Generalized Nonparametric Mixed Effects Models

Generalized linear mixed effects models (GLMM) provide useful tools for correlated and/or over-dispersed non-Gaussian data. This article considers generalized nonparametric mixed effects models (GNMM), which relax the rigid linear assumption on the conditional predictor in a GLMM. We use smoothing splines to model fixed effects. The random effects are general and may also contain stochastic processes corresponding to smoothing splines. We show how to construct smoothing spline ANOVA (SS ANOVA) decompositions for the predictor function. Components in a SS ANOVA decomposition have nice interpretations as main effects and interactions. Experimental design considerations help determine which components are fixed or random. We estimate all parameters and spline functions using stochastic approximation with Markov chain Monte Carlo (MCMC). As iteration increases we increase the MCMC sample size and decrease the step-size of the parameter update. This approach guarantees convergence of the estimates to the expected fixed points. We evaluate our methods through a simulation study.     

变系数混合模型的光滑样条推断

为了拟合纵向数据和其他相关数据,本文提出了变系数混合效应模型(VCMM).该模型运用变系数线性部分来表示协变量对响应变量的影响,而用随机效应来描述纵向数据组内的相关性, 因此,该模型允许协变量和响应变量之间存在十分灵活的泛函关系.文中运用光滑样条来估计均值部分的系数函数,而用限制最大似然的方法同时估计出光滑参数和方差成分,我们还得到了所提估计的计算方法.大量的模拟研究表明对于具有各种协方差结构的变系数混合效应模型,运用本文所提出的方法都能够十分有效地估计出模型中的系数函数和方差成分.     

Spline Smoothing on Surfaces

This article presents a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a finite element method. The article has a substantial tutorial component: we start by reviewing smoothness measures for functions defined on surfaces, simplicial surfaces and differentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, we show results of an experiment comparing finite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an effective way of determining the optimal degree of smoothing for function estimation on surfaces.     

Sizer for smoothing splines

Smoothing splines are an attractive method for scatterplot smoothing. The SiZer approach to statistical inference is adapted to this smoothing method, named SiZerSS. This allows quick and sure inference as to “which features in the smooth are really there” as opposed to “which are due to sampling artifacts”, when using smoothing splines for data analysis. Applications of SiZerSS to mode, linearity, quadraticity and monotonicity tests are illustrated using a real data example. Some small scale simulations are presented to demonstrate that the SiZerSS and the SiZerLL (the original local linear version of SiZer) often give similar performance in exploring data structure but they can not replace each other completely. Marron’s research was supported by the Dept. of Stat. and Appl. Prob., National Univ. of Singapore, and by the National Science Foundation Grant DMS-9971649. Zhang’s research was supported by the National Univ. of Singapore Academic Research grant R-155-000-023-112. The Editor, the Associate Editor, and the referees are appreciated for their invaluable comments and suggestions that help improve the article significantly.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

用广义交互确认方法选择良好参数进行多元多项式散乱数据自然样条光顺

1．简介多元样条函数在多元逼近中发挥很大作用，已有数量相当多的综合报告和研究论文正式发表，就在1996年6月在法国召开的第三届国际曲线与曲面会议上便有不少多元样条方面的报告，不过总的感觉是仍然缺乏对噪声数据特别是散乱数据的有效光顺方法．李岳生、崔锦泰、关履泰、胡日章等讨论广义调配样条与张量积函数，并用希氏空间样条方法处理多元散乱数据样条插值与光顺，提出多元多项式自然样条，推广了相应一元的结果．我们知道，在样条光顺中有一个如何选择参数的问题，用广义交互确认方法（generalizedcross－validation，以下简称GC…     

散乱数据曲面拟合的局部加权最小二乘插值方法及权函数的选择讨论

本文首先用局部加权最小二乘法将三维空间内任意散乱数据点集均匀,再估计出立方体网格点上的偏导数值及混合偏导数值,最后仅用网格点数据进行快速光滑插值加密计算,从而可得到任意点处的函数值。通过对已知函数的随机数据点集进行计算,取得了令人满意的效果。同时,在最小二乘逼近过程中,本文提供了一种权函数,并与其它二种权函数进行分析比较,给出了各种情况下的误差。     

On value efficiency

Value efficiency analysis (VEA) is a data envelopment analysis (DEA)-based approach that measures the efficiency of decision-making units, incorporating the decision maker's preferences explicitly. If the functional form of the value function is assumed pseudoconcave, then a tangent cone where the most preferred point is its vertex can be used to recognize value inefficient points. In this article, we consider the value function assumptions and show that the pseudoconcavity assumption can be extended to also cover nonsmooth value functions in VEA using a known class of limiting subdifferentials in nonsmooth analysis. The focus of this article is to show that the tangent cone approach to recognize value inefficient points is a valid approach even if the value function assumptions are more general than pseudoconcavity. Moreover, we show that continuity is needed in the case of true value efficiency.     

The bidimensional interproximation problem and mixed splines

Interproximation methods for surfaces can be used to construct a smooth surface interpolating some data points and passing through specified regions. In this paper we study the use of mixed splines, that is smoothing splines with additional interpolation constraints, to solve the interproximation problem for surfaces in the case of scattered data. The solution is obtained by solving a linear system whose structure can be improved by using “bell-shaped” thin plate splines.     

