| With the advance of science and technology,fields such as e-commerce,finance,information retrieval,wireless communications,and biomedical imaging must account for an increasing number of variables in the available data.This data can be characterized as high-dimensional,which means the number of observed features exceeds the available observations.Traditional multivariate statistical analysis is mostly applicable to low dimensional data sets.When dealing with high-dimensional data,many multivariate analysis tools based on sample covariance matrix,such as likelihood ratio test,factor analysis(FA),principal component analysis(PCA),and so on,are facing huge challenges.The problem of covariance matrix estimation has always played a key role not only for multivariate analysis but also when inferring the underlying latent structures from the observed data.In addition to the covariance matrix,the precision matrix under gaussian graph model also conveys the correlation information.Analyzing the correlation network can help us better understand the role of each feature in the system.Especially in the field of biology,the association network analysis of high-dimensional omics data(genomics,metagenomics,proteomics and metabolomics,etc.)provides a more comprehensive perspective for understanding living organisms and ecological environment,and plays a crucial role in maintaining individual health and promoting the development of medical treatment.In practical application,we often encounter distinct classes of high-dimensional data,such as different tissues and different age groups.Since there is always an internal relationship among classes,how to exploit the similarity between different classes and identify the class-specific association has become the focus of our research.In this thesis,we study the joint estimation of covariance matrices and precision matrices of multi-class data,not only discussing the paired correlation between features,but also exploring the conditional correlation among features,and comprehensively analyze the structural characteristics of multi-class data association networks.In addition,timevarying data is also a hot topic in recent years,such as sensor monitoring data,stock trading data and clinical treatment data.Compared with static data,time-varying data has a greater amount of data and needs more complex processing methods.We intend to build dynamic networks of time-varying data to reconstruct the course of events and discover the key factors that cause dynamic changes.This thesis aims at the correlation analysis and network construction for high dimensional data.It can be summarized into five chapters as follows.In Chapter 1,we review firstly four kinds of covariance matrix estimation methods for high-dimensional data,including Banding,Tapering,Shrinkage and Thresholding.Then we present the existing precision matrix estimation methods in the gaussian graph model,and emphasize the joint estimation for precision matrices based on the likelihood function.At the end of this chapter,statistical methods for differential network analysis are mainly introduced,which include not only the differential network between multiple classes but also the dynamic network for time-varying data.In Chapter 2,we propose a joint adaptive thresholding estimate(JATE)for multiclass high-dimensional data covariance matrices,and the asymptotic results of the estimation are derived.Compared with the thresholding method for a single class,JATE takes more account of the structural similarity between distinct classes and improves the accuracy to detect the differential edges.Although the joint adaptive thresholding estimator has been shown to enjoy good asymptotic properties for estimating multiple large covariance matrices,its positive definiteness property can be easily violated in practice.In order to solve the problem,we further propose a joint positive-definite lasso estimation and give an effective alternating direction method of multipliers(ADMM)algorithm.The statistical properties of the estimation under regularity conditions are studied.In Chapter 3,we study the joint network inference for multi-class high-dimensional data with common and specific hub features.Some studies have shown that scale-free topology exists in many complex networks in the real world,such as metabolic networks,protein-protein interactions and genetic regulatory networks.The number of edges incident to each vertex(degree)obeys the power-law distribution.In view of the structural characteristics,we propose the extended joint hub graphical lasso(EDOHA)to capture more complex interactions and identify class-specific hubs.The ADMM algorithm to solve the optimization problem is given and its convergence is proved.We also derive the theoretical results of reducing the tuning parameter space to improve computational efficiency.Simulation results show that EDOHA has remarkable advantages in recognizing class-specific hubs over the existing comparable methods.The result of this chapter indicates that EDOHA is the proper choice if it is not sure whether the real networks have class-specific hubs or not.In Chapter 4,we focus on time-varying data and directly estimate the differential network between adjacent stages.For different types of dynamic network structures,we encode different types of penalty functions to capture network evolutionary patterns.Kernel density estimation and regularization are combined to construct the model and an effective ADMM algorithm is established.We also deduce the consistency of the estimator under exponential-tail and polynomial-tail conditions.Simulation results show that the model outperforms the existing methods.In Chapter 5,we summarize the main content and innovation points of this thesis and put forward the prospects of the work in the future. |
