2012A New Covariance Matrix Estimation Method in

Package‘dccmidas’October13,2022Type PackageTitle DCC Models with GARCH-MIDAS Specifications in the UnivariateStepVersion0.1.0Description Estimates a variety of Dynamic Conditional Correlation(DCC)models.More in de-tail,the'dccmidas'package allows the estimation of the cor-rected DCC(cDCC)of Aielli(2013)<doi:10.1080/07350015.2013.771027>,the DCC-MIDAS of Colacito et al.(2011)<doi:10.1016/j.jeconom.2011.02.013>,the Asymmet-ric DCC of Cappiello et al.<doi:10.1093/jjfinec/nbl005>,and the Dynamic Equicorrela-tion(DECO)of Engle and Kelly(2012)<doi:10.1080/07350015.2011.652048>.'dccmidas'of-fers the possibility of including standard GARCH<doi:10.1016/0304-4076(86)90063-1>,GARCH-MIDAS<doi:10.1162/REST_a_00300>and Double Asymmetric GARCH-MIDAS<doi:10.1016/j.econmod.2018.07.025>models in the univariate estimation.Fi-nally,the package calculates also the var-cov matrix under two non-parametric models:the Mov-ing Covariance and the RiskMetrics specifications.License GPL-3LinkingTo Rcpp,RcppArmadilloEncoding UTF-8LazyData trueRoxygenNote7.1.1RdMacros RdpackDepends R(>=4.0.0)Imports Rcpp(>=1.0.5),maxLik(>=1.3-8),rumidas(>=0.1.1),rugarch(>=1.4-4),roll(>=1.1.4),xts(>=0.12.0),tseries(>=0.10.47),Rdpack(>=1.0.0),lubridate(>=1.7.9),zoo(>=1.8.8),stats(>=4.0.2),utils(>=4.0.2)Suggests knitr,rmarkdownNeedsCompilation yesAuthor Vincenzo Candila[aut,cre]Maintainer Vincenzo Candila<****************************>Repository CRANDate/Publication2021-03-1510:00:07UTC12cov_eval R topics documented:cov_eval (2)dcc_fit (3)Det (7)ftse100 (7)indpro (8)Inv (9)nasdaq (9)plot_dccmidas (10)sp500 (11)Index13 cov_eval Var-cov matrix evaluationDescriptionEvaluates the estimated var-cov matrix H_t with respect to a covariance proxy,under different robust loss functions(Laurent et al.2013).The losses considered are also used in Amendola et al.(2020).Usagecov_eval(H_t,cov_proxy=NULL,r_t=NULL,loss="FROB")ArgumentsH_t Estimated covariance matrix,formatted as arraycov_proxy optional Covariance matrix,formatted as arrayr_t optional List of daily returns used to calculate H_t.If parameter’cov_proxy’is not provided,then r_t must be included.In this case,a(noise)proxy will beautomatically usedloss Robust loss function to use.Valid choices are:"FROB"for Frobenius(by de-fault),"SFROB"for Squared Frobenius,"EUCL"for Euclidean,"QLIKE"forQLIKE and"RMSE"for Root Mean Squared ErrorsValueThe value of the loss for each tReferencesAmendola A,Braione M,Candila V,Storti G(2020).“A Model Confidence Set approach to the combination of multivariate volatility forecasts.”International Journal of Forecasting,36(3),873-891.doi:10.1016/j.ijforecast.2019.10.001.Laurent S,Rombouts JV,Violante F(2013).“On loss functions and ranking forecasting perfor-mances of multivariate volatility models.”Journal of Econometrics,173(1),1–10.doi:10.1016/ j.jeconom.2012.08.004.Examplesrequire(xts)#open to close daily log-returnsr_t_s<-log(sp500[ 2010/2019 ][,3])-log(sp500[ 2010/2019 ][,1])r_t_n<-log(nasdaq[ 2010/2019 ][,3])-log(nasdaq[ 2010/2019 ][,1])r_t_f<-log(ftse100[ 2010/2019 ][,3])-log(ftse100[ 2010/2019 ][,1])db_m<-merge.xts(r_t_s,r_t_n,r_t_f)db_m<-db_m[complete.cases(db_m),]colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")#list of