| Graphical model is a common method to estimate network structure and describe the dependence relationship between variables.It is closely related to the inverse matrix of covariance matrix.In the case of higher dimensions,the sample size of the data is generally relatively small,and the graphical model based on a single study(dataset)typically bears large uncertainty and low power in detecting the significant connections in the corresponding graph.However,with the development of big data storage technology,it is very possible to obtain datasets in similar fields,and the same set of variables from different data sources may have some similar dependency structures.Therefore,we can use transfer learning to mine information from similar or related auxiliary studies(datasets)to improve the statistical efficiency of target study.In this paper,we consider transfer learning problem in estimating undirected semiparametric graphical model.We propose an algorithm called Trans-Copula-CLIME for estimating an undirected graphical model while uncovering information from similar auxiliary studies,characterizing the similarity between the target graph and each auxiliary graph by the sparsity of a divergence matrix.The proposed method extends the assumption of Gaussian distribution to the assumption of Gaussian Copula distribution,and uses the non-parametric rank estimation of correlation coefficients to achieve the robustness.It has a wide range of application and robust performance.We expounds the performance of Trans-Copula-CLIME from three aspects:theoretical property,numerical simulation and empirical study.First,we establish the convergence rate of the Trans-Copula-CLIME estimator under some mild conditions,which provides the statistical guarantee of the algorithm.The theoretical results show that if the similarity between the auxiliary studies and the target study is sufficiently high and the number of informative auxiliary samples is sufficiently large,the Trans-Copula-CLIME estimator shows great advantage over the existing non-transfer-learning ones.Secondly,we conducted a variety of numerical simulations to compare the statistical efficiency of TransCopula-CLIME and five other graphical model estimation methods.Simulation studies show that Trans-Copula-CLIME estimator has better performance especially when data are not from Gaussian distribution.Finally,we explore the performance of Trans-CopulaCLIME on real datasets.We applied this algorithm to functional magnetic resonance imaging datasets related to Attention Deficit Hyperactivity Disorder(ADHD)from various sites of the ADHD-200 Global Competition to study brain connectivity patterns in patients with ADHD,and found some important brain regions.In summary,we use transfer learning to integrate the information in the auxiliary studies and propose a method to estimate the high-dimensional semiparametric graphical model with statistical guarantee which relaxes the restrictive Gaussian distribution assumption of the data.The work in this paper extends the research of graphical model on transfer learning.The theoretical results and numerical simulations show that the TransCopula-CLIME estimator achieves a faster convergence rate than the estimator from a single study.Hence,if we have a sufficient number of auxiliary samples similar to the target study and are not sure whether the data satisfies the Gaussian assumption,we can always choose the Trans-Copula-CLIME to estimate the precision matrix of the target study and this estimator performs no worse than the existing ones. |
