南方科技大学知识苑(SUSTech KC): ALGORITHM EXPLORATION FOR MODELING A STATIONARY COUNT TIME SERIES

题名	ALGORITHM EXPLORATION FOR MODELING A STATIONARY COUNT TIME SERIES
其他题名	平稳计数时间序列建模的算法探索
姓名	刘晓晓
姓名拼音	LIU Xiaoxiao
学号	12032012
学位类型	硕士
学位专业	0701 数学
学科门类/专业学位类别	07 理学
导师	焦熙云
导师单位	统计与数据科学系
论文答辩日期	2022-05-05
论文提交日期	2022-06-21
学位授予单位	南方科技大学
学位授予地点	深圳
摘要	This project aims to propose a particle-filtering-based Markov Chain Monte Carlo (MCMC) algorithm to estimate the parameters of a new type of count time series models. Count time series model is vital in random process and time series analysis. However, the corresponding parameter estimation can be challenging. Jia et al. (2021) developed a more general category of count time series models by transforming a Gaussian process. Specifically, this technique makes distributional transformation on a latent Gaussian process. The resulting stationary time series has flexible correlation structures and can specify as its marginal any common count distribution, such as the classical Poisson, generalized Poisson, negative binomial, and binomial. For estimating the parameters in the new type of models, Jia et al. deployed three routine methods, which are Gaussian pseudo-likelihood estimation, Yule-Walker estimation, and particle filtering/sequential Monte Carlo (PF/SMC) likelihood estimation respectively. The Gaussian pseudo-likelihood and Yule-Walker estimations are derived from a new type of Hermite expansion. The results of simulation and real-data studies show that the PF/SMC method is less biased and more accurate than the other two methods. Furthermore, because PF/SMC does not require the distribution of the data to be close to Gaussian, it can be applied to more general scenarios. However, PF/SMC still faces challenges in many common situations, such as when trying to improve the efffficiency, when the parameter space is high-dimensional, etc. MCMC algorithms are widely used methods for sampling from the posterior distributions under the Bayesian statistical framework, and Metropolis-Hastings (MH) is one of the most popular MCMC methods. Due to the use of prior distributions, Bayesian inference naturally avoids the problem of overfitting, and provides a more flexible framework. MCMC algorithms do not need high-dimensional integration, easy to implement and suitable for the inference of high-dimensional parameters and complex models. It is not easy to directly apply MCMC for sampling from the posterior of the new count times series model. Therefore, in this project, we combine PF with the MCMC algorithm, that is, using PF to approximate the likelihood of the model and construct the MH algorithm based on this approximated likelihood. This PF-based MH algorithm proposed by us would be a new and more flexible inference method for the new count times series model of Jia et al. We did simulation studies based on the model with AR(1) as the latent Gaussian process and classical Poisson as the marginal distribution. The results show that our newly proposed PF-base MHmethod has good convergence properties and is accurate in estimation. However, there is still space for it to improve in computational efffficiency.
其他摘要	本项目旨在建立一种基于粒子滤波 (particle filtering, PF) 的马尔可夫链蒙特卡罗（Markov chain Monte Carlo, MCMC）算法，以对一类新型的计数时间序列模型进行参数估计。计数时间序列模型在随机过程和时间序列分析中具有重要意义，但此类模型的参数估计颇具挑战性。Jia et al. (2021) 通过对高斯过程进行转换，建立了一类更具一般性的平稳计数时间序列模型。具体而言，他们是对一个潜在的高斯过程进行概率密度分布函数变换，由此产生的平稳时间序列具备灵活的相关性结构，能够使用任何一种常见的技术分布为序列的边际分布，如经典泊松、广义泊松、负二项以及二项分布等。Jia et al. 采用了三种常规算法对这类新模型进行参数估计。这三种算法分别是高斯伪似然方法、Yule-Walker 估计方法、以及粒子滤波/序列蒙特卡罗（sequential Monte Carlo, SMC）似然方法，其中高斯伪似然和 Yule-Walker 方法是在一种新式 Hermite 展开基础上推导的。仿真和实证分析显示， PF/SMC 方法较其他两种方法估计的偏差更小，准确性更高，且由于没有数据分布与高斯相近的要求，适用范围更广；但 PF/SMC 在计算效率以及高维数据空间应用等方面仍存在改进空间。MCMC 算法是在贝叶斯统计框架下从后验分布中抽样的常用算法，Metropolis-Hastings (MH) 是最常见的 MCMC 算法之一。贝叶斯统计由于使用了先验分布，自然地避免了过拟合问题，提供了一个更灵活的框架。而 MCMC 算法避免了高维积分，且计算简便，适用于高维参数和复杂模型推断。将 MCMC 直接用于计数时间序列模型的后验抽样有一定困难，因此，本项目将 PF 与 MCMC 算法结合，用 PF 近似模型的似然函数，而基于此近似似然函数建立 MH 算法，由此为 Jia et al. 提出的新型平稳计数时间序列模型提供了一种新的更为灵活的推断算法。我们用潜在高斯过程为 AR(1)、边际分布为经典泊松的新型时间计数序列为模型进行了仿真实验，结果显示我们的新式基于 PF 的 MCMC 算法收敛效果良好、估计准确性高，但在计算效率上仍有提升空间。
关键词	Markov Chain Monte Carlo Algorithm Stationary Series Particle Filtering PF-based Markov Chain Monte Carlo
语种	英语
培养类别	独立培养
入学年份	2020
学位授予年份	2022-07
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所在学位评定分委会	统计与数据科学系
国内图书分类号	O212.1
来源库	人工提交
成果类型	学位论文
条目标识符	http://sustech.caswiz.com/handle/2SGJ60CL/336385
专题	理学院_统计与数据科学系
推荐引用方式 GB/T 7714	Liu XX. ALGORITHM EXPLORATION FOR MODELING A STATIONARY COUNT TIME SERIES[D]. 深圳. 南方科技大学,2022.

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