returnsr_t<-list(db_m[,1],db_m[,2],db_m[,3])#estimationK_c<-144N_c<-36cdcc_est<-dcc_fit(r_t,univ_model="sGARCH",distribution="norm",corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)cov_eval(cdcc_est$H_t,r_t=r_t)[(K_c+1):dim(cdcc_est$H_t)[3]]dcc_fit DCCfit(first and second steps)DescriptionObtains the estimation of a variety of DCC models,using as univariate models both GARCH and GARCH-MIDAS specifications.Usagedcc_fit(r_t,univ_model="sGARCH",distribution="norm",MV=NULL,K=NULL,corr_model="cDCC",lag_fun="Beta",N_c=NULL,K_c=NULL)Argumentsr_t List of daily returns on which estimate a DCC model.Each daily return must bean’xts’object.Note that the sample period of the returns should be the same.Otherwise,a merge is performeduniv_model Specification of the univariate model.Valid choices are:some of the specifica-tions used in the rugarch(ugarchspec)and rumidas(ugmfit)packages.Morein detail,the models coming from rugarch are:model Valid models(currentlyimplemented)are’sGARCH’,’eGARCH’,’gjrGARCH’,’iGARCH’,and’cs-GARCH’.The models implemented from rumidas are:’GM_skew’,’GM_noskew’,’DAGM_skew’,and’DAGM_noskew’distribution optional Distribution chosen for the univariate estimation.Valid choices are:"norm"(by default)and"std",respectively,for the Normal and Student’s t dis-tributionsMV optional MIDAS variable to include in the univariate estimation,if the modelspecificied is a GARCH-MIDAS(GM,Engle et al.(2013))or a Double Asym-metric GM(DAGM,Engle et al.(2013)).In the case of MIDAS-based models,please provide a list of the MIDAS variables obtained from the mv_into_matfunction.If the same MV variable is used,then provide always a list,with thesame(transformed)variable repeatedK optional The number of lagged realization of MV variable to use,if’univ_model’has a MIDAS termcorr_model Correlation model used.Valid choices are:"cDCC"(the corrected DCC ofAielli(2013)),"aDCC"(the asymmetric DCC model of Cappiello et al.(2006)),"DECO"(Dynamic equicorrelation of Engle and Kelly(2012)),and"DCCMI-DAS"(the DCC-MIDAS of Colacito et al.(2011)).By detault,it is"cDCC"lag_fun g function to use.Valid choices are"Beta"(by default)and"Al-mon",for the Beta and Exponential Almon lag functions,respectively,if’univ_model’has a MIDAS term and/or if’corr_model’is"DCCMIDAS"N_c optional Number of(lagged)realizations to use for the standarized residualsforming the long-run correlation,if’corr_model’is"DCCMIDAS"K_c optional Number of(lagged)realizations to use for the long-run correlation,if’corr_model’is"DCCMIDAS"DetailsFunction dcc_fit implements the two-steps estimation of the DCC models.In thefirst step,a variety of univariate models are considered.These models can be selected using for the pa-rameter’univ_model’one of the following choices:’sGARCH’(standard GARCH of Bollerslev (1986)),’eGARCH’of Nelson(1991),’gjrGARCH’of Glosten et al.(1993),’iGARCH’(Integrated GARCH of Engle and Bollerslev(1986)),’csGARCH’(the Component GARCH of Engle and Lee (1999)),’GM_noskew’and’GM_skew’(the GARCH-MIDAS model of Engle et al.(2013),respec-tively,without and with the asymmetric term in the short-run component),and’DAGM_noskew’and’DAGM_skew’(the Double Asymmetric GARCH-MIDAS model of Amendola et al.(2019), respectively,without and with the asymmetric term in the short-run component).Valuedcc_fit returns an object of class’dccmidas’.The function summary.dccmidas can be used to print a summary of the results.Moreover,an object of class’dccmidas’is a list containing the following components:•assets:Names of the assets considered.