半参数双重Tweedie复合泊松回归模型的贝叶斯分析

为了分析健康保险行业中出现的半连续卫生保健费用数据,本文提出一类半参数双重Tweedie复合泊松回归模型.在分析中,首先采用修正鞍点逼近的数值方法去近似Tweedie复合泊松分布的密度函数;其次,利用Gibbs抽样技术和Metropolis-Hastings(MH)算法的混合算法获得了模型参数的联合贝叶斯估计;最后,给出了几个模拟研究以及把这些方法用来分析兰德健康保险实验中的卫生保健费用数据.     

零膨胀泊松混合回归中的变量选择

来源于不同总体的数据异质性较大,数据“零取值”较多且离散度大,可利用零膨胀泊松(ZIP)混合回归模型建模分析,然而混合模型中自变量较多.为了筛选出重要变量,本文利用自适应LASSO对ZIP混合回归模型进行变量选择,即在似然函数中加入惩罚项,再利用EM算法估计参数.通过模拟,验证了该方法在变量选择和参数估计中的有效性.同时,将ZIP混合回归模型应用于预测借贷失败次数的实际数据分析,筛选出对借贷失败有重要影响的因素.最后,通过比较各模型的预测效果,得到ZIP混合回归模型优于泊松(Poisson),负二项(NB)和ZIP回归模型.     

以截尾伽玛分布为先验分布的产品失效率的贝叶斯估计

本文研究了基于泊松分布的产品失效率估计问题.利用贝叶斯统计推断方法,获得了以截尾伽玛分布为先验分布时,产品失效率的贝叶斯估计和相关性质,推广了以伽玛分布为先验分布的贝叶斯估计结果.     

零—膨胀泊松回归模型的非参数统计分析及其应用

零膨胀计数数据是当今数据分析的热点问题之一,该类数据的特点是零点过多,目前对这类数据的研究已经比较全面。另外还有些计数数据不仅会出现零点过多的现象,也会同时存在零、一点都过多的情形,如果再用零膨胀计数数据的统计方法去研究,产生的误差较大。目前国内外对零和一都膨胀的数据的研究还比较少,针对这种现象,本文引入零一膨胀泊松回归模型,并用局部多项式核回归法这种非参数统计分析方法对零一膨胀泊松回归模型进行参数估计,这是本文的创新点也是难点,并在求解参数的过程中引进了EM算法和Newton-Raphson迭代对参数近似求解。通过模拟结果可以得出此方法的可行性,最后通过对糖尿病患者数据的实例分析,可以验证此方法的有效性。     

IIRCT下泊松分布参数多变点模型的贝叶斯估计

通过添加缺损的寿命变量数据得到了带有不完全信息随机截尾试验下泊松分布参数多变点模型的完全数据似然函数,研究了变点位置参数和其它参数的满条件分布.利用Gibbs抽样与Metropolis-Hastings算法相结合的MCMC方法对各参数的满条件分布分别进行了抽样,把Gibbs样本的均值作为各参数的贝叶斯估计,并且详细介绍了MCMC方法的实施步骤.最后进行了随机模拟试验,试验结果表明各参数贝叶斯估计的精度都较高.     

零膨胀集群数据层次回归模型的贝叶斯推断

零膨胀Poisson回归模型是研究零观测值过多的计数数据的常用工具,本文提出了一类拟合具有这类特征的集群数据的层次零膨胀泊松回归模型,并给出了相应的贝叶斯推断方法,参数估计通过Gibbs抽样获得,模型比较与选择则通过拟合优度检验与BIC准则实现.最后,利用一个船舶受损事故数据来展示本文方法的实现及应用.     

纵向数据下半参数联合均值协方差模型的贝叶斯分析

基于改进的Cholesky分解,研究分析了纵向数据下半参数联合均值协方差模型的贝叶斯估计和贝叶斯统计诊断,其中非参数部分采用B样条逼近.主要通过应用Gibbs抽样和Metropolis-Hastings算法相结合的混合算法获得模型中未知参数的贝叶斯估计和贝叶斯数据删除影响诊断统计量.并利用诊断统计量的大小来识别数据的异常点.模拟研究和实例分析都表明提出的贝叶斯估计和诊断方法是可行有效的.     

