/ss/1038425655
Gill, 1981, Practical Optimization
Gu, 1991, Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method, SIAM J. Sci. Statist. Comput., 12, 383, 10.1137/0912021
1993, Semiparametric analysis of variance with tensor product thin plate splines, J. R. Statist. Soc., 55, 353
Hastie, 1990, Generalized Additive Models
Hutchinson, 1985, Smoothing noisy data with spline functions, Numer. Math., 47, 99, 10.1007/BF01389878
Parker, 1985, Discussion on ‘Some aspects of the spline smoothing approach to non-parametric regression curve fitting, J. R. Statist. Soc., 47, 40
Silverman, 1985, Some aspects of the spline smoothing approach to non-parametric regression curve fitting (with discussion), J. R. Statist. Soc., 47, 1
Smola, 2000, Proc. 17th Int. Conf. Machine Learning
Wahba, 1980, Approximation Theory III, 905
1983, Bayesian ‘‘confidence intervals’’ for the cross-validated smoothing spline, J. R. Statist. Soc., 45, 133
1990, Spline models for observational data. CBMS–NSF Regl, Conf. Ser. Appl. Math., 59
Wahba, 1995, Smoothing spline ANOVA for exponential families, with application to the Wisconsin epidemiological study of diabetic retinopathy, Ann. Statist., 23, 1865, 10.1214/aos/1034713638
Watkins, 1991, Fundamentals of Matrix Computation
Williams, 2001, Using the Nyström method to speed up kernel machines, Advances in Neural Information Processing Systems
Wood, 2000, Modelling and smoothing parameter estimation with multiple quadratic penalties, J. R. Statist. Soc., 62, 413, 10.1111/1467-9868.00240