Generalized Nonparametric Mixed-Effect Models: Computation and Smoothing Parameter Selection

Generalized linear mixed-effect models are widely used for the analysis of correlated non-Gaussian data such as those found in longitudinal studies. In this article, we consider extensions with nonparametric fixed effects and parametric random effects. The estimation is through the penalized likelihood method, and our focus is on the efficient computation and the effective smoothing parameter selection. To assist efficient computation, the joint likelihood of the observations and the latent variables of the random effects is used instead of the marginal likelihood of the observations. For the selection of smoothing parameters and correlation parameters, direct cross-validation techniques are employed; the effectiveness of cross-validation with respect to a few loss functions are evaluated through simulation studies. Real data examples are presented to illustrate potential applications of the methodology. Open-source R code is demonstrated in the Appendix.     

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.     

A simple smoothing spline,III

Summary  Computational methods for spline smoothing are studied in the context of the linear smoothing spline. Comparisons are made between two efficient methods for computing the estimator using band-limited basis functions and the Kalman filter. In particular, the Kalman filter approach is shown to be an efficient method for computing under the Kimeldorf-Wahba representation for the estimator. Run time comparisons are made between band-limited B-spline and Kalman filter based algorithms.     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

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??In the last few decades, longitudinal data was deeply research in statistics science and widely used in many field, such as finance, medical science, agriculture and so on. The characteristic of longitudinal data is that the values are independent from different samples but they are correlate from one sample. Many nonparametric estimation methods were applied into longitudinal data models with development of computer technology. Using Cholesky decomposition and Profile least squares estimation, we will propose a effective spline estimation method pointing at nonparametric model of longitudinal data with covariance matrix unknown in this paper. Finally, we point that the new proposed method is more superior than Naive spline estimation in the covariance matrix is unknown case by comparing the simulated results of one example.     

Algorithms with Adaptive Smoothing for Finite Minimax Problems

We present a new feedback precision-adjustment rule for use with a smoothing technique and standard unconstrained minimization algorithms in the solution of finite minimax problems. Initially, the feedback rule keeps a precision parameter low, but allows it to grow as the number of iterations of the resulting algorithm goes to infinity. Consequently, the ill-conditioning usually associated with large precision parameters is considerably reduced, resulting in more efficient solution of finite minimax problems.The resulting algorithms are very simple to implement, and therefore are particularly suitable for use in situations where one cannot justify the investment of time needed to retrieve a specialized minimax code, install it on one's platform, learn how to use it, and convert data from other formats. Our numerical tests show that the algorithms are robust and quite effective, and that their performance is comparable to or better than that of other algorithms available in the Matlab environment.     

二次锥规划的预估-校正光滑方法

基于光滑Fischer-Burmeister函数,给出一个求解二次锥规划的预估-校正光滑牛顿法.该算法构造一个等价于最优性条件的非线性方程组,再用牛顿法求解此方程组的扰动.在适当的假设下,证明算法是全局收敛且是局部二阶收敛的.数值试验表明算法的有效性.     

Spline Multiscale Smoothing to Control FDR for Exploring Features of Regression Curves

SiZer (significant zero crossing of the derivatives) is a multiscale smoothing method for exploring trends, maxima, and minima in data. In this article, a regression spline version of SiZer is proposed in a nonparametric regression setting by the fiducial method. The number of knots for spline interpolation is used as the scale parameter of the new SiZer, which controls the smoothness of estimate. In the construction of the new SiZer, multiple testing adjustment is made to control the row-wise false discovery rate (FDR) of SiZer. This adjustment is appealing for exploratory data analysis and has potential to increase the power. A special map is also produced on a continuous scale using p-values to assess the significance of features. Simulations and a real data application are carried out to investigate the performance of the proposed SiZer, in which several comparisons with other existing SiZers are presented. Supplementary materials for this article are available online.     

A Geometric Programming Framework for Univariate Cubic L 1 Smoothing Splines

Univariate cubic L ₁ smoothing splines are capable of providing shape-preserving C ¹-smooth approximation of multi-scale data. The minimization principle for univariate cubic L ₁ smoothing splines results in a nondifferentiable convex optimization problem that, for theoretical treatment and algorithm design, can be formulated as a generalized geometric program. In this framework, a geometric dual with a linear objective function over a convex feasible domain is derived, and a linear system for dual to primal conversion is established. Numerical examples are given to illustrate this approach. Sensitivity analysis for data with uncertainty is presented. This work is supported by research grant #DAAG55-98-D-0003 of the Army Research Office, USA.     

Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques

We develop two implementable algorithms, the first for the solution of finite and the second for the solution of semi-infinite min-max-min problems. A smoothing technique (together with discretization for the semi-infinite case) is used to construct a sequence of approximating finite min-max problems, which are solved with increasing precision. The smoothing and discretization approximations are initially coarse, but are made progressively finer as the number of iterations is increased. This reduces the potential ill-conditioning due to high smoothing precision parameter values and computational cost due to high levels of discretization. The behavior of the algorithms is illustrated with three semi-infinite numerical examples.     

On use of the Kalman filter for spatial smoothing

A state-space model to perform discrete thin plate smoothing for data on a two-dimensional rectangular lattice is proposed with the use of the Kalman filter. The use of the Kalman filter reduces computational difficulties in the maximum likelihood estimation of a smoothing parameter. A procedure to reduce computational difficulties in the estimation of trend is given also. Numerical illustration is provided using two sets of artificial data.     

Sequential smoothing for turning point detection with application to financial decisions

A fundamental problem in financial trading is the correct and timely identification of turning points in stock value series. This detection enables to perform profitable investment decisions, such as buying‐at‐low and selling‐at‐high. This paper evaluates the ability of sequential smoothing methods to detect turning points in financial time series. The novel idea is to select smoothing and alarm coefficients on the gain performance of the trading strategy. Application to real data shows that recursive smoothers outperform two‐sided filters at the out‐of‐sample level. Copyright © 2012 John Wiley & Sons, Ltd.     

The <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Approximation Order of Surface Spline Interpolation for 1 \leq <Emphasis Type="Italic">p</Emphasis> \leq 2

We show that the L_p-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + &frac; and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain , we show specifically that the L_p()-norm of the error between f and its surface spline interpolant is O(h^{m + 1/p}) provided that f belongs to an appropriate Sobolev or Besov space and that \subset R^d is open, bounded, and has the C^2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h^2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the L_p-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.