Choice Of Smoothing Parameter Research Articles

Restoration of the Product Consumption Rate with Integral Cubic Smoothing Spline, Study of the Best Smoothing Parameter Choice

Restoration of the Product Consumption Rate with Integral Cubic Smoothing Spline, Study of the Best Smoothing Parameter Choice

Variable selection for ultra-high dimensional quantile regression with missing data and measurement error.

In this paper, we consider variable selection for ultra-high dimensional quantile regression model with missing data and measurement errors in covariates. Specifically, we correct the bias in the loss function caused by measurement error by applying the orthogonal quantile regression approach and remove the bias caused by missing data using the inverse probability weighting. A nonconvex Atan penalized estimation method is proposed for simultaneous variable selection and estimation. With the proper choice of the regularization parameter and under some relaxed conditions, we show that the proposed estimate enjoys the oracle properties. The choice of smoothing parameters is also discussed. The performance of the proposed variable selection procedure is assessed by Monte Carlo simulation studies. We further demonstrate the proposed procedure with a breast cancer data set.

Posterior contraction and credible sets for filaments of regression functions

A filament consists of local maximizers of a smooth function $f$ when moving in a certain direction. A filamentary structure is an important feature of the shape of an object and is also considered as an important lower dimensional characterization of multivariate data. There have been some recent theoretical studies of filaments in the nonparametric kernel density estimation context. This paper supplements the current literature in two ways. First, we provide a Bayesian approach to the filament estimation in regression context and study the posterior contraction rates using a finite random series of B-splines basis. Compared with the kernel-estimation method, this has a theoretical advantage as the bias can be better controlled when the function is smoother, which allows obtaining better rates. Assuming that $f:\mathbb{R}^{2}\mapsto \mathbb{R}$ belongs to an isotropic Hölder class of order $\alpha \geq 4$, with the optimal choice of smoothing parameters, the posterior contraction rates for the filament points on some appropriately defined integral curves and for the Hausdorff distance of the filament are both $(n/\log n)^{(2-\alpha )/(2(1+\alpha ))}$. Secondly, we provide a way to construct a credible set with sufficient frequentist coverage for the filaments. We demonstrate the success of our proposed method in simulations and one application to earthquake data.

A New Smoothing Method for Time Series Data in the Presence of Autocorrelated Error

Spline Smoothing is used to filter out noise or disturbance in an observation, its performance depends on the choice of smoothing parameters. There are many methods of estimating smoothing parameters; most popular among them are; Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR), this methods tend to overfit smoothing parameters in the presence of autocorrelation error. A new Spline Smoothing estimation method is proposed and compare with three existing methods in order to eliminate the problem of over fitting associated with the presence of Autocorrelation in the error term. It is demonstrated through a simulation study performed by using a program written in R based on the predictive Mean Score Error criteria. The result indicated that the predictive mean square error (PMSE) of the four smoothing methods decreases as the smoothing parameters increases and decreases as the sample sizes increases. This study discovered that the proposed smoothing method is the best for time series observations with Autocorrelated error because it doesn’t over fit and works well for large sample sizes. This study will help researchers overcome the problem of over fitting associated with applying Smoothing spline method time series observation.

Variable selection for high dimensional partially linear varying coefficient errors-in-variables models

In this paper, we consider variable selection procedure for the high dimensional partially linear varying coefficient models where the parametric part covariates are measured with additive errors. The penalized bias-corrected profile least squares estimators are conducted, and their asymptotic properties are also studied under some regularity conditions. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle property. Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies.

Intermediate shapes for incremental sheet forming

Incremental sheet forming has limitations such as when forming a steep wall angle with a small radius tool. These limits need to be anticipated by using a multistage strategy with intermediate shapes, that avoids excessive strains. In designing methods to smooth a formed shape to achieve an intermediate shape, work on digital image processing is informative. In particular, K. Crane [6] and others have used solution of Poisson’s diffusion equation to obtain a smoothing of a surface. An alternative approach labelled false elasticity smoothing to estimate intermediate shapes is presented here. Two parameters control smoothing of the final shape: one to scale back curvatures and one to scale back membrane strains. Membrane strains are estimated to give nodal forces. The curvature tensor at any element centre is estimated from the final shape to predict nodal moments. Scaling back moments will reduce local curvature, and scaling back forces will reduce the membrane strains. An elastic finite element solution predicts an intermediate shape, typically giving priority to reducing the local peak curvatures. This can be repeated until the sheet is flattened. The application of this approach to trial geometries is presented, and the choice of smoothing parameters discussed.

