高维矩阵数据的稳健差分网络估计及分类

[14]，经过对比，我们发现稳健估计方法较非稳健估计方法的分类结果更加精确。图6非稳健方法分类展示图图7稳健方法分类展示图综合分析对比非稳健估计和稳健估计方法得到的差分网络和分类结果，我们发现稳健估计方法受异常数据影响较小，能更稳定地对AD组和NC组进行估计，且分类结果更加精确。结论本文聚焦于高维矩阵数据的精度矩阵估计和分类问题，回顾了Kronecker结构下特征相关矩阵的Kendall相关估计、矩阵正态分布和稀疏矩阵图模型方法等内容，提出了稳健差分网络估计及分类方法，使用模拟数据和fMRI数据集进行实验，验证了该方法的正确性和有效性。我们放松正态分布假设，将Kronecker乘积结构应用于矩阵数据，使用基于Kendall相关性理论改进的CLIME稳健估计方法同时得到差分网络并对数据进行分类。模拟数据实验证明，在AR模型，枢纽结构和AR模型，小世界结构两种场景的和维度下，稳健估计方法和非稳健估计方法的TPR平均值具有可比性，但稳健估计方法的TNR平均值显著高于非稳健估计方法，随着异常程度增大，稳健估计方法的TPR和TNR平均值下降更缓慢，受异常影响较小。fMRI数据实验证明，在面对异常值时，稳健估计方法能更加稳定地估计AD组和NC组的数据，且分类结果更精确。模拟数据实验和fMRI数据实验均证明了稳健差分网络估计及分类方法地有效性。未来，我们可以拓展模型至张量数据框架以捕捉多模态信息，优化CLIME算法并行计算架构以应对超高维基因组数据挑战，也可推动本方法在个性化医疗多组学整合等领域的跨学科应用，助力高维矩阵数据和医学研究进一步发展，为社会贡献力量。参考文献Lauritzen,S.L.GraphicalModels,Oxford:ClarendonPress,1996.Yuan,M.,andLin,Y.“ModelSelectionandEstimationintheGaussianGraphicalModel,”Biometrika,2007,94,19–35.Fan,J.,Feng,Y.,andWu,Y.“NetworkExplorationviatheAdaptiveLASSOandSCADPenalties,”TheAnnalsofAppliedStatistics,2009,3,521–541.Peng,J.,Wang,P.,Zhou,N.,andZhu,J.“PartialCorrelationEstimationbyJointSparseRegressionModel,”JournaloftheAmericanStatisticalAssociation,2009,104,735–746.]L.Wallbacks,U.E.B.Norden,Multivariatedataanalysisofinsitupulpkineticsusing13ccp/masnmr,J.WoodChem.Technol,2006,9(2):235–249.T.Sejnowski,S.Makeig,A.Delorme,EnhanceddetectionofartifactsinEEGdatausinghigher-orderstatisticsandindependentcomponentanalysis,Neuroimage,2007,34(34):1443–1449.M.D.Fox,M.E.Raichle,Spontaneousfluctuationsinbrainactivityobservedwithfunctionalmagneticresonanceimaging.,Nat.Rev.Neurosci,2007,8(9):700–711.G.K.Smyth,Limma:Linearmodelsformicroarraydata,Bioinform.Comput.Biol.Sol.UsingRBiocond,2011,397–420.D.Kong,B.An,J.Zhang,H.Zhu,L2rm:Low-ranklinearregressionmodelsforhigh-dimensionalmatrixresponses,J.Amer.Statist.Assoc.(just-accepted),2018,1–55.S.E.Medland,N.Jahanshad,B.M.Neale,P.M.Thompson,Whole-genomeanalysesofwhole-braindata:workingwithinanexpandedsearchspace,NatureNeurosci,2014,17(6):791.H.Hung,C.-C.Wang,MatrixvariatelogisticregressionmodelwithapplicationtoEEGdata,Biostatistics,2012,14(1):189–202.S.L.Teng,H.Huang,Astatisticalframeworktoinferfunctionalgenerelationshipsfrombiologicallyinterrelatedmicroarrayexperiments,J.Amer.Statist.Assoc,2009,104(486):465–473.F.Han,H.Liu,Scale-invariantsparsePCAonhighdimensionalmeta-ellipticaldata.,J.Amer.Statist.Assoc,2014,109(505):275–287.Kendall,M.G.(1962).RankCorrelationMethods,3rded.Griffin&Co,London.Sen,P.K.