A new estimation procedure for a partially nonlinear model via a mixed‐effects approach

The authors consider the estimation of the parametric component of a partially nonlinear semiparametric regression model whose nonparametric component is viewed as a nuisance parameter. They show how estimation can proceed through a nonlinear mixed‐effects model approach. They prove that under certain regularity conditions, the proposed estimate is consistent and asymptotically Gaussian. They investigate its finite‐sample properties through simulations and illustrate its use with data on the relation between the photosynthetically active radiation and the net ecosystem‐atmosphere exchange of carbon dioxide.     

Laplace based approximate posterior inference for differential equation models

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Makov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.     

陕西省二氧化碳库兹涅茨曲线的形状、拐点与影响因素

以中国首批低碳试点地区——陕西省为研究对象,利用1978—2010年的时间序列数据,对人均二氧化碳排放的库兹涅茨曲线进行了经验估计,发现曲线形状为N型,两个拐点分别在人均GDP为3 370.31元和4 070.06元处。对影响人均二氧化碳排放的因素进行研究,利用迪式分解法进行短期分析,发现经济收入和能源消耗强度对人均二氧化碳的排放影响较大,而能源结构的影响很小;利用协整方程进行长期分析,结果表明,能源消耗强度、能源结构和第二产业结构对人均二氧化碳排放量的影响作用都比较大,其中最大的是能源结构。     

Stochastic models for greenhouse gas emission rate estimation from hydroelectric reservoirs: a Bayesian hierarchical approach

Herein, we propose a fully Bayesian approach to the greenhouse gas emission problem. The goal of this work is to estimate the emission rate of polluting gases from the area flooded by hydroelectric reservoirs. We present models for gas concentration evolution in two ways: first, by proposing them from ordinary differential equation solutions and, second, by using stochastic differential equations with a discretization scheme. Finally, we present techniques to estimate the emission rate for the entire reservoir. In order to carry out the inference, we use the Bayesian framework with Monte Carlo via Markov Chain methods. Discretization schemes over continuous differential equations are used when necessary. These models applied to greenhouse gas emission and Bayesian inference for this purpose are completely new in statistical literature, as far as we know, and contribute to estimate the amount of polluting gases released from hydroelectric reservoirs in Brazil. The proposed models are applied in a real data set and results are presented.     

基于时空地理加权回归模型的中国碳排放驱动因素实证研究

基于2006-2011年中国省级区域面板数据,应用时空加权回归模型(GTWR)实证考察了各驱动因素对碳排放规模和碳排放强度影响的时空差异。研究结果表明:大部分解释变量的时空系数估计值显著,波动性较为稳定,符号与预期一致,各驱动因素及其外溢效应在不同区域存在较强的空间异质性,且表现出一定的空间梯度分布。若实现区域差异化碳减排,需要充分考虑空间异质性和外溢性。     

Differential equation models for statistical functions

Differential equations have been used in statistics to define functions such as probability densities. But the idea of using differential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, differential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods.     

Quantile regression methods for left-truncated and right-censored data

Left-truncated and right-censored (LTRC) data are encountered frequently due to a prevalent cohort sampling in follow-up studies. Because of the skewness of the distribution of survival time, quantile regression is a useful alternative to the Cox's proportional hazards model and the accelerated failure time model for survival analysis. In this paper, we apply the quantile regression model to LTRC data and develops an unbiased estimating equation for regression coefficients. The proposed estimation methods use the inverse probabilities of truncation and censoring weighting technique. The resulting estimator is uniformly consistent and asymptotically normal. The finite-sample performance of the proposed estimation methods is also evaluated using extensive simulation studies. Finally, analysis of real data is presented to illustrate our proposed estimation methods.     

Classical testing in functional linear models

We extend four tests common in classical regression – Wald, score, likelihood ratio and F tests – to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.     

An ordinary differential equation based solution path algorithm

Efron, Hastie, Johnstone and Tibshirani (2004) proposed Least Angle Regression (LAR), a solution path algorithm for the least squares regression. They pointed out that a slight modification of the LAR gives the LASSO (Tibshirani, 1996) solution path. However it is largely unknown how to extend this solution path algorithm to models beyond the least squares regression. In this work, we propose an extension of the LAR for generalized linear models and the quasi-likelihood model by showing that the corresponding solution path is piecewise given by solutions of ordinary differential equation systems. Our contribution is twofold. First, we provide a theoretical understanding on how the corresponding solution path propagates. Second, we propose an ordinary differential equation based algorithm to obtain the whole solution path.     