0-1膨胀几何分布回归模型及其应用

在制造缺陷、专利申请、道路安全和公共卫生等应用领域,经常会出现较多的零观测值和一观测值.采用传统的泊松回归或负二项回归模型往往会过低地估计零观测值和一观测值出现的概率,数据拟合的效果欠佳.文章提出了0-1膨胀几何分布回归模型,巧妙地引入隐变量并进行极大似然估计和贝叶斯估计,基于数据扩充策略分别采用最大期望(EM)算法和Metropolis-Hastings抽样算法对回归参数向量进行估计.在不同的样本容量下进行数值模拟,并对两种估计方法的性能进行评价.研究表明,对于博士研究生发表论文数量的数据集,0-1膨胀几何分布回归模型能够达到更好的拟合效果.     

零调整分位数回归模型在车辆保险索赔额中的研究与应用

车辆保险产品的定价一般会考虑保单持有人的索赔概率和期望索赔额等两个因素,零调整逆高斯回归模型作为解决这类问题的一个有力工具,由于变量分布的限定,从而具有一定的局限性.针对该问题,本文基于零调整逆高斯回归模型和分位数回归模型的思想,提出零调整分位数回归模型,并结合实际数据进行了拟合分析.与零调整逆高斯回归模型拟合的结果比较表明,零调整分位数回归模型可以作为研究车辆保险中索赔额的一个有力工具.     

第六讲 扫除算法

1.引言 扫除算子(Sweep operator)是对矩阵的一种变换运算,也称为扫除变换或扫除算法.其实质是高斯──约唐消去法求逆矩阵的一种改进算法. 扫除算法可用于求解线代数方程组,计算矩阵的逆阵(包括广义道),也可以用于计算行列式的值.在统计计算中,扫除算法有很丰富的统计含义,它是回归分析、判别分析及各种逐步算法的基础.本文将从矩阵代数运算和统计含义两个方面对扫除算法作一个简要的介绍.最后还给出FORTRAN程序. 2.从回归计算谈起 设线性回归模型为Y=Xβ+e　　　(2.1)其中 X,Y可为观测数据,β为回归系数,e为随机误差.通常假设有m个自…     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

A Gaussian Mixed Model for Learning Discrete Bayesian Networks

In this paper, we address the problem of learning discrete Bayesian networks from noisy data. A graphical model based on a mixture of Gaussian distributions with categorical mixing structure coming from a discrete Bayesian network is considered. The network learning is formulated as a maximum likelihood estimation problem and performed by employing an EM algorithm. The proposed approach is relevant to a variety of statistical problems for which Bayesian network models are suitable—from simple regression analysis to learning gene/protein regulatory networks from microarray data.     

Robust regression for mixed Poisson–Gaussian model

This paper focuses on efficient computational approaches to compute approximate solutions of a linear inverse problem that is contaminated with mixed Poisson–Gaussian noise, and when there are additional outliers in the measured data. The Poisson–Gaussian noise leads to a weighted minimization problem, with solution-dependent weights. To address outliers, the standard least squares fit-to-data metric is replaced by the Talwar robust regression function. Convexity, regularization parameter selection schemes, and incorporation of non-negative constraints are investigated. A projected Newton algorithm is used to solve the resulting constrained optimization problem, and a preconditioner is proposed to accelerate conjugate gradient Hessian solves. Numerical experiments on problems from image deblurring illustrate the effectiveness of the methods.     

