Semiparametric regression with shape-constrained penalized splines

In semiparametric regression models, penalized splines can be used to describe complex, non-linear relationships between the mean response and covariates. In some applications it is desirable to restrict the shape of the splines so as to enforce properties such as monotonicity or convexity on regression functions. We describe a method for imposing such shape constraints on penalized splines within a linear mixed model framework. We employ Markov chain Monte Carlo (MCMC) methods for model fitting, using a truncated prior distribution to impose the requisite shape restrictions. We develop a computationally efficient MCMC sampler by using a correspondingly truncated multivariate normal proposal distribution, which is a restricted version of the approximate sampling distribution of the model parameters in an unconstrained version of the model. We also describe a cheap approximation to this methodology that can be applied for shape-constrained scatterplot smoothing. Our methods are illustrated through two applications, the first involving the length of dugongs and the second concerned with growth curves for sitka spruce trees.     

A case-deletion diagnostic for penalized calibration estimators and BLUP under linear mixed models in survey sampling

The penalized calibration technique in survey sampling combines usual calibration and soft calibration by introducing a penalty term. Certain relevant estimates in survey sampling can be considered as penalized calibration estimates obtained as particular cases from an optimization problem with a common basic structure. In this framework, a case deletion diagnostic is proposed for a class of penalized calibration estimators including both design-based and model-based estimators. The diagnostic compares finite population parameter estimates and can be calculated from quantities related to the full data set. The resulting diagnostic is a function of the residual and leverage, as other diagnostics in regression models, and of the calibration weight, a singular feature in survey sampling. Moreover, a particular case, which includes the basic unit level model for small area estimation, is considered. Both a real and an artificial example are included to illustrate the diagnostic proposed. The results obtained clearly show that the proposed diagnostic depends on the calibration and soft-calibration variables, on the penalization term, as well as on the parameter to estimate.     

A class of template splines

A common frame of template splines that unifies the definitions of various spline families, such as smoothing, regression or penalized splines, is considered. The nonlinear nonparametric regression problem that defines the template splines can be reduced, for a large class of Hilbert spaces, to a parameterized regularized linear least squares problem, which leads to an important computational advantage. Particular applications of template splines include the commonly used types of splines, as well as other atypical formulations. In particular, this extension allows an easy incorporation of additional constraints, which is generally not possible in the context of classical spline families.     

A class of template splines

A common frame of template splines that unifies the definitions of various spline families, such as smoothing, regression or penalized splines, is considered. The nonlinear nonparametric regression problem that defines the template splines can be reduced, for a large class of Hilbert spaces, to a parameterized regularized linear least squares problem, which leads to an important computational advantage. Particular applications of template splines include the commonly used types of splines, as well as other atypical formulations. In particular, this extension allows an easy incorporation of additional constraints, which is generally not possible in the context of classical spline families.     

Secure computation with horizontally partitioned data using adaptive regression splines

When several data owners possess data on different records but the same variables, known as horizontally partitioned data, the owners can improve statistical inferences by sharing their data with each other. Often, however, the owners are unwilling or unable to share because the data are confidential or proprietary. Secure computation protocols enable the owners to compute parameter estimates for some statistical models, including linear regressions, without sharing individual records’ data. A drawback to these techniques is that the model must be specified in advance of initiating the protocol, and the usual exploratory strategies for determining good-fitting models have limited usefulness since the individual records are not shared. In this paper, we present a protocol for secure adaptive regression splines that allows for flexible, semi-automatic regression modeling. This reduces the risk of model mis-specification inherent in secure computation settings. We illustrate the protocol with air pollution data.     

Secure computation with horizontally partitioned data using adaptive regression splines

When several data owners possess data on different records but the same variables, known as horizontally partitioned data, the owners can improve statistical inferences by sharing their data with each other. Often, however, the owners are unwilling or unable to share because the data are confidential or proprietary. Secure computation protocols enable the owners to compute parameter estimates for some statistical models, including linear regressions, without sharing individual records’ data. A drawback to these techniques is that the model must be specified in advance of initiating the protocol, and the usual exploratory strategies for determining good-fitting models have limited usefulness since the individual records are not shared. In this paper, we present a protocol for secure adaptive regression splines that allows for flexible, semi-automatic regression modeling. This reduces the risk of model mis-specification inherent in secure computation settings. We illustrate the protocol with air pollution data.     

Parametrization and penalties in spline models with an application to survival analysis

A simple parametrization, built from the definition of cubic splines, is shown to facilitate the implementation and interpretation of penalized spline models, whatever configuration of knots is used. The parametrization is termed value-first derivative parametrization. Inference is Bayesian and explores the natural link between quadratic penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some penalty functionals. Alternatives include empirical Bayes inference methods involving model selection type criteria. The proposed methodology is illustrated by an application to survival analysis where the usual Cox model is extended to allow for time-varying regression coefficients.     