Functional regression concurrent model with spatially correlated errors: application to rainfall ground validation

In this paper, we give an extension of the functional regression concurrent model to the case of spatially correlated errors. We propose estimating the spatial correlation structure by using functional geostatistics. The estimation of the regression parameters is carried out by feasible generalized least squares. This modeling approach is motivated by the problem of validating rainfall data retrieved from satellite sensors. In this sense, we use the methodology to study the relationship between satellite and ground rainfall time series recorded in 82 weather stations from Department of Valle del Cauca, Colombia. The model obtained allows predicting pentadal rainfall curves in many sites of the region of interest by using as input the satellite information. A residual analysis shows a good performance of the methodology proposed.     

A note on variable selection in functional regression via random subspace method

Variable selection problem is one of the most important tasks in regression analysis, especially in a high-dimensional setting. In this paper, we study this problem in the context of scalar response functional regression model, which is a linear model with scalar response and functional regressors. The functional model can be represented by certain multiple linear regression model via basis expansions of functional variables. Based on this model and random subspace method of Mielniczuk and Teisseyre (Comput Stat Data Anal 71:725–742, 2014), two simple variable selection procedures for scalar response functional regression model are proposed. The final functional model is selected by using generalized information criteria. Monte Carlo simulation studies conducted and a real data example show very satisfactory performance of new variable selection methods under finite samples. Moreover, they suggest that considered procedures outperform solutions found in the literature in terms of correctly selected model, false discovery rate control and prediction error.     

基于连续性动态分布方法的中国碳排放收敛分析

本文采用连续性动态分布方法结合2002-2011年286个地级以上城市的面板数据分析了中国碳排放强度和人均碳排放的动态演进和长期趋势。研究发现,虽然所研究时期碳排放强度和人均碳排放均为单峰分布,但按照目前的发展趋势,二者都将出现明显的俱乐部收敛现象,且这种俱乐部收敛并不由地域因素决定。为了避免出现俱乐部收敛的极化现象,政府需要重点加强碳排放强度约为4.05、4.7倍均值和人均碳排放约为3.9、5.0倍均值城市的碳减排。     

Robust location estimation with missing data

In a missing data setting, we have a sample in which a vector of explanatory variables ${\bf x}_i$ is observed for every subject i, while scalar responses $y_i$ are missing by happenstance on some individuals. In this work we propose robust estimators of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response's distribution. In particular, strongly consistent estimators of any continuous location functional, such as the median, L‐functionals and M‐functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimators is also derived. The proofs of the theorems are presented in Supplementary Material available online. The Canadian Journal of Statistics 41: 111–132; 2013 © 2012 Statistical Society of Canada     

二氧化碳排放强度的实证研究

 二氧化碳(CO2)排放与减排问题已成为全球关注的焦点,而作为全球最大CO2排放量的中国,尤其值得研究。本文运用基于投入产出模型的结构因素分解法对中国1997-2007年CO2排放强度进行了因素分析。与已有研究相比,本文既测度了直接CO2排放,也测度了间接CO2排放;并考虑了各类不同种类能源的碳排放因子的差别,因此更具准确性和合理性。研究结果表明,生产模式的转变是CO2排放强度降低的主要原因,尤其是能源强度;而需求模式的作用较小。在此基础上,提出中国在未来发展中不仅要注重生产领域的减排,更应重视培养清洁需求的模式。     

Parameter estimation of complex mixed models based on meta-model approach

Complex biological processes are usually experimented along time among a collection of individuals, longitudinal data are then available. The statistical challenge is to better understand the underlying biological mechanisms. A standard statistical approach is mixed-effects model where the regression function is highly-developed to describe precisely the biological processes (solutions of multi-dimensional ordinary differential equations or of partial differential equation). A classical estimation method relies on coupling a stochastic version of the EM algorithm with a Monte Carlo Markov Chain algorithm. This algorithm requires many evaluations of the regression function. This is clearly prohibitive when the solution is numerically approximated with a time-consuming solver. In this paper a meta-model relying on a Gaussian process emulator is proposed to approximate the regression function, that leads to what is called a mixed meta-model. The uncertainty of the meta-model approximation can be incorporated in the model. A control on the distance between the maximum likelihood estimates of the mixed meta-model and the maximum likelihood estimates of the exact mixed model is guaranteed. Eventually, numerical simulations are performed to illustrate the efficiency of this approach.     

Functional logistic regression with fused lasso penalty

This study considers the binary classification of functional data collected in the form of curves. In particular, we assume a situation in which the curves are highly mixed over the entire domain, so that the global discriminant analysis based on the entire domain is not effective. This study proposes an interval-based classification method for functional data: the informative intervals for classification are selected and used for separating the curves into two classes. The proposed method, called functional logistic regression with fused lasso penalty, combines the functional logistic regression as a classifier and the fused lasso for selecting discriminant segments. The proposed method automatically selects the most informative segments of functional data for classification by employing the fused lasso penalty and simultaneously classifies the data based on the selected segments using the functional logistic regression. The effectiveness of the proposed method is demonstrated with simulated and real data examples.     

基于空间关联网络结构的中国省域协同碳减排研究

利用1997-2014年的省域数据,运用社会网络分析方法对中国省域二氧化碳排放的空间关联结构特征进行实证考察,研究认为:在样本期内,中国碳排放空间关联网络的紧密程度呈递增趋势,网络结构的不平等程度在降低,网络复杂性和稳定性逐渐增强。上海、江苏、天津、北京、浙江、广东等省份位于网络的中心位置,在网络中发挥核心中介作用,中西部的部分省份在网络中的作用逐渐增强。在碳排放空间关联网络中,长三角、环渤海和珠三角地区属于"受益者"板块,其他地区属于"贡献者"板块。碳排放的空间关联网络结构为省际间实现"引领-跟随"型的协同碳减排奠定了基础。     

Spatio-temporal statistical analysis of the carbon budget of the terrestrial ecosystem

The Net Ecosystem Exchange describes the net carbon dioxide flux between an ecosystem and the atmosphere and is a key quantity in climate change studies and in political negotiations. This paper provides a spatio-temporal statistical framework, which is able to infer the Net Ecosystem Exchange from remotely-sensed carbon dioxide ground concentrations together with data on the Normalized Difference Vegetation Index, the Gross Primary Production and the land cover classification. The model is based on spatial and temporal latent random effects, that act as space–time varying coefficients, which allows for a flexible modeling of the spatio-temporal auto- and cross-correlation structure. The intra- and inter-annual variations of the Net Ecosystem Exchange are evaluated and dynamic maps are provided on a nearly global grid and in intervals of 16 days.     

Curve prediction and clustering with mixtures of Gaussian process functional regression models

Shi, Wang, Murray-Smith and Titterington (Biometrics 63:714–723, 2007) proposed a Gaussian process functional regression (GPFR) model to model functional response curves with a set of functional covariates. Two main problems are addressed by their method: modelling nonlinear and nonparametric regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very good results for curve fitting and prediction but side-steps the problem of heterogeneity. In this paper we present a new method for modelling functional data with ‘spatially’ indexed data, i.e., the heterogeneity is dependent on factors such as region and individual patient’s information. For data collected from different sources, we assume that the data corresponding to each curve (or batch) follows a Gaussian process functional regression model as a lower-level model, and introduce an allocation model for the latent indicator variables as a higher-level model. This higher-level model is dependent on the information related to each batch. This method takes advantage of both GPFR and mixture models and therefore improves the accuracy of predictions. The mixture model has also been used for curve clustering, but focusing on the problem of clustering functional relationships between response curve and covariates, i.e. the clustering is based on the surface shape of the functional response against the set of functional covariates. The model is examined on simulated data and real data.     

Sparse group lasso for multiclass functional logistic regression models

Sparsity-inducing penalties are useful tools for variable selection and are also effective for regression problems where the data are functions. We consider the problem of selecting not only variables but also decision boundaries in multiclass logistic regression models for functional data, using sparse regularization. The parameters of the functional logistic regression model are estimated in the framework of the penalized likelihood method with the sparse group lasso-type penalty, and then tuning parameters for the model are selected using the model selection criterion. The effectiveness of the proposed method is investigated through simulation studies and the analysis of a gene expression data set.