(1968).EstimatesoftheregressioncoefficientbasedonKendall’stau.J.Amer.Statist.Assoc.631379–1389.MR0258201CHENH,GUOY,HEY,etal,Simultaneousdifferentialnetworkanalysisandclassificationformatrix-variatedatawithapplicationtobrainconnectivity,Biostatistics,2022,23(3):967-989.RYALI,S.,CHEN,T.,SUPEKAR,K.ANDMENON,V.,EstimationoffunctionalconnectivityinfMRIdatausingstabilityselection-basedsparsepartialcorrelationwithelasticnetpenalty.NeuroImage,2012,59,3852–3861.HE,Y.,ZHANG,L.,JI,J.ANDZHANG,X.,Robustfeaturescreeningforellipticalcopularegressionmodel.JournalofMultivariateAnalysis,2019,173,568–582.XIA,Y.ANDLI,L.,Matrixgraphhypothesistestingandapplicationinbrainconnectivityalternationdetection.StatisticaSinica,2019,29,303–328.LIUH,HANF,YUANM,etal.HighdimensionalsemiparametricGaussiancopulagraphicalmodels[J].AnnalsofStatistics,2012,40(4):2293-2326.LENGC,TANGCY.Sparsematrixgraphicalmodels[J].JournaloftheAmericanStatisticalAssociation,2012,107(499):1187-1200.NINGY,LIUH,High-dimensionalsemiparametricbigraphicalmodels,Biometrika,2013,100(3):655–670.LIM,YUANB,2d-lda:Astatisticallineardiscriminantanalysisforimagematrix,PatternRecognit.Lett.,2005,26(5):527–532.YUAN,H.,XI,R.ANDDENG,M.,DifferentialnetworkanalysisviathelassopenalizedD-traceloss.Biometrika,2015,104,755–770.TIAN,D.,GU,Q.ANDMA,J.,Identifyinggeneregulatorynetworkrewiringusinglatentdifferentialgraphicalmodels.NucleicAcidsResearch,2016,44,e140.GRIMES,T.,POTTER,S.S.ANDDATTA,S.,Integratinggeneregulatorypathwaysintodifferentialnetworkanalysisofgeneexpressiondata.ScientificReports,2019,9,5479.BICKEL,P.J.ANDLEVINA,E.,SometheoryforFisher’slineardiscriminantfunction,‘naiveBayes’,andsomealternativeswhentherearemanymorevariablesthanobservations.Bernoulli,2004,10,989–1010.ZHU,J.ANDHASTIE,T.,Classificationofgenemicroarraysbypenalizedlogisticregression.Biostatistics,2004,5,427–443.MAI,Q.,ZOU,H.ANDYUAN,M.,Adirectapproachtosparsediscriminantanalysisinultra-highdimensions.Biometrika,2012,99,29–42.ZHOU,H.ANDLI,L.,Regularizedmatrixregression.JournaloftheRoyalStatisticalSociety:SeriesB(StatisticalMethodology),2014,76,463–483.ZHONG,W.ANDSUSLICK,K.S.,Matrixdiscriminantanalysiswithapplicationtocolorimetricsensorarraydata.Technometrics,2015,57,524–534.MOLSTAD,A.J.ANDROTHMAN,A.J.,Apenalizedlikelihoodmethodforclassificationwithmatrix-valuedpredictors.JournalofComputational&GraphicalStatistics,2019,28,11–22.BICKELPJ,LEVINAE.Regularizedestimationoflargecovariancematrices[J].AnnalsofStatistics,2008,36(1):199-227.FANJ,LIAOY,LIUH.Anoverviewontheestimationoflargecovarianceandprecisionmatrices[J].ProceedingsoftheIEEEInternationalConferenceonMi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