摘要
本文综述近年来因子模型研究的最新进展及其在统计机器学习中的应用.因子模型通过较少的因子实现降维,并为协方差矩阵提供了一种低秩加稀疏的结构,不仅受到高维数据分析领域的关注,也被广泛应用于计量经济学、数量金融学、基因组学、神经科学和图像处理等许多科学、工程及人文社科领域的研究中.本文系统阐述利用主成分分析方法提取潜在因子、估计因子载荷、异质结构与整体协方差矩阵的统计推断方法,这套方法被证明可以有效应对当前大数据所表现出的高维性、强相关性、厚尾性和异质性等重大挑战;另外,还重点介绍了高维因子模型在处理协方差矩阵估计、模型选择和多重检验等高维统计学习问题中的作用;最后,通过几个应用实例说明因子模型与现代机器学习问题之间的密切联系,其中包括当下流行的网络分析和低秩矩阵还原等.
This paper reviews the recent developments on factor model and its applications to statistical machine learning.The factor model reduces the dimensionality of variables,and provides a low-rank plus sparse structure for the high-dimensional covariance matrices.Therefore,it attracts much attention in high-dimensional data analysis,and has been widely applied in many fields of sciences,engineering,humanities and social sciences,including economics,finance,genomics,neuroscience,machine learning,and so on.We elaborate how to use principal component analysis method to extract latent factors,estimate their associated factor loadings,idiosyncratic components,and their associated covariance matrices.These methods have been proven to effectively cope with the challenges of big data,such as high dimensionality,strong dependence,heavy-tailed variables,and heterogeneity.In addition,we also focus on the role of the factor model in dealing with high-dimensional statistical learning problems such as covariance matrix estimation,model selection,multiple testing,and prediction.Finally,we illustrate the innate relationships between factor models and modern machine learning problems through several applications,including network analysis,matrix completion,ranking,and mixture models.
作者
陈钊
范剑青
王丹
Zhao Chen;Jianqing Fan;Christina Dan Wang
出处
《中国科学：数学》
CSCD
北大核心
2020年第4期447-490,共44页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:11690014,11690015,71991471,71991475,U1811461和11901395)
上海浦江计划(批准号:19PJ1408200和19PJ1400900)资助项目。