•model:Univariate model used in thefirst step.•est_univ_model:List of matrixes of estimated coefficients of the univariate model,with the QML(Bollerslev and Wooldridge1992)standard errors.•corr_coef_mat:Matrix of estimated coefficients of the correlation model,with the QML stan-dard errors.•mult_model:Correlation model used in the second step.•obs:The number of daily observations used for the estimation.•period:The period of the estimation.•H_t:Conditional covariance matrix,reported as an array.•R_t:Conditional correlation matrix,reported as an array.•R_t_bar:Conditional long-run correlation matrix,reported as an array,if the correlation ma-trix includes a MIDAS specification.•est_time:Time of estimation.•Days:Days of the sample period.•llk:The value of the log-likelihood(for the second step)at the maximum.ReferencesAielli GP(2013).“Dynamic conditional correlation:on properties and estimation.”Journal of Business&Economic Statistics,31(3),282–299.doi:10.1080/07350015.2013.771027.Amendola A,Candila V,Gallo GM(2019).“On the asymmetric impact of macro–variables on volatility.”Economic Modelling,76,135–152.doi:10.1016/j.econmod.2018.07.025.Bollerslev T(1986).“Generalized autoregressive conditional heteroskedasticity.”Journal of Econo-metrics,31(3),307–327.doi:10.1016/03044076(86)900631.Bollerslev T,Wooldridge JM(1992).“Quasi-maximum likelihood estimation and inference in dy-namic models with time-varying covariances.”Econometric Reviews,11,143–172.doi:10.1080/ 07474939208800229.Cappiello L,Engle RF,Sheppard K(2006).“Asymmetric dynamics in the correlations of global equity and bond returns.”Journal of Financial Econometrics,4(4),537–572.doi:10.1093/jjfinec/ nbl005.Colacito R,Engle RF,Ghysels E(2011).“A component model for dynamic correlations.”Journalof Econometrics,164(1),45–59.doi:10.1016/j.jeconom.2011.02.013.Engle R,Kelly B(2012).“Dynamic equicorrelation.”Journal of Business&Economic Statis-tics,30(2),212–228.doi:10.1080/07350015.2011.652048.Engle RF,Bollerslev T(1986).“Modelling the persistence of conditional variances.”Econometric Reviews,5(1),1–50.doi:10.1080/07474938608800095.Engle RF,Ghysels E,Sohn B(2013).“Stock market volatility and macroeconomic fundamen-tals.”Review of Economics and Statistics,95(3),776–797.doi:10.1162/REST_a_00300.Engle RF,Lee GJ(1999).“A Long-run and Short-run Component Model of Stock Return V olatil-ity.”In Engle RF,White H(eds.),Cointegration,Causality,and Forecasting:A Festschrift in Honor of Clive W.J.Granger,475–497.Oxford University Press,Oxford.Glosten LR,Jagannathan R,Runkle DE(1993).“On the relation between the expected value and the volatility of the nominal excess return on stocks.”The Journal of Finance,48(5),1779–1801.doi:10.1111/j.15406261.1993.tb05128.x.Nelson DB(1991).“Conditional heteroskedasticity in asset returns:A new approach.”Econo-metrica,59,347–370.doi:10.2307/2938260.Examplesrequire(xts)#open to close daily log-returnsr_t_s<-log(sp500[ 2005/2008 ][,3])-log(sp500[ 2005/2008 ][,1])r_t_n<-log(nasdaq[ 2005/2008 ][,3])-log(nasdaq[ 2005/2008 ][,1])r_t_f<-log(ftse100[ 2005/2008 ][,3])-log(ftse100[ 2005/2008 ][,1])db_m<-merge.xts(r_t_s,r_t_n,r_t_f)db_m<-db_m[complete.cases(db_m),]colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")#list of returnsr_t<-list(db_m[,1],db_m[,2],db_m[,3])#MV transformation(same MV for all the stocks)require(rumidas)mv_m<-mv_into_mat(r_t[[1]],diff(indpro),K=12,"monthly")#list of MVMV<-list(mv_m,mv_m,mv_m)#estimationK_c<-144N_c<-36dccmidas_est<-dcc_fit(r_t,univ_model="GM_noskew",distribution="norm",MV=MV,K=12,corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)dccmidas_estsummary.dccmidas(dccmidas_est)Det7 Det Matrix determinantDescriptionCalculates the determinant of a numeric matrix.UsageDet(x)Argumentsx a numeric matrixValueThe determinant of x.Examplesx<-matrix(sample(1:25,25,replace=TRUE),ncol=5)Det(x)ftse100FTSE100dataDescriptionDaily data on FTSE100collected from the realized library of the Oxford-Man Institute(Heber et al.2009).Usagedata(ftse100)FormatAn object of class"xts".Detailsftse100includes the open price(open_price),the realized variance(rv5),and the close price(close_price).The realized variance has been calculated using intradaily intervals offive minutes(Andersen and Bollerslev1998).8indproSourceRealized library of the Oxford-Man InstituteReferencesAndersen TG,Bollerslev T(1998).“Answering the Skeptics:Yes,Standard V olatility Models do Provide Accurate Forecasts.”International Economic Review,39,885–905.doi:10.2307/2527343.Heber G,Lunde A,Shephard N,Sheppard K(2009).“OMI’s realised library,version0.1.”Oxford–Man Institute,University of Oxford.Exampleshead(ftse100)summary(ftse100)indpro Monthly U.S.Industrial ProductionDescriptionMonthly data on the U.S.Industrial Production index(IP,index2012=100,seasonally adjusted) collected from the Federal Reserve Economic Data(FRED)archive.The IP has been used as MIDAS term in different contributions(see,for instance,Engle et al.(2013),Conrad and Loch (2015),and Amendola et al.(2017)).Usagedata(indpro)FormatAn object of class"xts".SourceArchive of the Federal Reserve Economic Data(FRED)ReferencesAmendola A,Candila V,Scognamillo A(2017).“On the influence of US monetary policy on crude oil price volatility.”Empirical Economics,52(1),155–178.doi:10.1007/s0018101610695.Conrad C,Loch K(2015).“Anticipating Long-Term Stock Market V olatility.”Journal of Applied Econometrics,30(7),1090–1114.doi:10.1002/jae.2404.Engle RF,Ghysels E,Sohn B(2013).“Stock market volatility and macroeconomic fundamentals.”Review of Economics and Statistics,95(3),776–797.doi:10.1162/REST_a_00300.Inv9Exampleshead(indpro)summary(indpro)plot(indpro)Inv Inverse of a matrixDescriptionCalculates the inverse of a numeric matrixUsageInv(x)Argumentsx a numeric matrixValueThe inverse of x.Examplesx<-matrix(sample(1:25,25,replace=TRUE),ncol=5)Inv(x)nasdaq NASDAQ dataDescriptionDaily data on NASDAQ collected from the realized library of the Oxford-Man Institute(Heber et al.2009).Usagedata(nasdaq)FormatAn object of class"xts".10plot_dccmidas Detailsnasdaq includes the open price(open_price),the realized variance(rv5),and the close price(close_price).The realized variance has been calculated using intradaily intervals offive minutes(Andersen and Bollerslev1998).SourceRealized library of the Oxford-Man InstituteReferencesAndersen TG,Bollerslev T(1998).“Answering the Skeptics:Yes,Standard V olatility Models do Provide Accurate Forecasts.”International Economic Review,39,885–905.doi:10.2307/2527343.Heber G,Lunde A,Shephard N,Sheppard K(2009).