Smoothing splines with varying smoothing parameter

This paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain. Two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty. The penalty is a function of the design points in order to accommodate local behaviour of the regression function. We show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator, and that the equivalent kernel is spatially dependent. The equivalent kernels for traditional smoothing splines are a special case of this general solution. With the aid of the Green's function for a two-point boundary value problem, explicit forms of the asymptotic mean and variance are obtained for any interior point. Thus, the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error. Simulation results and an application illustrate the performance of the proposed estimator. Copyright 2013, Oxford University Press.

[1]  Vincent N. LaRiccia,et al.  Maximum Penalized Likelihood Estimation: Volume II Regression , 2011 .

[2]  G. Vachtsevanos,et al.  Epileptic Seizures May Begin Hours in Advance of Clinical Onset A Report of Five Patients , 2001, Neuron.

[3]  Robert Kohn,et al.  On the Smoothness Properties of the Best Linear Unbiased Estimate of a Stochastic Process Observed with Noise , 1983 .

[4]  David Ruppert,et al.  Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting , 2000 .

[5]  Young K. Truong,et al.  Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture , 1997 .

[6]  G. Wahba.  A Comparison of GCV and GML for Choosing the Smoothing Parameter in the Generalized Spline Smoothing Problem , 1985 .

[7]  G. Wahba,et al.  Some results on Tchebycheffian spline functions , 1971 .

[8]  David Ruppert,et al.  On the asymptotics of penalized spline smoothing , 2011 .

[9]  M. Nussbaum.  Spline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$ , 1985 .

[10]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[11]  R. Eubank.  Nonparametric Regression and Spline Smoothing , 1999 .

[12]  Christopher Holmes,et al.  Spatially adaptive smoothing splines , 2006 .

[13]  Curtis B. Storlie,et al.  A Locally Adaptive Penalty for Estimation of Functions With Varying Roughness , 2010 .

[14]  D. Ruppert,et al.  Local Asymptotics of P-Spline Smoothing , 2009, 0912.1824.

[15]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[16]  Peter Dalgaard,et al.  R Development Core Team (2010): R: A language and environment for statistical computing , 2010 .

[17]  David Ruppert,et al.  Local polynomial variance-function estimation , 1997 .

[18]  Sally Wood,et al.  Bayesian mixture of splines for spatially adaptive nonparametric regression , 2002 .

[19]  Lie Wang,et al.  Adaptive variance function estimation in heteroscedastic nonparametric regression , 2008, 0810.4780.

[20]  G. Wahba,et al.  Hybrid Adaptive Splines , 1997 .

[21]  J. Friedman,et al.  FLEXIBLE PARSIMONIOUS SMOOTHING AND ADDITIVE MODELING , 1989 .

[22]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[23]  A. Antoniadis,et al.  Variance Function Estimation in Regression by Wavelet Methods , 1995 .

[24]  Uniform error bounds for smoothing splines , 2006, math/0612776.

[25]  H. Müller,et al.  Variable Bandwidth Kernel Estimators of Regression Curves , 1987 .

[26]  M. Rosenblatt,et al.  Smoothing Splines: Regression, Derivatives and Deconvolution , 1983 .

[27]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[28]  Raymond J. Carroll,et al.  Variance Function Estimation in Regression: the Effect of Estimating the Mean , 1988 .

[29]  M. Hansen,et al.  Spline Adaptation in Extended Linear Models , 1998 .

[30]  Jianqing Fan,et al.  Efficient Estimation of Conditional Variance Functions in Stochastic Regression , 1998 .

[31]  I. Johnstone,et al.  Minimax estimation via wavelet shrinkage , 1998 .

[32]  Charles Kooperberg,et al.  Spline Adaptation in Extended Linear Models (with comments and a rejoinder by the authors , 2002 .

[33]  F. Abramovich,et al.  Derivation of equivalent kernel for general spline smoothing: a systematic approach , 1999 .

[34]  Karen Messer,et al.  A Comparison of a Spline Estimate to its Equivalent Kernel Estimate , 1991 .

[35]  R. Kohn,et al.  Nonparametric regression using Bayesian variable selection , 1996 .

[36]  R. Kass,et al.  Bayesian curve-fitting with free-knot splines , 2001 .

[37]  Douglas W. Nychka,et al.  Splines as Local Smoothers , 1995 .

[38]  Wensheng Guo,et al.  DATA DRIVEN ADAPTIVE SPLINE SMOOTHING , 2010 .

[39]  Linda H. Zhao,et al.  Free‐knot polynomial splines with confidence intervals , 2003 .

[40]  Chong Gu.  Smoothing Spline Anova Models , 2002 .

[41]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[42]  D. Cox.  Asymptotics for $M$-Type Smoothing Splines , 1983 .

[43]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[44]  Li Qin,et al.  A Time-Frequency Functional Model for Locally Stationary Time Series Data , 2009, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[45]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .