局部权重调节的自适应惩罚样条回归模型

传统惩罚样条回归模型中的惩罚是均匀惩罚未考虑数据的局部异质性,因而对复杂数据的拟合缺乏自适应性.本文针对约束回归模型惩罚项的设置特点,设计一种局部惩罚权重向量并将其加入到模型中,构造基于B样条基的自适应惩罚样条回归模型.新模型在观测数据波动较大的区域,给予拟合曲线较小的惩罚,而在观测数据波动较小的区域,给予拟合曲线较大的惩罚,从而使拟合曲线能自适应的反映观测数据的局部变化特征.模拟和应用的结果显示新模型的拟合效果显著优于传统的惩罚样条回归模型.     

极差调节的局部惩罚样条回归方法

在惩罚样条回归模型中,根据截断幂基函数系数的直观意义,以结点两边数据点极差的线性递减函数作为局部惩罚权重,构造了一种新的局部惩罚样条回归模型.不同于整体惩罚样条,该方法使得当数据点集在局部具有较大的波动性时,能给予拟合曲线较小的惩罚,从而能更好地控制曲线在拟合优度与光滑度之间的平衡.模拟结果显示,当数据具有空间异质性时,采用该方法的回归模型相比整体惩罚模型有更好的信息准则得分.     

基于惩罚回归样条的积分近似计算与应用

提出一种利用惩罚回归样条拟合被积函数f(x),从而计算复杂积分∫baf(x)dx的新方法.在仅知f(x)带随机扰动的离散数据点集的情况下,利用基于截断幂形式的样条基函数,通过惩罚样条回归,给出函数的多项式拟合结果,再根据该多项式形式便捷计算出积分.模拟和实际应用结果显示该方法计算简单快捷,并具有较好的准确度.     

生长曲线模型的变量选择

生长曲线模型是一个典型的多元线性模型, 在现代统计学上占有重要地位. 文章首先基于Potthoff-Roy变换后的生长曲线模型, 采用自适应LASSO为惩罚函数给出了参数矩阵的惩罚最小二乘估计, 实现了变量的选择. 其次, 基于局部渐近二次估计, 对生长曲线模型的惩罚最小二乘估计给出了统一的近似估计表达式. 接着, 讨论了经过Potthoff-Roy变换后模型的惩罚最小二乘估计, 证明了自适应LASSO具有Oracle性质. 最后对几种变量选择方法进行了数据模拟. 结果表明自适应LASSO效果比较好. 另外, 综合考虑, Potthoff-Roy变换优于拉直变换.     

动态面板数据的自适应惩罚分位回归方法研究

传统均值角度下研究的动态面板数据模型会受经典假设条件的约束,将动态面板数据与分位回归数模型相结合,不仅可以解决约束问题,而且能更加全面地描述响应变量条件分布的全貌.文章引入自适应惩罚项,并应用工具变量构造了自适应惩罚的动态面板分位回归方法,证明了该方法得到的估计量具有大样本性质.同时蒙特卡洛模拟结果表明自适应惩罚的方法相较于传统的方法更加有效.文章最后对中国大中城市商品房销售价格与各地人均国民生产总值的关系进行案例分析,发现两者之间存在正反馈机制.     

纵向数据非参数模型的惩罚修正二次推断函数估计

基于纵向数据研究非参数模型y=f(t)+ε,其中f(·)为未知平滑函数,ε为零均值随机误差项.利用截断幂函数基对f(·)进行基函数展开近似,并且结合惩罚样条的方法构造关于基函数系数的惩罚修正二次推断函数.然后利用割线法迭代得到基函数系数估计的数值解,从而得到未知平滑函数的估计.理论证明,应用此方法所得到的基函数系数估计具有相合性和渐近正态性.最后通过数值方法得到了较好的拟合结果.     

微观噪音下扩散模型波动函数的非参数设定检验及其应用

由于观测噪音的影响,现有的绝大多数扩散模型波动函数的识别检验在高频环境下会失效.本文采用局部平均方法对存在观测噪音的数据进行平滑处理,基于平滑后的观测值,结合条件矩和非参数核估计方法构造U统计量对扩散模型波动函数进行识别.所构造的检验统计量在波动函数形式设定正确时,收敛到标准正态分布.蒙特卡罗模拟结果显示,与现有检验方法相比,该统计量具有更合理的检验水平和更强的检验功效.将构造的检验统计量应用于中国银行股价对数化序列的识别过程,得到更为合理的检验结果.     

面板数据的自适应惩罚分位回归方法

传统的面板数据是从均值角度进行研究,但这会受经典假设条件的约束.而考虑面板数据的分位回归模型,可以更加全面地描述响应变量条件分布的全貌.文章引入自适应惩罚函数构造了自适应惩罚的分位回归面板数据方法,并证明所提出的估计量具有大样本性质.蒙特卡洛模拟结果显示该方法相对于均值回归更具优势,是处理面板数据的有效手段.文章最后对我国居民交通通讯消费进行案例分析,得到了有利于决策的参考信息.     

生长曲线模型的惩罚最小二乘估计

主要考虑了生长曲线模型中的参数矩阵的估计.首先基于Potthoff-Roy变换后的生长曲线模型,采用不同的惩罚函数:Hard Thresholding函数,LASSO,ENET,改进LASSO,SACD给出了参数矩阵的惩罚最小二乘估计.接着对不做变换的生长曲线模型,直接定义其惩罚最小二乘估计,基于Nelder-Mead法给出了估计的数值解算法.最后对提出的参数估计方法进行了数据模拟.结果表明自适应LASSO在估计方面效果比较好.     

众数自适应Lasso回归的统计推断

本文给出了自适应Lasso的众数回归模型,用来对众数回归模型的变量进行选择.对比传统的均值回归模型和中位数回归模型,众数回归在解决重尾、多峰分布问题时更加稳健.众数回归模型的主要估计方法是核估计方法,当自变量的数目较大时,该方法会产生难以忽略的计算误差.本文在核估计方法的众数回归模型基础上添加惩罚项,并通过自适应Lasso方法进行参数估计,有效的剔除了贡献率低的自变量,同时提高了计算的准确性.本文详细阐述了该计算方法,并在一些正则条件下,给出了模型的参数的估计方法和估计值的渐近正态性.模拟实验和实证分析研究了所提方法在有限样本下的性质.对比均值回归模型和传统的众数回归模型,添加自适应Lasso惩罚项的众数回归模型极大地提高了参数估计的准确性.     

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??Inspired by intuitive meanings of truncated power basis's coefficients, the local penalization based on range's linear decreasing function is given in penalized spline regression model. This method gives less penalization to fitting curve where data is with more volatility, which makes fitted curve controls tradeoff between goodness-of-fit and smoothness better. Simulations show that regression models with local penalized spline obtain lower information rules' scores than global penalized spline when the data is with heteroskedasticity.     

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.     

基于联合估计法的债券信用利差期限结构拟合及应用

基于2016~2018年月度数据,通过独立估计的单曲线样条模型和SV 模型、联合估计的多曲线样条模型和SV模型拟合公司债信用利差期限结构,进而对模型拟合效果进行比较,讨论模型在宏观经济预测中的应用,得到以下结论:(1)拟合模型的函数形式是导致理论信用利差期限结构曲线翻折的原因。样条模型和SV模型拟合的信用利差曲线形状完全不同,且模型函数变动引起的误差变动大于曲线变动引起的误差变动。(2)联合估计模型可以修正独立估计模型的人为扭曲形式。多曲线模型的结果更接近实际信用利差,误差波动性明显减小,曲线更为平滑,且联合估计的多曲线样条模型优于独立估计的单曲线样条模型、独立估计的SV 模型和联合估计的SV模型。(3)公司债信用利差期限结构在一定程度上蕴含了市场对未来宏观经济的预期信息,且在短期内预测结果随先行期限延长而改善。因此,宏观经济政策制定者需关注信用利差和期限结构模型拟合研究,重视对信用利差期限结构的深度信息挖掘,从而提高中国宏观政策制定者调控手段的前瞻性和有效性。     

Nonparametric Regression on a Graph

The ‘Signal plus Noise’ model for nonparametric regression can be extended to the case of observations taken at the vertices of a graph. This model includes many familiar regression problems. This article discusses the use of the edges of a graph to measure roughness in penalized regression. Distance between estimate and observation is measured at every vertex in the L₂ norm, and roughness is penalized on every edge in the L₁ norm. Thus the ideas of total variation penalization can be extended to a graph. The resulting minimization problem presents special computational challenges, so we describe a new and fast algorithm and demonstrate its use with examples.

The examples include image analysis, a simulation applicable to discrete spatial variation, and classification. In our examples, penalized regression improves upon kernel smoothing in terms of identifying local extreme values on planar graphs. In all examples we use fully automatic procedures for setting the smoothing parameters. Supplemental materials are available online.     

MARS: selecting basis functions and knots with an empirical Bayes method

An empirical Bayes method to select basis functions and knots in multivariate adaptive regression spline (MARS) is proposed, which takes both advantages of frequentist model selection approaches and Bayesian approaches. A penalized likelihood is maximized to estimate regression coefficients for selected basis functions, and an approximated marginal likelihood is maximized to select knots and variables involved in basis functions. Moreover, the Akaike Bayes information criterion (ABIC) is used to determine the number of basis functions. It is shown that the proposed method gives estimation of regression structure that is relatively parsimonious and more stable for some example data sets.     

Bootstrapping for Penalized Spline Regression

We describe and contrast several different bootstrap procedures for penalized spline smoothers. The bootstrap methods considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered as an estimation technique to find an unknown smooth function. The smooth function is represented in a high-dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrap methods can be implemented under both frameworks, and we show theoretically and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrap ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives.

Datasets and computer code are available in the online supplements.     

Bayesian Free-Knot Monotone Cubic Spline Regression

This article proposes a new Bayesian approach for monotone curve fitting based on the isotonic regression model. The unknown monotone regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones. We treat the number and locations of knots as free parameters and use reversible jump Markov chain Monte Carlo to obtain posterior samples of knot configurations. Given the number and locations of the knots, second-order cone programming is used to estimate the remaining parameters. Simulation results suggest the method performs well and we illustrate the approach using the ASA car data.     

Comparison of Radial Basis Functions

A survey of algorithms for approximation of multivariate functions with radial basis function (RBF) splines is presented. Algorithms of interpolating, smoothing, selecting the smoothing parameter, and regression with splines are described in detail. These algorithms are based on the feature of conditional positive definiteness of the spline radial basis function. Several families of radial basis functions generated by means of conditionally completely monotone functions are considered. Recommendations for the selection of the spline basis and preparation of initial data for approximation with the help of the RBF spline are given.     

基于最小一乘准则的三次样条对利率期限结构的拟合

将基于最小一乘准则的三次样条函数法应用于拟合在上海证券交易所交易的国债的利率期限结构,并与传统的最小二乘法进行比较。样本外预测结果显示,稳健的最小一乘方法能有效的降低异常点的干扰,弥补最小二乘法的不足,提高预测的精度。     

纵向数据半参数Beta回归模型的影响分析

本将随机效应当作是缺失数据,基于Q函数和EM算法并利用P-样条拟合非参数部分,得到了纵向数据半参数Beta回归模型估计方法.基于数据删除模型,我们得到了模型参数部分的广义Cook距离以及非参数部分的广义DFIT.此外,本文还研究了在四种不同扰动情形下模型的局部影响分析,得到了相应的影响矩阵.最后,我们通过两个数值实例验证了所得诊断统计量的有效性.