【作者】 李立；

【导师】 孙东初；

【作者基本信息】 华东师范大学 ， 统计学， 2023， 硕士

【摘要】 贝叶斯光滑样条模型借助贝叶斯方法的“先验-观测-后验”的推断思想、MCMC等高效抽样方法、以及光滑样条在拟合复杂函数时广泛的应用与灵活性,其已经成为金融、生物统计、医学等领域中有力的统计工具。贝叶斯光滑样条模型得到越来越多的关注得益于贝叶斯方法以及光滑样条的快速发展。首先,贝叶斯方法在近几十年得到长足发展:不仅在先验选取上,客观贝叶斯方法降低了主观性带来的弊端;另外,随着计算能力的提高和许多高效算法的提出(如MCMC算法,Gibbs抽样等),后验抽样速度也得到飞速提升。其次,因为其适用场景广泛性与灵活性,即利用特殊条件的控制点求解光滑曲线或曲面,光滑样条方法成为了使用非参数方法拟合求解复杂函数的重要手段。首先,本文介绍了贝叶斯学派在后验抽样速度提升、客观性准则上取得的长足发展,也举例说明了贝叶斯在统计推断和解决小样本量问题时的优势;随后,分别介绍了两种光滑样条的频率学方法,即自然光滑样条方法和可再生希尔伯特空间法;同时,本章也介绍了两种贝叶斯光滑样条方法,即高斯过程先验法和部分信息正态先验方法。最后,本章着重研究了频率学派和贝叶斯学派光滑样条方法的等价性,证明了Speckman&Sun于2003年提出的部分信息正态先验方法与其他三种方法的等价性。其次,本文提出了两种不同的多元光滑样条模型:(1)基于收缩的逆威沙特先验分布的多元光滑样条模型,本文放宽同方差假设条件,结合实际应用场景,在模型中考虑异方差条件,提出基于收缩的逆威沙特先验分布的异方差多元光滑样条模型。(2)本文还提出可交换先验分布方法求解多元光滑样条模型。使用该先验分布可以降低计算维度,并且其满足许多特殊模型,拓宽了实际应用场景。最后,本文基于2020年部分省份的新型冠状病毒疫情数据,使用贝叶斯方法和自然光滑样条等非贝叶斯方法进行疫情初期发展趋势曲线拟合,并且利用不同评价指标进行比较,从而得出了使用贝叶斯方法优于自然光滑样条方法等非贝叶斯方法的结论。另外,本文使用数值模拟的方式验证了基于逆威沙特先验的异方差贝叶斯光滑样条模型具备很好的拟合优度,在均方误差、拟合优度上优于自然光滑样条方法。更多还原

【Abstract】 On one hand,Bayesian smoothing spline model is based on the idea of“a priori-observation-posteriori”of Bayesian method,and usually combines with efficient sam-pling methods such as Markov Chain Monte Carlo(MCMC).On the other hand,it has a wide application in fitting complex functions.Therefore,Bayesian smoothing spline model has become a powerful statistical tool in many fields,such as finance,biostatis-tics and medical fields.The Bayesian smoothing spline model becomes more and more popular.This is due to the developments of the Bayesian methods and the smoothing spline methods.Firstly,Bayesian methods have been developed significantly in recent decades:In the priori selection,the use of objective Bayesian methods avoid the disa-vantages cauded by the subjectivity;The improvement of computing efficiencies and the proposal of many efficient algorithms(such as MCMC algorithm,Gibbs sampling,etc.)improve the speed of a posterior sampling.Secondly,smoothing spline methods have become a powerful statistical tool for fitting complex functions using non-parametric methods,because of their wide application and flexibility,i.e.,using nodes with special conditions to fit smoothing curves or surfaces.In this paper,we introduce the significant development made by the Bayesians in the efficiency of sampling the posterior and objectivity criterion.Also,we give ex-amples to illustrate the advantages of Bayesian in statistical inference and solving the problems of small sample size.We then introduce not only two smoothing spline meth-ods on the framework of frequency which are natural smoothing spline and reproducing kernel Hilbert space method but also two bayesian smppthing spline methods which are Gaussian process prior method(Wahba,1978)and partially normal prior distribution method(Speckman&Sun,2003).Last but not least,we focus on the equivalence of these four classical methods.In this chapter,we mainly prove that the partially normal prior distribution method(Speckman&Sun,2003)is equivalent with the other three methods.Furthermore,we propose two different kinds of multivariate smoothing spline models:One is the heteroskedasticity multivariate smoothing spline model based on the shrinkage inverse Wishart prior distribution(HSIW).In this method,we improve the multivariate smoothing spline model with the inverse Wishart prior distribution by relaxing the homovariance assumption,combining the real-world application scenarios,and considering the heteroscedasticity condition.The other one is multivariate smooth-ing spline with the commutative prior.The use of the commutative prior can not only reduce the calculation dimension,but also it meets many special models,which broad-ens the practical application scenarios.Finally,we analyse a data set which contains the numbers of confirmed,cured,suspected,and dead people in some provinces at the beginning of COVID-19.We use Bayesian methods and non-Bayesian methods such as natural smooth spline to fit the curve of the initial epidemic development trend,and compares them using different evaluation indexes,so as to conclude that using Bayesian methods is better than non-Bayesian methods.In addition,we conduct several Monte carlo simulations to compare the behavior of HSIW and the natural smoothing spline method in fitting models using the mean squared error(MSE)and R squares(R~2).The results demonstrate that HSIW has a good fit and outperforms the natural smoothing spline method.更多还原