Robustness and sensitivity of a blueprint for on-farm estimation of dairy cow energy balance

Excessive negative energy balance (EB) has been associated with decreased reproductive performance and increased risk of lameness and metabolic diseases. On-farm, automated EB estimates for individual cows would enable dairy farmers to detect excessive negative EB early and act to minimize its extent and duration by altering feeding. Previously, we have shown that EB can be estimated from frequent measurements of body weight (BW) and body condition score (BCS) changes, referred to as EBbody. In this study, we investigated the robustness and sensitivity of the EBbody method to assess its genericity and on-farm applicability. We used 5 data sets with BW of lactating cows (name of data set in parenthesis): 65 Holstein cows in a French feeding trial (INRA); 6 Holstein cows in a British feeding trial (Friggens); 31 Holstein cows and 17 Jersey cows in a Danish feeding trial (DCRC); 140 Holstein cows in a British feeding trial (Scotland's Rural College, SRUC); and 1,592 Holstein cows on 9 Danish farms with milking robots (automatic milking system). We used the INRA and Friggens data sets to develop a dynamic formula to correct BW for increasing residual gut-fill (RGF) during early lactation. With the DCRC data, we tested the effect of smoothing parameters and weighing frequency on EBbody. Also, 2 robustness tests were performed using the SRUC data to test the effect of diet change on BW and the automatic milking system data to test the effect of farm on BW variation. Finally, we combined the results into a blueprint describing different ways to calculate EBbody depending on the purpose and on the availability of BCS. The dynamic RGF adjustment resulted in a lower empty BW during early lactation than that obtained with the previously used constant RGF. The double-exponential smoothing method used to correct for meal-related gut-fill was robust to choice of smoothing parameters. Cows should be weighed at least once every 4 d during early lactation to capture the duration of negative EBbody. Our EBbody method proved robust to diet changes. Finally, although cow BW varied significantly between farms, the quantile regression smoothing of BW did not bias the estimation of weight differences between herds. In conclusion, these results validate the applicability of the EBbody method to estimate EB across a range of farm conditions, and we provided a blueprint that enables the estimation of EBbody for individual cows on-farm using only frequent BW, in combination with BCS when available.

Bias correction in estimation of public health risk attributable to short‐term air pollution exposure

Numerous epidemiologic studies have reported associations between short‐term air pollution exposure and mortality. Such short‐term risk models include smooth functions of time to control for unmeasured confounding variables. We demonstrate bias in these short‐term Generalized Additive Model estimates because of lack of accounting for long timescale variations and propose a family of improved time smoothers to reduce and control the bias. The strengths of the proposed smoother are twofold: a clear separating of short‐term and long‐term effects and an obvious choice of smoothing parameters from pre‐determined timescales of interest. We demonstrate improvements through simulations and analysis of examples of air pollution and mortality data from Chicago, Il. from the National Morbidity, Mortality and Air Pollution Study database, showing reduced bias in the risk estimates. © 2015 The Authors. Environmetrics Published by John Wiley & Sons Ltd.

Nonnegative bias reduction methods for density estimation using asymmetric kernels

Two classes of multiplicative bias correction (“MBC”) methods are applied to density estimation with support on [0,∞). It is demonstrated that under sufficient smoothness of the true density, each MBC technique reduces the order of magnitude in bias, whereas the order of magnitude in variance remains unchanged. Accordingly, the mean integrated squared error of each MBC estimator achieves a faster convergence rate of O(n−8/9) when best implemented, where n is the sample size. Furthermore, MBC estimators always generate nonnegative estimates by construction. Plug-in smoothing parameter choice rules for the estimators are proposed, and their finite sample performance is examined via Monte Carlo simulations.

New inference methods for quantile regression based on resampling

Summary Inference for quantile regression parameters presents two problems. First, it is computationally costly because estimation requires optimising a non-differentiable objective function which is a formidable numerical task, specially with many number of observations and regressors. Second, it is controversial because standard asymptotic inference requires the choice of smoothing parameters and different choices may lead to different conclusions. Bootstrap methods solve the latter problem at the price of enlarging the former. We give a theoretical justification for a new inference method consisting of the construction of asymptotic pivots based on a small number of bootstrap replications. The procedure still avoids smoothing and reduces usual bootstrap methods' computational cost. We show its usefulness to draw inferences on linear or non-linear functions of the parameters of quantile regression models. © 2013 Royal Economic Society.

