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.     

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.     

Fast covariance estimation for sparse functional data

Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline smoother that is designed for covariance smoothing and can be used for sparse functional or longitudinal data. We propose a fast algorithm for covariance smoothing using leave-one-subject-out cross-validation. Our simulations show that the proposed method compares favorably against several commonly used methods. The method is applied to a study of child growth led by one of coauthors and to a public dataset of longitudinal CD4 counts.     

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).     

Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.     

Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression

Cross-validation as a means of choosing the smoothing parameter in spline regression has achieved a wide popularity. Its appeal comprises of an automatic method based on an attractive criterion and along with many other methods it has been shown to minimize predictive mean square error asymptotically. However, in practice there may be a substantial proportion of applications where a cross-validation style choice may lead to drastic undersmoothing often as far as interpolation. Furthermore, because the criterion is so appealing the user may be misled by an inappropriate, automatically-chosen value. In this paper we investigate the nature of cross-validatory methods in spline smoothing regression and suggest variants which provide small sample protection against undersmoothing.     

Smoothing spline Gaussian regression: more scalable computation via efficient approximation

Summary.  Smoothing splines via the penalized least squares method provide versatile and effective nonparametric models for regression with Gaussian responses. The computation of smoothing splines is generally of the order O ( n ³), n being the sample size, which severely limits its practical applicability. We study more scalable computation of smoothing spline regression via certain low dimensional approximations that are asymptotically as efficient. A simple algorithm is presented and the Bayes model that is associated with the approximations is derived, with the latter guiding the porting of Bayesian confidence intervals. The practical choice of the dimension of the approximating space is determined through simulation studies, and empirical comparisons of the approximations with the exact solution are presented. Also evaluated is a simple modification of the generalized cross-validation method for smoothing parameter selection, which to a large extent fixes the occasional undersmoothing problem that is suffered by generalized cross-validation.     

A frequency domain comparison of estimation methods for gaussian fields

This paper demonstrates that well-known parameter estimation methods for Gaussian fields place different emphasis on the high and low frequency components of the data. As a consequence, the relative importance of the frequencies under the objective of the analysis should be taken into account when selecting an estimation method, in addition to other considerations such as statistical and computational efficiency. The paper also shows that when noise is added to the Gaussian field, maximum pseudolikelihood automatically sets the smoothing parameter of the model equal to one. A simulation study then indicates that generalised cross-validation is more robust than maximum likelihood un-

der model misspecification in smoothing and image restoration problems. This has implications for Bayesian procedures since these use the same weightings of the frequencies as the likelihood.     

Resistant selection of the smoothing parameter for smoothing splines

Robust automatic selection techniques for the smoothing parameter of a smoothing spline are introduced. They are based on a robust predictive error criterion and can be viewed as robust versions of C _p and cross-validation. They lead to smoothing splines which are stable and reliable in terms of mean squared error over a large spectrum of model distributions.     

A method for choosing the smoothing parameter in a semi-parametric model for detecting change-points in blood flow

In a smoothing spline model with unknown change-points, the choice of the smoothing parameter strongly influences the estimation of the change-point locations and the function at the change-points. In a tumor biology example, where change-points in blood flow in response to treatment were of interest, choosing the smoothing parameter based on minimizing generalized cross-validation (GCV) gave unsatisfactory estimates of the change-points. We propose a new method, aGCV, that re-weights the residual sum of squares and generalized degrees of freedom terms from GCV. The weight is chosen to maximize the decrease in the generalized degrees of freedom as a function of the weight value, while simultaneously minimizing aGCV as a function of the smoothing parameter and the change-points. Compared with GCV, simulation studies suggest that the aGCV method yields improved estimates of the change-point and the value of the function at the change-point.     

On algorithms for ordinary least squares regression spline fitting: A comparative study

Regression spline smoothing is a popular approach for conducting nonparametric regression. An important issue associated with it is the choice of a "theoretically best" set of knots. Different statistical model selection methods, such as Akaike's information criterion and generalized cross-validation, have been applied to derive different "theoretically best" sets of knots. Typically these best knot sets are defined implicitly as the optimizers of some objective functions. Hence another equally important issue concerning regression spline smoothing is how to optimize such objective functions. In this article different numerical algorithms that are designed for carrying out such optimization problems are compared by means of a simulation study. Both the univariate and bivariate smoothing settings will be considered. Based on the simulation results, recommendations for choosing a suitable optimization algorithm under various settings will be provided.     

Global optimization of the generalized cross-validation criterion

Generalized cross-validation is a method for choosing the smoothing parameter in smoothing splines and related regularization problems. This method requires the global minimization of the generalized cross-validation function. In this paper an algorithm based on interval analysis is presented to find the globally optimal value for the smoothing parameter, and a numerical example illustrates the performance of the algorithm.     

Estimating curves and derivatives with parametric penalized spline smoothing

Accurate estimation of an underlying function and its derivatives is one of the central problems in statistics. Parametric forms are often proposed based on the expert opinion or prior knowledge of the underlying function. However, these strict parametric assumptions may result in biased estimates when they are not completely accurate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smoothing methods. It also uses the prior knowledge of the underlying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accurate estimates of the function and its derivatives than the penalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data.     

Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

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 Simulation Study of Ridge Regression Estimators with Autocorrelated Errors

In this paper we run a large number of simulations to study the effects of collinearity and autocorrelated disturbances in the performance of several Ridge Regression estimators. The results suggest that with a fair amount of multicollinearity and of autocorrelation the Ridge Regression estimators which take the autocorrelation into account can perform better than the other methods. Also if the error term is only moderately autocorrelated; then the performance of the Ridge Regression estimators built upon ignoring the autocorrelation can outperform the other estimators.     

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.     

Statistical Monitoring of Autocorrelated Simple Linear Profiles Based on Principal Components Analysis

In this article, a transformation method using the principal component analysis approach is first applied to remove the existing autocorrelation within each profile in Phase I monitoring of autocorrelated simple linear profiles. This easy-to-use approach is independent of the autocorrelation coefficient. Moreover, since it is a model-free method, it can be used for Phase I monitoring procedures. Then, five control schemes are proposed to monitor the parameters of the profile with uncorrelated error terms. The performances of the proposed control charts are evaluated and are compared through simulation experiments based on different values of autocorrelation coefficient as well as different shift scenarios in the parameters of the profile in terms of probability of receiving an out-of-control signal.     

Monitoring the mean of autocorrelated observations with one generally weighted moving average control chart

Traditionally, using a control chart to monitor a process assumes that process observations are normally and independently distributed. In fact, for many processes, products are either connected or autocorrelated and, consequently, obtained observations are autocorrelative rather than independent. In this scenario, applying an independence assumption instead of autocorrelation for process monitoring is unsuitable. This study examines a generally weighted moving average (GWMA) with a time-varying control chart for monitoring the mean of a process based on autocorrelated observations from a first-order autoregressive process (AR(1)) with random error. Simulation is utilized to evaluate the average run length (ARL) of exponentially weighted moving average (EWMA) and GWMA control charts. Numerous comparisons of ARLs indicate that the GWMA control chart requires less time to detect various shifts at low levels of autocorrelation than those at high levels of autocorrelation. The GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in a process mean.