Penalized spline estimation for functional coefficient regression models

The functional coefficient regression models assume that the regression coefficients vary with some “threshold” variable, providing appreciable flexibility in capturing the underlying dynamics in data and avoiding the so-called “curse of dimensionality” in multivariate nonparametric estimation. We first investigate the estimation, inference, and forecasting for the functional coefficient regression models with dependent observations via penalized splines. The P-spline approach, as a direct ridge regression shrinkage type global smoothing method, is computationally efficient and stable. With established fixed-knot asymptotics, inference is readily available. Exact inference can be obtained for fixed smoothing parameter λ, which is most appealing for finite samples. Our penalized spline approach gives an explicit model expression, which also enables multi-step-ahead forecasting via simulations. Furthermore, we examine different methods of choosing the important smoothing parameter λ: modified multi-fold cross-validation (MCV), generalized cross-validation (GCV), and an extension of empirical bias bandwidth selection (EBBS) to P-splines. In addition, we implement smoothing parameter selection using mixed model framework through restricted maximum likelihood (REML) for P-spline functional coefficient regression models with independent observations. The P-spline approach also easily allows different smoothness for different functional coefficients, which is enabled by assigning different penalty λ accordingly. We demonstrate the proposed approach by both simulation examples and a real data application.     

Comparing measures of model selection for penalized splines in Cox models

This article presents an application and a simulation study of model fit criteria for selecting the optimal degree of smoothness for penalized splines in Cox models. The criteria considered were the Akaike information criterion, the corrected AIC, two formulations of the Bayesian information criterion, and a generalized cross-validation method. The estimated curves selected by the five methods were compared to each other in a study of rectal cancer mortality in autoworkers. In the stimulation study, we estimated the fit of the penalized spline models in six exposure-response scenarios, using the five model fit criteria. The methods were compared on the basis of a mean squared error score and the power and size of hypothesis tests for any effect and for detecting nonlinearity. All comparisons were made across a range in the total sample size and number of cases.     

Classification of pathological stage of prostate cancer patients using penalized splines

We propose a penalized splines (P-splines) based method to predict the pathological stage of localized prostate cancer. A combination of prostate-specific antigen, Gleason histological score, and clinical stage from a cohort study of 834 prostate cancer patients are used to build the P-splines model. It turns out that the proposed methodology results in improved prediction of pathological stage compared to usual logistic regression after removing a few outliers. The improvement is shown to be statistically significant. Receiver-operating characteristic (ROC) curve is drawn and we show that the increase in area under the ROC curve over the commonly used logistic regression based classification method is also statistically significant.     

Analytical determination and classification of pollutant concentration fields using air pollution monitoring network data: Methodology and application in the Paris area,during episodes with peak nitrogen dioxide levels

During the last years, an important number of episodes with peak nitrogen dioxide levels occurred in the Paris region. Modelling air pollution is necessary to predict future episodes and to take decisions for the protection of populations. Our study proposes a methodology for urban air pollution analysis, as a preliminary stage of its modelling.This paper focuses on two major points: the estimation of pollutant concentration fields using measures from a monitoring network, and the definition of a reduced number of concentration spatial patterns. Interpolation by thin plate splines is proposed and the approximation quality is estimated by leave-one-out cross validation tests. Analysis of large numbers of pollutant fields is difficult: cluster analysis of spatial distributions is proposed. For this study, Ward's algorithm is selected and applied to AIRPARIF nitrogen dioxide data during peak episodes from January 1993 to December 1997. Finally, some results and conclusions are presented.     

Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models

Several tests for a zero random effect variance in linear mixed models are compared. This testing problem is non-regular because the tested parameter is on the boundary of the parameter space. Size and power of the different tests are investigated in an extensive simulation study that covers a variety of important settings. These include testing for polynomial regression versus a general smooth alternative using penalized splines. Among the test procedures considered, three are based on the restricted likelihood ratio test statistic (RLRT), while six are different extensions of the linear model F-test to the linear mixed model. Four of the tests with unknown null distributions are based on a parametric bootstrap, the other tests rely on approximate or asymptotic distributions. The parametric bootstrap-based tests all have a similar performance. Tests based on approximate F-distributions are usually the least powerful among the tests under consideration. The chi-square mixture approximation for the RLRT is confirmed to be conservative, with corresponding loss in power. A recently developed approximation to the distribution of the RLRT is identified as a rapid, powerful and reliable alternative to computationally intensive parametric bootstrap procedures. This novel method extends the exact distribution available for models with one random effect to models with several random effects.     