Combining Strategies for the Estimation of Treatment Effects

The estimation of the average effect of a program or treatment on a variable of interest is an important tool for the assessment of economic policies. In general, assignment of potential participants to treatment does not occur at random and could thus generate a selection bias in absence of some correction. A way to get around this problem is by assuming that the econometrician observes a set of determinant characteristics of participation up to a strictly random component. Under such an assumption, the literature contains semiparametric estimators of the average treatment effect that are consistent and can asymptotically reach the semiparametric efficiency bound. However, in frequently available samples, the performance of these methods is not always satisfactory.The aim of this paper is to investigate how the combination of two strategies may generate estimators with better properties in small samples. Therefore, we consider two ways of combining these approaches, based on the double robustness literature developed by James Robins et al. We analyze the properties of these combined estimators and discuss why they can outperform the separate use of each method. Finally, using a Monte Carlo simulation, we compare the performance of these estimators with that of the imputation and reweighting techniques. Our results show that the combination of strategies can reduce bias and variance, but this improvement depends on adequate implementation. We conclude that the choice of smoothing parameters is decisive for the performance of estimators in medium-sized samples.

Misspecification Testing in a Class of Conditional Distributional Models

We propose a specification test for a wide range of parametric models for the conditional distribution function of an outcome variable given a vector of covariates. The test is based on the Cramer–von Mises distance between an unrestricted estimate of the joint distribution function of the data and a restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has nontrivial power against local deviations of order n − 1/2 from the null hypothesis, and does not require the choice of smoothing parameters. In an empirical application, we use our test to study the validity of various models for the conditional distribution of wages in the United States.

Misspecification Testing in a Class of Conditional Distributional Models

We propose a specification test for a wide range of parametric models for the conditional distribution function of an outcome variable given a vector of covariates. The test is based on the Cramer-von Mises distance between an unrestricted estimate of the joint distribution function of the data, and a restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has non-trivial power against local deviations of order n^-1/2 from the null hypothesis, and does not require the choice of smoothing parameters. In an empirical application, we use our test to study the validity of various models for the conditional distribution of wages in the US.

Functional Singular Component Analysis

Summary Aiming at quantifying the dependence of pairs of functional data (X,Y), we develop the concept of functional singular value decomposition for covariance and functional singular component analysis, building on the concept of ‘canonical expansion’ of compact operators in functional analysis. We demonstrate the estimation of the resulting singular values, functions and components for the practically relevant case of sparse and noise-contaminated longitudinal data and provide asymptotic consistency results. Expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes. A natural application of the functional singular value decomposition is a measure of functional correlation. Owing to the involvement of an inverse operation, most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters. These problems are exacerbated for the case of sparse longitudinal data, on which we focus. The functional correlation measure that is derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error, as we demonstrate in a simulation study. Practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and on-line auctions.

Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters

This paper considers the problem of estimating curve and surface functions when the structures of an unknown function vary spatially. Classical approaches such as using smoothing splines, which are controlled by a single smoothing parameter, are inefficient in estimating the underlying function that consists of different spatial structures. In this paper, we propose a blockwise method of fitting smoothing splines wherein the smoothing parameter λ varies spatially, in order to accommodate possible spatial nonhomogeneity of the regression function. A key feature of the proposed blockwise method is the parameterization of a smoothing parameter function λ ( x ) that produces a continuous spatially adaptive fit over the entire range of design points. The proposed parameterization requires two important ingredients: (1) a blocking scheme that divides the data into several blocks according to the degree of spatial variation of the data; and (2) a method for choosing smoothing parameters of blocks. We propose a block selection approach that is based on the adaptive thinning algorithm and a choice of smoothing parameters that minimize a newly defined blockwise risk. The results obtained from numerical experiments validate the effectiveness of the proposed method.