“OMI’s realised library,version0.1.”Oxford–Man Institute,University of Oxford.Exampleshead(nasdaq)summary(nasdaq)plot_dccmidas Plot method for’dccmidas’classDescriptionPlots of the conditional volatilities on the main diagonal and of the conditional correlations on the extra-diagonal elements.Usageplot_dccmidas(x,K_c=NULL,vol_col="black",long_run_col="red",cex_axis=0.75,LWD=2,asset_sub=NULL)sp50011Argumentsx An object of class’dccmidas’,that is the result of a call to dcc_fit.K_c optional Number of(lagged)realizations to use for the long-run correlation,,if ’corr_model’is"DCCMIDAS"vol_col optional Color of the volatility and correlation plots."black"by defaultlong_run_col optional Color of the long-run correlation plots,if present."red"by default cex_axis optional Size of the x-axis.Default to0.75LWD optional Width of the plotted lines.Default to2asset_sub optional Numeric vector of selected assets to consider for the plot.NULL by defaultValueNo return value,called for side effectsExamplesrequire(xts)#open to close daily log-returnsr_t_s<-log(sp500[ 2010/2019 ][,3])-log(sp500[ 2010/2019 ][,1])r_t_n<-log(nasdaq[ 2010/2019 ][,3])-log(nasdaq[ 2010/2019 ][,1])r_t_f<-log(ftse100[ 2010/2019 ][,3])-log(ftse100[ 2010/2019 ][,1])db_m<-merge.xts(r_t_s,r_t_n,r_t_f)db_m<-db_m[complete.cases(db_m),]colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")#list of returnsr_t<-list(db_m[,1],db_m[,2],db_m[,3])#estimationK_c<-144N_c<-36cdcc_est<-dcc_fit(r_t,univ_model="sGARCH",distribution="norm",corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)plot_dccmidas(cdcc_est,K_c=144)sp500S&P500dataDescriptionDaily data on S&P500collected from the realized library of the Oxford-Man Institute(Heber et al.2009).Usagedata(sp500)12sp500FormatAn object of class"xts".Detailssp500includes the open price(open_price),the realized variance(rv5),and the close price(close_price).The realized variance has been calculated using intradaily intervals offive minutes(Andersen and Bollerslev1998).SourceRealized library of the Oxford-Man InstituteReferencesAndersen TG,Bollerslev T(1998).“Answering the Skeptics:Yes,Standard V olatility Models do Provide Accurate Forecasts.”International Economic Review,39,885–905.doi:10.2307/2527343.Heber G,Lunde A,Shephard N,Sheppard K(2009).“OMI’s realised library,version0.1.”Oxford–Man Institute,University of Oxford.Exampleshead(sp500)summary(sp500)Index∗datasetsftse100,7indpro,8nasdaq,9sp500,11cov_eval,2dcc_fit,3,11Det,7ftse100,7indpro,8Inv,9mv_into_mat,4nasdaq,9plot_dccmidas,10sp500,11summary.dccmidas,5ugarchspec,4ugmfit,413。

Package‘spcov’October14,2022Type PackageTitle Sparse Estimation of a Covariance MatrixVersion1.3Date2022-09-21Author Jacob Bien and Rob TibshiraniMaintainer Jacob Bien<*************>Description Provides a covariance estimator for multivariate normaldata that is sparse and positive definite.Implements themajorize-minimize algorithm described in Bien,J.,andTibshirani,R.(2011),``Sparse Estimation of a CovarianceMatrix,''Biometrika.98(4).807--820.License GPL-2LazyLoad yesEncoding UTF-8RoxygenNote7.2.1NeedsCompilation noRepository CRANDate/Publication2022-09-2320:20:02UTCR topics documented:spcov-package (2)GenerateCliquesCovariance (2)ProxADMM (3)spcov (5)Index812GenerateCliquesCovariance spcov-package Sparse Estimation of a Covariance MatrixDescriptionPerforms the majorize-minimize algorithm described inDetailsBien,J.,and Tibshirani,R.