Restricted Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models

The goal of our article is to provide a transparent, robust, and computationally feasible statistical platform for restricted likelihood ratio testing (RLRT) for zero variance components in linear mixed models. This problem is nonstandard because under the null hypothesis the parameter is on the boundary of the parameter space. Our proposed approach is different from the asymptotic results of Stram and Lee who assumed that the outcome vector can be partitioned into many independent subvectors. Thus, our methodology applies to a wider class of mixed models, which includes models with a moderate number of clusters or nonparametric smoothing components. We propose two approximations to the finite sample null distribution of the RLRT statistic. Both approximations converge weakly to the asymptotic distribution obtained by Stram and Lee when their assumptions hold. When their assumptions do not hold, we show in extensive simulation studies that both approximations outperform the Stram and Lee approximation and the parametric bootstrap. We also identify and address numerical problems associated with standard mixed model software. Our methods are motivated by and applied to a large longitudinal study on air pollution health effects in a highly susceptible cohort. Relevant software is posted as an online supplement.     

On the problems of smoothing and near-interpolation

In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.

     

Mixed Model-Based Hazard Estimation

This article proposes a new method for estimation of the hazard function from a set of censored failure time data, with a view to extending the general approach to more complicated models. The approach is based on a mixed model representation of penalized spline hazard estimators. One payoff is the automation of the smoothing parameter choice through restricted maximum likelihood. Another is the option to use standard mixed model software for automatic hazard estimation.     

Genetic algorithms for the selection of smoothing parameters in additive models

Summary  Additive models of the type y=f ₁(x ₁)+...+f _p(x _p)+ε where f _j , j=1,..,p, have unspecified functional form, are flexible statistical regression models which can be used to characterize nonlinear regression effects. One way of fitting additive models is the expansion in B-splines combined with penalization which prevents overfitting. The performance of this penalized B-spline (called P-spline) approach strongly depends on the choice of the amount of smoothing used for components f _j . In particular for higher dimensional settings this is a computationaly demanding task. In this paper we treat the problem of choosing the smoothing parameters for P-splines by genetic algorithms. In several simulation studies this approach is compared to various alternative methods of fitting additive models. In particular functions with different spatial variability are considered and the effect of constant respectively local adaptive smoothing parameters is evaluated.     

Identifying local smoothness for spatially inhomogeneous functions

We consider a problem of estimating local smoothness of a spatially inhomogeneous function from noisy data under the framework of smoothing splines. Most existing studies related to this problem deal with estimation induced by a single smoothing parameter or partially local smoothing parameters, which may not be efficient to characterize various degrees of smoothness of the underlying function when it is spatially varying. In this paper, we propose a new nonparametric method to estimate local smoothness of the function based on a moving local risk minimization coupled with spatially adaptive smoothing splines. The proposed method provides full information of the local smoothness at every location on the entire data domain, so that it is able to understand the degrees of spatial inhomogeneity of the function. A successful estimate of the local smoothness is useful for identifying abrupt changes of smoothness of the data, performing functional clustering and improving the uniformity of coverage of the confidence intervals of smoothing splines. We further consider a nontrivial extension of the local smoothness of inhomogeneous two-dimensional functions or spatial fields. Empirical performance of the proposed method is evaluated through numerical examples, which demonstrates promising results of the proposed method.     

Dynamic crack analysis in 2D elastic solids with the singular edge-based smoothed finite element method

The singular edge-based smoothed finite element method (sES-FEM) is developed for stationary dynamic crack analysis in two-dimensional (2D) elastic solids. The paper aims at providing a better understanding of the dynamic fracture behaviors in linear elastic solids by means of the strain smoothing technique. The strains are smoothed and the system stiffness matrix is performed using the strain smoothing over the smoothing domains associated with the element edges. A two-layer singular five-node crack-tip element is employed while the standard implicit time integration scheme is used for solving the discrete sES-FEM equation system. Dynamic stress intensity factors (DSIFs) are extracted using the domain-form of interaction integrals in terms of the smoothing technique. The normalized DSIFs are compared with reference solutions showing a high accuracy of the sES-FEM. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variable Selection in Varying-Coefficient Models Using P-Splines

In this article, we consider nonparametric smoothing and variable selection in varying-coefficient models. Varying-coefficient models are commonly used for analyzing the time-dependent effects of covariates on responses measured repeatedly (such as longitudinal data). We present the P-spline estimator in this context and show its estimation consistency for a diverging number of knots (or B-spline basis functions). The combination of P-splines with nonnegative garrote (which is a variable selection method) leads to good estimation and variable selection. Moreover, we consider APSO (additive P-spline selection operator), which combines a P-spline penalty with a regularization penalty, and show its estimation and variable selection consistency. The methods are illustrated with a simulation study and real-data examples. The proofs of the theoretical results as well as one of the real-data examples are provided in the online supplementary materials.     

Smoothing noisy data for irregular regions using penalized bivariate splines on triangulations

The penalized spline method has been widely used for estimating univariate smooth functions based on noisy data. This paper studies its extension to the two-dimensional case. To accommodate the need of handling data distributed on irregular regions, we consider bivariate splines defined on triangulations. Penalty functions based on the second-order derivatives are employed to regularize the spline fit and generalized cross-validation is used to select the penalty parameters. A simulation study shows that the penalized bivariate spline method is competitive to some well-established two-dimensional smoothers. The method is also illustrated using a real dataset on Texas temperature.     

Sobolev smoothing of SVD-based Fourier continuations

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.     

Robust penalized regression spline fitting with application to additive mixed modeling

An increasingly popular method for smoothing noisy data is penalized regression spline fitting. In this paper a new procedure is proposed for fitting robust penalized regression splines. This procedure is computationally fast, straightforward to implement, and can be paired with any smoothing parameter selection method. In addition, it can also be extended to other settings, such as additive mixed modeling. Both simulated and real data examples are used to illustrate the effectiveness of the procedure.     

Recursive approximation of the high dimensional max function

This paper proposes a smoothing method for the general n-dimensional max function, based on a recursive extension of smoothing functions for the two-dimensional max function. A theoretical framework is introduced, and some applications are discussed. Finally, a numerical comparison with a well-known smoothing method is presented.