带参数的四次Hermite插值样条

为了克服标准三次Hermite插值样条的不足,给出了一种带参数的四次Hermite插值样条,具有标准三次Hermite插值样条完全相同的性质。在插值条件给定时,四次Hermite插值样条的形状可通过改变参数的取值进行调控。通过选择合适的参数,四次Hermite曲线能达到C2连续,而且其整体逼近效果要好于标准三次Hermite插值样条。所提出的新样条进一步丰富了Hermite插值样条理论,也为工程中插值曲线曲面的构造提供了一种新方法。     

Exploring US Business Cycles with Bivariate Loops Using Penalized Spline Regression

The phrase business cycle is usually used for short term fluctuations in macroeconomic time series. In this paper we focus on the estimation of business cycles in a bivariate manner by fitting two series simultaneously. The underlying model is thereby nonparametric in that no functional form is prespecified but smoothness of the functions are assumed. The functions are then estimated using penalized spline estimation. The bivariate approach will allow to compare business cycles, check and compare phase lengths and visualize this in forms of loops in a bivariate way. Moreover, the focus is on separation of long and short phase fluctuation, where only the latter is the classical business cycle while the first is better known as Friedman or Goodwin cycle, respectively. Again, we use nonparametric models and fit the functional shape with penalized splines. For the separation of long and short phase components we employ an Akaike criterion.     

Generalized profiling estimation for global and adaptive penalized spline smoothing

We propose the generalized profiling method to estimate the multiple regression functions in the framework of penalized spline smoothing, where the regression functions and the smoothing parameter are estimated in two nested levels of optimization. The corresponding gradients and Hessian matrices are worked out analytically, using the Implicit Function Theorem if necessary, which leads to fast and stable computation. Our main contribution is developing the modified delta method to estimate the variances of the regression functions, which include the uncertainty of the smoothing parameter estimates. We further develop adaptive penalized spline smoothing to estimate spatially heterogeneous regression functions, where the smoothing parameter is a function that changes along with the curvature of regression functions. The simulations and application show that the generalized profiling method leads to good estimates for the regression functions and their variances.     

Spline-regression estimation of the influence of exposure dose on cancer probability

Spline regression is used to analyze the influence of radiation on the cancer probability in a group of participants of Chernobyl accident mitigation, depending on the exposure dose. A new method is proposed to approximate modified polygons by linear splines with two nodes. An algorithm for the identification of a transition point is outlined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 168–176, May–June 2006.     

双曲抛物面上的一类G2连续逼近样条

在双曲抛物面上,仿射坐标系下,通过带逼近控制因子的双参数化方法,以及研究其参数间的函数关系构造出一类G2连续样条曲线。当控制多边形是平形四边形时,样条曲线段在逼近控制因子大于某个数时具有保形性质。对这类样条曲线段的逼近问题进行了一定的理论分析。     

利用Box样条构造完全重构模型的研究

针对图象处理中的图象重构问题,结合多频带DFT滤波器库和双变量Box样条构造了一个完全重构模型,在介绍了DFT滤波器库后,给出了形成完全重构模型的条件,通过双变量Box样条及其在图象处理极为有用的几条性质的分析,提出了一类双变量Box样条,使用该样条即可构造分解/重构模型,实验证明,该模型能满足完全重构条件,并可有效分解合成图象,最后,给出了应用此模型的实验结果。     

Mixed model-based inference in geoadditive hazard regression for interval-censored survival times

Mixed model-based estimation of additive or geoadditive regression models has become popular throughout recent years. It provides a unified and modular framework that facilitates joint estimation of nonparametric covariate effects and the corresponding smoothing parameters. Therefore, extensions of mixed model-based inference to a Cox-type regression model for the hazard rate are considered, allowing for a combination of general censoring schemes for the survival times and a flexible, geoadditive predictor. In particular, the proposed methodology allows for arbitrary combinations of right, left, and interval censoring as well as left truncation. The geoadditive predictor comprises time-varying effects, nonparametric effects of continuous covariates, spatial effects, and potentially a number of extensions such as cluster-specific frailties or interaction surfaces. In addition, all covariates are allowed to be piecewise constant time-varying. Nonlinear and time-varying effects as well as the baseline hazard rate are modeled by penalized splines. Spatial effects can be included based on either Markov random fields or stationary Gaussian random fields. Estimation is based on a reparametrization of the model as a variance component mixed model. The variance parameters, corresponding to inverse smoothing parameters, can then be determined using an approximate marginal likelihood approach. An analysis on childhood mortality in Nigeria serves as an application, where the interval censoring framework additionally allows to deal with the problem of heaped survival times. The effect of ignoring the impact of interval-censored observations is investigated in a simulation study.     

基于粒子群三次样条优化的移动机器人路径规划算法

针对移动机器人路径规划问题,提出了一种基于粒子群三次样条优化的路径规划方法．借助三次样条 连接描述路径,这样将路径规划问题转化为三次样条曲线的参数优化问题．借助粒子群优化算法快速收敛和全局寻 优特性实现最优路径规划．实验结果表明：所提算法可以快速有效地实现障碍环境下机器人的无碰撞路径规划,规 划路径平滑,利于机器人的运动控制．