Penalized Spline Estimation for Varying-Coefficient Models

Varying-coefficient models are useful extensions of classical linear models. They arise from multivariate nonparametric regression, nonlinear time series modeling and forecasting, longitudinal data analysis, and others. This article proposes the penalized spline estimation for the varying-coefficient models. Assuming a fixed but potentially large number of knots, the penalized spline estimators are shown to be strong consistency and asymptotic normality. A systematic optimization algorithm for the selection of multiple smoothing parameters is developed. One of the advantages of the penalized spline estimation is that it can accommodate varying degrees of smoothness among coefficient functions due to multiple smoothing parameters being used. Some simulation studies are presented to illustrate the proposed methods.     

Spatially-adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the p th derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.     

Modified spline regression based on randomly right-censored data: A comparative study

ABSTRACT

In this paper, we propose modified spline estimators for nonparametric regression models with right-censored data, especially when the censored response observations are converted to synthetic data. Efficient implementation of these estimators depends on the set of knot points and an appropriate smoothing parameter. We use three algorithms, the default selection method (DSM), myopic algorithm (MA), and full search algorithm (FSA), to select the optimum set of knots in a penalized spline method based on a smoothing parameter, which is chosen based on different criteria, including the improved version of the Akaike information criterion (AICc), generalized cross validation (GCV), restricted maximum likelihood (REML), and Bayesian information criterion (BIC). We also consider the smoothing spline (SS), which uses all the data points as knots. The main goal of this study is to compare the performance of the algorithm and criteria combinations in the suggested penalized spline fits under censored data. A Monte Carlo simulation study is performed and a real data example is presented to illustrate the ideas in the paper. The results confirm that the FSA slightly outperforms the other methods, especially for high censoring levels.     

Free-knot polynomial splines with confidence intervals

Summary.  We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free-knot locations. The number of knots is determined by generalized cross-validation. The estimates of knot locations and coefficients are obtained through a non-linear least squares solution that corresponds to the maximum likelihood estimate. Confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator. Average coverage probabilities and the accuracy of the estimate are examined via simulation. This includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a Bayesian version of the variable knots method. Simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions. It also performs well for fairly small sample sizes.     

A non-iterative optimization method for smoothness in penalized spline regression

Typically, an optimal smoothing parameter in a penalized spline regression is determined by minimizing an information criterion, such as one of the C _p , CV and GCV criteria. Since an explicit solution to the minimization problem for an information criterion cannot be obtained, it is necessary to carry out an iterative procedure to search for the optimal smoothing parameter. In order to avoid such extra calculation, a non-iterative optimization method for smoothness in penalized spline regression is proposed using the formulation of generalized ridge regression. By conducting numerical simulations, we verify that our method has better performance than other methods which optimize the number of basis functions and the single smoothing parameter by means of the CV or GCV criteria.     

A model-averaging approach for smoothing spline regression

ABSTRACT

This article considers nonparametric regression problems and develops a model-averaging procedure for smoothing spline regression problems. Unlike most smoothing parameter selection studies determining an optimum smoothing parameter, our focus here is on the prediction accuracy for the true conditional mean of Y given a predictor X. Our method consists of two steps. The first step is to construct a class of smoothing spline regression models based on nonparametric bootstrap samples, each with an appropriate smoothing parameter. The second step is to average bootstrap smoothing spline estimates of different smoothness to form a final improved estimate. To minimize the prediction error, we estimate the model weights using a delete-one-out cross-validation procedure. A simulation study has been performed by using a program written in R. The simulation study provides a comparison of the most well known cross-validation (CV), generalized cross-validation (GCV), and the proposed method. This new method is straightforward to implement, and gives reliable performances in simulations.     

Nonparametric density estimation for censored survival data: Regression-spline approach

A method for nonparametric estimation of density based on a randomly censored sample is presented. The density is expressed as a linear combination of cubic M -splines, and the coefficients are determined by pseudo-maximum-likelihood estimation (likelihood is maximized conditionally on data-dependent knots). By using regression splines (small number of knots) it is possible to reduce the estimation problem to a space of low dimension while preserving flexibility, thus striking a compromise between parametric approaches and ordinary nonparametric approaches based on spline smoothing. The number of knots is determined by the minimum AIC. Examples of simulated and real data are presented. Asymptotic theory and the bootstrap indicate that the precision and the accuracy of the estimates are satisfactory.     