(2011),"Sparse Estimation of a Covariance Matrix,"Biometrika.98(4).807–820.Generalized gradient descent is used to minimize each majorizer.Package:spcovType:PackageVersion: 1.01Date:2012-03-04License:GPL-2LazyLoad:yesSee the function spcov.Author(s)Jacob Bien and Rob TibshiraniMaintainer:Jacob Bien<******************>ReferencesBien,J.,and Tibshirani,R.(2011),"Sparse Estimation of a Covariance Matrix,"Biometrika.98(4).807–820.See AlsospcovGenerateCliquesCovarianceGenerate a block diagonal covariance matrixDescriptionThis function is included in the package so that it can be used in the example code provided in spcov.UsageGenerateCliquesCovariance(ncliques,cliquesize,theta)Argumentsncliques number of blockscliquesize size of each blocktheta magnitude of non-zerosDetailsThis function generates a block diagonal positive definite matrix with randomly-signed,non-zero elements.A shift is added to the diagonal of the matrix so that its condition number equals p,the number of variables.ValueSigma the covariance matrixA symmetric square root of Sigmashift how much the eigenvalues were shifted.See details.Author(s)Jacob Bien and Rob TibshiraniReferencesBien,J.,and Tibshirani,R.(2011),"Sparse Estimation of a Covariance Matrix,"accepted for pub-lication in Biometrika.ProxADMM Solving penalized Frobenius problem.DescriptionThis function solves the optimization problemUsageProxADMM(A,del,lam,P,rho=0.1,tol=1e-06,maxiters=100,verb=FALSE)ArgumentsA A symmetric matrix.del A non-negative scalar.Lower bound on eigenvalues.lam A non-negative scalar.L1penalty parameter.P Matrix with non-negative elements and dimension of A.Allows for differing L1 penalty parameters.rho ADMM parameter.Can affect rate of convergence a lot.tol Convergence threshold.maxiters Maximum number of iterations.verb Controls whether to be verbose.DetailsMinimize_X(1/2)||X-A||_F^2+lam||P*X||_1s.t.X>=del*I.This is the prox function for the generalized gradient descent of Bien&Tibshirani2011(see full reference below).This is the R implementation of the algorithm in Appendix3of Bien,J.,and Tibshirani,R.(2011), "Sparse Estimation of a Covariance Matrix,"Biometrika.98(4).807–820.It uses an ADMM approach to solve the problemMinimize_X(1/2)||X-A||_F^2+lam||P*X||_1s.t.X>=del*I.Here,the multiplication between P and X is elementwise.The inequality in the constraint is a lower bound on the minimum eigenvalue of the matrix X.Note that there are two variables X and Z that are outputted.Both are estimates of the optimal X.However,Z has exact zeros whereas X has eigenvalues at least del.Running the ADMM algorithm long enough,these two are guaranteed to converge.ValueX Estimate of optimal X.Z Estimate of optimal X.obj Objective values.Author(s)Jacob Bien and Rob TibshiraniReferencesBien,J.,and Tibshirani,R.(2011),"Sparse Estimation of a Covariance Matrix,"Biometrika.98(4).807–820.See AlsospcovExamplesset.seed(1)n<-100p<-200#generate a covariance matrix:model<-GenerateCliquesCovariance(ncliques=4,cliquesize=p/4,1)#generate data matrix with x[i,]~N(0,model$Sigma):x<-matrix(rnorm(n*p),ncol=p)%*%model$AS<-var(x)#compute sparse,positive covariance estimator:P<-matrix(1,p,p)diag(P)<-0lam<-0.1aa<-ProxADMM(S,0.01,lam,P)spcov Sparse Covariance EstimationDescriptionProvides a sparse and positive definite estimate of a covariance matrix.This