| 摘要: |
| 目的 解决因子分析模型后验分布函数复杂计算的问题。 方法 首先利用变分贝叶斯(Variational Bayes,VB)
方法,对服从指数分布的因子分析模型进行参数估计,其中 VB 方法使用坐标上升变分推断(Coordinate Ascent
Variational Inference,CAVI)算法对参数进行迭代求解,然后与 MCMC(Makov Chain Monte Carlo)方法进行比较,通
过随机模拟揭示了当样本量为 300 时,VB 方法和 MCMC 方法的有效性,绘制 ELBO 图判断了 VB 方法的收敛性,
绘制参数追踪图、自相关图判断了 MCMC 方法的收敛性,并根据预测值和真实值之间的偏差判断两种方法的好
坏,最后对样本量为 713 时的实际数据集进行了实证分析。 结果 通过模拟和实证分析可以看到两种方法都是收敛
的,偏差绝对值均小于 0. 1,但是 VB 方法在估计精度、计算复杂度和运行时间上均优于 MCMC 方法,特别是当样本
容量很大时,VB 方法的优势更加明显。 结论 在因子分析模型中,因子服从指数分布有时是一种合理的选择,与
MCMC 方法相比,变分贝叶斯方法可以有效减少因子分析模型后验分布函数的计算量,估计参数的效果更好。 VB
方法的 3 个优势:第一,由于 VB 方法没有涉及复杂后验分布积分的计算,因此计算上更加简便;第二,由于 VB 方
法基于一个近似分布,特别是当样本量很大时,VB 方法的运行时间远远小于 MCMC 方法;第三,从运算结果来看,
VB 方法的估计精度要优于 MCMC 方法。 |
| 关键词: 变分贝叶斯方法 MCMC 方法 因子模型 参数估计 |
| DOI： |
| 分类号: |
| 基金项目: |
|
| Parameter Estimation of Factor Model Using Variational Bayes Approach |
|
LI Yalei CHEN Bowen LI Xingping
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|
School of Mathematics Yunnan Normal University Kunming 650500 China
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| Abstract: |
| Objective This study aims to address the challenge of complex calculations for the posterior distribution
function in the factor analysis model. Methods Initially the variational Bayes VB method is employed to estimate the
parameters of the factor analysis model where the factors follow an exponential distribution. The VB method utilizes the
coordinate ascent variational inference CAVI algorithm for iterative parameter solving. Subsequently a comparison is
made with the MCMC Makov Chain Monte Carlo method. Through random simulations the effectiveness of both the VB
and MCMC methods is evaluated when the sample size is 300. The ELBO plot is used to assess the convergence of the VB
method while the parameter trace plot and autocorrelation plot are used to determine the convergence of the MCMC
method. The performance of the two methods is judged based on the deviation between the predicted and true values.Finally an empirical analysis is carried out on an actual dataset with a sample size of 713. Results Simulation and
empirical analyses indicate that both methods converge with the absolute deviations being less than 0. 1. Nevertheless
the VB method outperforms the MCMC method in terms of estimation accuracy computational complexity and running
time. The advantages of the VB method become more prominent when dealing with large sample sizes. Conclusion In the
factor analysis model assuming that factors follow an exponential distribution can be a reasonable option. Compared to the
MCMC method the variational Bayes method effectively reduces the computational burden associated with the posterior
distribution function of the factor analysis model and provides more accurate parameter estimates. The VB method offers
three key advantages First it simplifies calculations as it avoids the integration of complex posterior distributions.
Second being based on an approximate distribution the VB method significantly reduces running time especially for
large-scale samples compared to the MCMC method. Third the VB method demonstrates higher estimation accuracy than
the MCMC method. |
| Key words: variational Bayes method MCMC method factor model parameter estimation |