Thin plate regression splines

Summary. 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.     

ON CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES

Generalized additive models represented using low rank penalized regression splines, estimated by penalized likelihood maximisation and with smoothness selected by generalized cross validation or similar criteria, provide a computationally efficient general framework for practical smooth modelling. Various authors have proposed approximate Bayesian interval estimates for such models, based on extensions of the work of Wahba, G. (1983) [Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B 45 , 133–150] and Silverman, B.W. (1985) [Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B 47 , 1–52] on smoothing spline models of Gaussian data, but testing of such intervals has been rather limited and there is little supporting theory for the approximations used in the generalized case. This paper aims to improve this situation by providing simulation tests and obtaining asymptotic results supporting the approximations employed for the generalized case. The simulation results suggest that while across‐the‐model performance is good, component‐wise coverage probabilities are not as reliable. Since this is likely to result from the neglect of smoothing parameter variability, a simple and efficient simulation method is proposed to account for smoothing parameter uncertainty: this is demonstrated to substantially improve the performance of component‐wise intervals.     

P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data

The P-splines of Eilers and Marx (Stat Sci 11:89–121, 1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as reduced rank smoothers: (i) the basis and the penalty are sparse, enabling efficient computation, especially for Bayesian stochastic simulation; (ii) it is possible to flexibly ‘mix-and-match’ the order of B-spline basis and penalty, rather than the order of penalty controlling the order of the basis as in spline smoothing; (iii) it is very easy to set up the B-spline basis functions and penalties. The discrete penalties are somewhat less interpretable in terms of function shape than the traditional derivative based spline penalties, but tend towards penalties proportional to traditional spline penalties in the limit of large basis size. However part of the point of P-splines is not to use a large basis size. In addition the spline basis functions arise from solving functional optimization problems involving derivative based penalties, so moving to discrete penalties for smoothing may not always be desirable. The purpose of this note is to point out that the three properties of basis-penalty sparsity, mix-and-match penalization and ease of setup are readily obtainable with B-splines subject to derivative based penalization. The penalty setup typically requires a few lines of code, rather than the two lines typically required for P-splines, but this one off disadvantage seems to be the only one associated with using derivative based penalties. As an example application, it is shown how basis-penalty sparsity enables efficient computation with tensor product smoothers of scattered data.     

Comparison of spline estimator at various levels of autocorrelation in smoothing spline non parametric regression for longitudinal data

The purpose of this research are: (1) to obtain spline function estimation in non parametric regression for longitudinal data with and without considering the autocorrelation between data of observation within subject, (2) to develop the algorithm that generates simulation data with certain autocorrelation level based on size of sample (N) and error variance (EV), and (3) to establish shape of spline estimator in non parametric regression for longitudinal data to simulation with various level of autocorrelation, as well as compare DM and TM approaches in predicting spline estimator in the data simulation with different of autocorrelation observational data on within subject. The results of the application are as follows: (a) implementation of smoothing spline with penalized weighted least square (PWLS) approach with or without consideration of autocorrelation in general (in all sizes and all error variances levels) provides significantly different spline estimator when the autocorrelation level >0.8; (b) based on size comparison, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall size, moderate and large sample size), and > 0.7 (in small sample size); (c) based on level of variance, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall level of variance, moderate and large variance), and > 0.7 (in small variance).     

Penalized regression with model‐based penalties

Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function ∫ μ based on observations ti, Yi is the minimizer of Σ{Yi ‐ μ(ti)}² + λ∫(μ′′)². Since ∫(μ″)² is zero when μ is a line, the cubic smoothing spline estimate favors the parametric model μ(t) = α_o + α₁t. Here the authors consider replacing ∫(μ″)² with the more general expression ∫(Lμ)² where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if Lμ is small. They present an O(n) algorithm for the computation of μ. This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study their estimates via simulation and apply them to several data sets.     