function performs the majorize-minimize algorithm described in Bien&Tibshirani2011(see full reference below). Usagespcov(Sigma,S,lambda,step.size,nesterov=TRUE,n.outer.steps=10000,n.inner.steps=10000,tol.outer=1e-04,thr.inner=0.01,backtracking=0.2,trace=0)ArgumentsSigma an initial guess for Sigma(suggestions:S or diag(diag(S))).S the empirical covariance matrix of the data.Must be positive definite(if it is not, add a small constant to the diagonal).lambda penalty parameter.Either a scalar or a matrix of the same dimension as Sigma.This latter choice should be used to penalize only off-diagonal elements.Allelements of lambda must be non-negative.step.size the step size to use in generalized gradient descent.Affects speed of algorithm.nesterov indicates whether to use Nesterov’s modification of generalized gradient de-scent.Default:TRUE.n.outer.steps maximum number of majorize-minimize steps to take(recall that MM is the outer loop).n.inner.steps maximum number of generalized gradient steps to take(recall that generalized gradient descent is the inner loop).tol.outer convergence threshold for outer(MM)loop.Stops when drop in objective be-tween steps is less than tol.outer.thr.inner convergence threshold for inner(i.e.generalized gradient)loop.Stops when mean absolute change in Sigma is less than thr.inner*mean(abs(S)).backtracking if FALSE,thenfixed step size used.If numeric and in(0,1),this is the parameter of backtracking that multiplies step.size on each ually,in range of(0.1,0.8).Default:0.2.trace controls how verbose output should be.DetailsThis is the R implementation of Algorithm1in Bien,J.,and Tibshirani,R.(2011),"Sparse Estima-tion of a Covariance Matrix,"Biometrika.98(4).807–820.The goal is to approximately minimize (over Sigma)the following non-convex optimization problem:minimize logdet(Sigma)+trace(S Sigma^-1)+||lambda*Sigma||_1subject to Sigma positive definite.Here,the L1norm and matrix multiplication between lambda and Sigma are elementwise.The empirical covariance matrix must be positive definite for the optimization problem to have bounded objective(see Section3.3of paper).We suggest adding a small constant to the diagonal of S if it is not.Since the above problem is not convex,the returned matrix is not guaranteed to be a global minimum of the problem.In Section3.2of the paper,we mention a simple modification of gradient descent due to Nesterov.The argument nesterov controls whether to use this modification(we suggest that it be used).We also strongly recommend using backtracking.This allows the algorithm to begin by taking large steps(the initial size is determined by the argument step.size)and then to gradually reduce the size of steps.At the start of the algorithm,a lower bound(delta in the paper)on the eigenvalues of the solution is calculated.As shown in Equation(3)of the paper,the prox function for our generalized gradient descent amounts to minimizing(over a matrix X)a problem of the formminimize(1/2)||X-A||_F^2+||lambda*X||_1subject to X>=delta IThis is implemented using an alternating direction method of multipliers approach given in Ap-pendix3.ValueSigma the sparse covariance estimaten.iter a vector giving the number of generalized gradient steps taken on each step of the MM algorithmobj a vector giving the objective values after each step of the MM algorithmAuthor(s)Jacob Bien and Rob TibshiraniReferencesBien,J.,and Tibshirani,R.