| The era of big data has produced a large amount of complex data.Common complex data includes:functional data,multi-source data,high-dimensional data,network data,etc.As an important unsupervised method,cluster analysis benefits exploratory analysis of these complex data to further understand the hidden common patterns and analyze the mechanism.This article constructs a clustering model for functional data,network data,and multi-source heterogeneous data,and studies the effects of the model from both theoretical and applied aspects.The main contents are as follows.(1)We study the biclustering model of functional data.In biological genetic data,clustering of samples can reveal the subtypes of a certain disease,while clustering of genes leads to different genomes,and different genomes can reflect different synergistic mechanisms.Biclustering is one unsupervised method to conduct clustering on samples and variables simultaneously.This article studies dynamic functional data,assuming that the observations of each variable in each sample are generated by the latent mean function,and there exists a biclustering structure across these functions.We propose a novel biclustering method via penalized fusion.By adopting two-level penalties,including a smoothness penalty and a fusion penalty,we can aggregate similar rows and similar columns and then a clustering problem is transformed into a penalized estimation problem.We have established theoretical guarantees to ensure cluster consistency and verified the good performance of the method in numerical simulations and application analysis.(2)We study the community influence model of network data.The network autoregressive(NAR)model is widely used to study the influences between nodes in social network.However,the NAR model can only identify the average i’nfluence effect within the network,while there may exist heterogeneity for real datasets,that is,different nodes show different influence effects,and these effects can be divided into several clusters(or communities).Therefore,to further explore the cluster structure of the influences within social network,we assume that the nodes belonging to the same community have the same influence parameter,and the nodes belonging to different communities have different influence parameters,so as to cluster the nodes.In the modeling process,firstly,we construct an objective function based on quasi-maximum likelihood estimation and fusion penalty,and then estimate unknown parameters and identify the number of communities;secondly,we establish the theoretical properties,including a homogeneous model(the number of communities is 1)and a heterogeneous model(the number of communities is greater than 1);finally,we verify the good performance of the model through numerical simulations and two real data analyses.(3)We study the clustered integrative model of multi-source data.Due to privacy constraints,it is difficult to take integrative analysis for multi-source data with raw data.This article adopts the DataSHIELD framework and constructs a new loss function by applying LSA to the original loss function.After that,we are able to take integrative analysis based on the summary statistics(such as local estimators,hessian matrices,gradients and so on)transferred by local machines.Besides,we simultaneously consider the sparsity of high-dimensional data and the heterogeneity of multi-source data.On one hand,we regard the same variables in each dataset as a group and assume that the variables in the same group are all important or unimportant variables,that is,"all in or all out";on the other hand,we assume there exists a cluster structure of multiple datasets,that is,the coefficient vectors of all datasets can be divided into several clusters,and the datasets belonging to the same cluster share the same coefficient vector.In theory,we have established the estimation consistency,model selection consistency,and cluster consistency of the proposed estimators.Numerically,we have verified the good performance of the proposed model by simulations and real data analysis. |