Period analysis of variable stars by robust smoothing

Summary.  The objective is to estimate the period and the light curve (or periodic function) of a variable star. Previously, several methods have been proposed to estimate the period of a variable star, but they are inaccurate especially when a data set contains outliers. We use a smoothing spline regression to estimate the light curve given a period and then find the period which minimizes the generalized cross-validation (GCV). The GCV method works well, matching an intensive visual examination of a few hundred stars, but the GCV score is still sensitive to outliers. Handling outliers in an automatic way is important when this method is applied in a 'data mining' context to a vary large star survey. Therefore, we suggest a robust method which minimizes a robust cross-validation criterion induced by a robust smoothing spline regression. Once the period has been determined, a nonparametric method is used to estimate the light curve. A real example and a simulation study suggest that the robust cross-validation and GCV methods are superior to existing methods.     

Function-on-function regression for two-dimensional functional data

ABSTRACT

We present methods for modeling and estimation of a concurrent functional regression when the predictors and responses are two-dimensional functional datasets. The implementations use spline basis functions and model fitting is based on smoothing penalties and mixed model estimation. The proposed methods are implemented in available statistical software, allow the construction of confidence intervals for the bivariate model parameters, and can be applied to completely or sparsely sampled responses. Methods are tested to data in simulations and they show favorable results in practice. The usefulness of the methods is illustrated in an application to environmental data.     

Penalized Spline Varying-Coefficient Single-Index Model

In this article, the varying-coefficient single-index model (VCSIM) is discussed based on penalized spline estimation method. All the coefficient functions are fitted by P-spline and all parameters in P-spline varying-coefficient model can be estimated simultaneously by penalized nonlinear least squares. The detailed algorithm is given, including choosing smoothing parameters and knots. The approach is rapid and computationally stable. √n consistency and asymptotic normality of the estimators of all the parameters are showed. Both simulated and real data examples are given to illustrate the proposed estimation methodology.     

Difference-based estimation and model identification for panel data semiparametric models with cross-section dependence

Abstract

In this article, we consider a panel data partially linear regression model with fixed effect and non parametric time trend function. The data can be dependent cross individuals through linear regressor and error components. Unlike the methods using non parametric smoothing technique, a difference-based method is proposed to estimate linear regression coefficients of the model to avoid bandwidth selection. Here the difference technique is employed to eliminate the non parametric function effect, not the fixed effects, on linear regressor coefficient estimation totally. Therefore, a more efficient estimator for parametric part is anticipated, which is shown to be true by the simulation results. For the non parametric component, the polynomial spline technique is implemented. The asymptotic properties of estimators for parametric and non parametric parts are presented. We also show how to select informative ones from a number of covariates in the linear part by using smoothly clipped absolute deviation-penalized estimators on a difference-based least-squares objective function, and the resulting estimators perform asymptotically as well as the oracle procedure in terms of selecting the correct model.     

Using recursive algorithms for the efficient identification of smoothing spline ANOVA models

In this paper we present a unified discussion of different approaches to the identification of smoothing spline analysis of variance (ANOVA) models: (i) the “classical” approach (in the line of Wahba in Spline Models for Observational Data, 1990; Gu in Smoothing Spline ANOVA Models, 2002; Storlie et al. in Stat. Sin., 2011) and (ii) the State-Dependent Regression (SDR) approach of Young in Nonlinear Dynamics and Statistics (2001). The latter is a nonparametric approach which is very similar to smoothing splines and kernel regression methods, but based on recursive filtering and smoothing estimation (the Kalman filter combined with fixed interval smoothing). We will show that SDR can be effectively combined with the “classical” approach to obtain a more accurate and efficient estimation of smoothing spline ANOVA models to be applied for emulation purposes. We will also show that such an approach can compare favorably with kriging.     

Variance Reduction in Smoothing Splines

Abstract.  We develop a variance reduction method for smoothing splines. For a given point of estimation, we define a variance-reduced spline estimate as a linear combination of classical spline estimates at three nearby points. We first develop a variance reduction method for spline estimators in univariate regression models. We then develop an analogous variance reduction method for spline estimators in clustered/longitudinal models. Simulation studies are performed which demonstrate the efficacy of our variance reduction methods in finite sample settings. Finally, a real data analysis with the motorcycle data set is performed. Here we consider variance estimation and generate 95% pointwise confidence intervals for the unknown regression function.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.