Bayesian Inversion of Piecewise Affine Operators in a Gaussian Framework

Piecewise affine inverse problems form a general class of nonlinear inverse problems. In particular inverse problems obeying certain variational structures, such as Fermat's principle in travel time tomography, are of this type. In a piecewise affine inverse problem a parameter is to be reconstructed when its mapping through a piecewise affine operator is observed, possibly with errors. A piecewise affine operator is defined by partitioning the parameter space and assigning a specific affine operator to each part. A Bayesian approach with a Gaussian random field prior on the parameter space is used. Both problems with a discrete finite partition and a continuous partition of the parameter space are considered.

The main result is that the posterior distribution is decomposed into a mixture of truncated Gaussian distributions, and the expression for the mixing distribution is partially analytically tractable. The general framework has, to the authors' knowledge, not previously been published, although the result for the finite partition is generally known.

Inverse problems are currently of large interest in many fields. The Bayesian approach is popular and most often highly computer intensive. The posterior distribution is frequently concentrated close to high-dimensional nonlinear spaces, resulting in slow mixing for generic sampling algorithms. Inverse problems are, however, often highly structured. In order to develop efficient sampling algorithms for a problem at hand, the problem structure must be exploited.

The decomposition of the posterior distribution that is derived in the current work can be used to develop specialized sampling algorithms. The article contains examples of such sampling algorithms. The proposed algorithms are applicable also for problems with exact observations. This is a case for which generic sampling algorithms tend to fail.     

A mixture of generalized hyperbolic factor analyzers

The mixture of factor analyzers model, which has been used successfully for the model-based clustering of high-dimensional data, is extended to generalized hyperbolic mixtures. The development of a mixture of generalized hyperbolic factor analyzers is outlined, drawing upon the relationship with the generalized inverse Gaussian distribution. An alternating expectation-conditional maximization algorithm is used for parameter estimation, and the Bayesian information criterion is used to select the number of factors as well as the number of components. The performance of our generalized hyperbolic factor analyzers model is illustrated on real and simulated data, where it performs favourably compared to its Gaussian analogue and other approaches.     

Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors

We study MCMC algorithms for Bayesian analysis of a linear regression model with generalized hyperbolic errors. The Markov operators associated with the standard data augmentation algorithm and a sandwich variant of that algorithm are shown to be trace-class.     

Bayesian quantile regression model for claim count data

Quantile regression model estimates the relationship between the quantile of a response distribution and the regression parameters, and has been developed for linear models with continuous responses. In this paper, we apply Bayesian quantile regression model for the Malaysian motor insurance claim count data to study the effects of change in the estimates of regression parameters (or the rating factors) on the magnitude of the response variable (or the claim count). We also compare the results of quantile regression models from the Bayesian and frequentist approaches and the results of mean regression models from the Poisson and negative binomial. Comparison from Poisson and Bayesian quantile regression models shows that the effects of vehicle year decrease as the quantile increases, suggesting that the rating factor has lower risk for higher claim counts. On the other hand, the effects of vehicle type increase as the quantile increases, indicating that the rating factor has higher risk for higher claim counts.     

Constitutive model error and uncertainty quantification

This paper proposes a novel Bayesian strategy for high-dimensional inverse problems in the context of elastostatics. Apart from parametric uncertainties, model inadequacies and, particularly, constitutive model errors, are also addressed. This is especially important in biomedical settings when the inferred material properties will be used to make decisions/diagnoses. Traditional approaches use an additional regression model (e.g., Gaussian process), added to the model output or within a submodel to account, for an underlying model error. This can violate physical constraints and becomes impractical in high dimensions. In this work we unfold conservation and constitutive laws to estimate model discrepancies. In addition, efficient Bayesian strategies are employed. In elastography, the accurate identification of the unknown mechanical properties of a tissue as well as the associated uncertainty, can greatly assist noninvasive, medical diagnosis. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poisson-Lindley回归模型基于EM算法的诊断分析(英文)

利用EM算法研究了来自于Lindley分布权重的混合Poisson模型,即Poisson-Lindley回归模型,从而利用基于完全数据似然函数的条件期望进行统计诊断和局部影响分析,得到了几个有用的诊断统计量,并用一个数值实例说明了所得统计量的有效性.