MO-EE-A4-06: Optimize the Smoothing Parameter in Penalized Weighted Least-Squares Algorithm for Noise Reduction of Low-Dose CBCT

Purpose: A statistical projection restoration algorithm based on the PWLS can substantially improve the image quality of low‐dose CBCTimages. The performance of PWLS is largely determined by the choice of smoothing parameters. In this work, we developed a technique to obtain the optimal parameter in PWLS algorithm for the noise suppression in low‐dose CBCT.Methods and Materials: In radiotherapy, daily CBCT is acquired for the same patient during a treatment course. In this work, we acquired the CBCT with a high mAs protocol for the first session and then lower mAs protocol for the subsequent sessions. The high‐mAs projections served as the goal (ideal) toward which the low‐mAs projections need be smoothed by minimizing the PWLS objective function. The optimal smoothing parameter was inversely calculated by plugging both the high and low mAs projections into the derivative of the objective function. Then the obtained smoothing parameter can be used for PWLS to eliminate the noise in low‐dose projections. An anthropomorphic head phantom was used to investigate the effectiveness of the proposed strategy. Results: The optimal smoothing parameter in PWLS was obtained for each projection in a CBCT set. The noise in the low‐dose CBCTimagesreconstructed by the FDK algorithm using the PWLS smoothed projections was greatly suppressed. Image quality in PWLS‐processed low‐dose CBCT is comparable to its corresponding high‐dose CBCT.Conclusion: A novel strategy was proposed to estimate the optimal smoothing parameter for the PWLS algorithm. The proposed strategy can be extended to any Bayesian‐based image restoration algorithms that require pre‐defined smoothing parameters.

Nonparametric Estimation of Risk-Neutral Densities

This chapter deals with nonparametric estimation of the risk neutral density. We present three different approaches which do not require parametric functional assumptions on the underlying asset price dynamics nor on the distributional form of the risk neutral density. The first estimator is a kernel smoother of the second derivative of call prices, while the second procedure applies kernel type smoothing in the implied volatility domain. In the conceptually different third approach we assume the existence of a stochastic discount factor (pricing kernel) which establishes the risk neutral density conditional on the physical measure of the underlying asset. Via direct series type estimation of the pricing kernel we can derive an estimate of the risk neutral density by solving a constrained optimization problem. The methods are compared using European call option prices. The focus of the presentation is on practical aspects such as appropriate choice of smoothing parameters in order to facilitate the application of the techniques.

Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval

This paper demonstrates that two classes of multiplicative bias correction (MBC) techniques, originally proposed for density estimation using symmetric second-order kernels by Terrell and Scott (1980) and Jones et al. (1995), can be applied to density estimation using the beta and modified beta kernels. It is shown that, under sufficient smoothness of the true density, both MBC techniques reduce the order of magnitude in bias, whereas the order of magnitude in variance remains unchanged. Accordingly, mean squared errors of these MBC estimators achieve a faster convergence rate of O ( n − 8 / 9 ) for the interior part, when best implemented. Furthermore, the estimators always generate nonnegative density estimates by construction. To implement the MBC estimators, a plug-in smoothing parameter choice method is proposed. Monte Carlo simulations indicate good finite sample performance of the estimators.

Semiparametric penalized generalized additive models for environmental research and epidemiology

AbstractFor more than a decade generalized additive models (GAMs) have been successfully applied in various environmental studies, for instance to evaluate the impact of air pollution on health. The air pollution measure is usually connected with the health indicator in a parametric fashion whilst the effects of other covariates are modelled through nonparametric smooth functions. This is the motivation for the widely used semiparametric GAMs. The backfitting‐GAM methodology, and its popular implementation in S‐Plus, constitutes the standard approach. Here we consider its limitations and offer an alternative penalized likelihood concept. The primary limitations are the lack of tools for multiple data‐driven smoothing parameter choice, slow convergence of the iterative backfitting algorithm when concurvity is present, and unstable biased estimates of the regression coefficients and their standard errors in the semiparametric case, even more pronounced under concurvity. The penalized likelihood methodology when combined with cubic spline smoothers allows for a computationally efficient and complete parametric representation of a GAM, either nonparametric or semiparametric. It helps circumvent most of the mentioned GAM flaws in environmental research and epidemiology (e.g. in studies of human exposure to particulate matter). Finally, we discuss various evidence from simulation experiments in the literature, concerning the proper use of the GAM methodology. Copyright © 2009 John Wiley & Sons, Ltd.

Variable Selection for Partially Linear Models With Measurement Errors

This article focuses on variable selection for partially linear models when the covariates are measured with additive errors. We propose two classes of variable selection procedures, penalized least squares and penalized quantile regression, using the nonconvex penalized principle. The first procedure corrects the bias in the loss function caused by the measurement error by applying the so-called correction-for-attenuation approach, whereas the second procedure corrects the bias by using orthogonal regression. The sampling properties for the two procedures are investigated. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle procedure as proposed by Fan and Li. Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application.