(2011),"Sparse Estimation of a Covariance Matrix,"Biometrika.98(4).807–820.See AlsoProxADMMExamplesset.seed(1)n<-100p<-20#generate a covariance matrix:model<-GenerateCliquesCovariance(ncliques=4,cliquesize=p/4,1)#generate data matrix with x[i,]~N(0,model$Sigma):x<-matrix(rnorm(n*p),ncol=p)%*%model$AS<-var(x)#compute sparse,positive covariance estimator:step.size<-100tol<-1e-3P<-matrix(1,p,p)diag(P)<-0lam<-0.06mm<-spcov(Sigma=S,S=S,lambda=lam*P,step.size=step.size,n.inner.steps=200,thr.inner=0,tol.outer=tol,trace=1)sqrt(mean((mm$Sigma-model$Sigma)^2))sqrt(mean((S-model$Sigma)^2))##Not run:image(mm$Sigma!=0)Index∗multivariateGenerateCliquesCovariance,2ProxADMM,3spcov,5spcov-package,2 GenerateCliquesCovariance,2ProxADMM,3spcov,2,5spcov-package,28。

Package‘nmw’May10,2023Version0.1.5Title Understanding Nonlinear Mixed Effects Modeling for PopulationPharmacokineticsDescription This shows how NONMEM(R)software works.NONMEM's classical estimation meth-ods like'First Order(FO)approximation','First Order Conditional Estima-tion(FOCE)',and'Laplacian approximation'are explained.Depends R(>=3.5.0),numDerivByteCompile yesLicense GPL-3Copyright2017-,Kyun-Seop BaeAuthor Kyun-Seop BaeMaintainer Kyun-Seop Bae<********>URL https:///package=nmwNeedsCompilation noRepository CRANDate/Publication2023-05-1003:40:02UTCR topics documented:nmw-package (2)AddCox (3)CombDmExPc (4)CovStep (5)EstStep (6)InitStep (7)TabStep (9)TrimOut (10)Index1112nmw-package nmw-package Understanding Nonlinear Mixed Effects Modeling for PopulationPharmacokineticsDescriptionThis shows how NONMEM(R)</innovation/nonmem/>software works. DetailsThis package explains’First Order(FO)approximation’method,’First Order Conditional Estima-tion(FOCE)’method,and’Laplacian(LAPL)’method of NONMEM software.Author(s)Kyun-Seop Bae<********>References1.NONMEM Users guide2.Wang Y.Derivation of various NONMEM estimation methods.J Pharmacokinet Pharmaco-dyn.2007.3.Kang D,Bae K,Houk BE,Savic RM,Karlsson MO.Standard Error of Empirical BayesEstimate in NONMEM(R)VI.K J Physiol Pharmacol.2012.4.Kim M,Yim D,Bae K.R-based reproduction of the estimation process hidden behind NON-MEM Part1:First order approximation method.2015.5.Bae K,Yim D.R-based reproduction of the estimation process hidden behind NONMEM Part2:First order conditional estimation.2016.ExamplesDataAll=Theophcolnames(DataAll)=c("ID","BWT","DOSE","TIME","DV")DataAll[,"ID"]=as.numeric(as.character(DataAll[,"ID"]))nTheta=3nEta=3nEps=2THETAinit=c(2,50,0.1)OMinit=matrix(c(0.2,0.1,0.1,0.1,0.2,0.1,0.1,0.1,0.2),nrow=nEta,ncol=nEta) SGinit=diag(c(0.1,0.1))LB=rep(0,nTheta)#Lower boundUB=rep(1000000,nTheta)#Upper boundFGD=deriv(~DOSE/(TH2*exp(ETA2))*TH1*exp(ETA1)/(TH1*exp(ETA1)-TH3*exp(ETA3))*(exp(-TH3*exp(ETA3)*TIME)-exp(-TH1*exp(ETA1)*TIME)),AddCox3 c("ETA1","ETA2","ETA3"),function.arg=c("TH1","TH2","TH3","ETA1","ETA2","ETA3","DOSE","TIME"),func=TRUE,hessian=TRUE)H=deriv(~F+F*EPS1+EPS2,c("EPS1","EPS2"),function.arg=c("F","EPS1","EPS2"),func=TRUE) PRED=function(THETA,ETA,DATAi){FGDres=FGD(THETA[1],THETA[2],THETA[3],ETA[1],ETA[2],ETA[3],DOSE=320,DATAi[,"TIME"]) Gres=attr(FGDres,"gradient")Hres=attr(H(FGDres,0,0),"gradient")if(e$METHOD=="LAPL"){Dres=attr(FGDres,"hessian")Res=cbind(FGDres,Gres,Hres,Dres[,1,1],Dres[,2,1],Dres[,2,2],Dres[,3,])colnames(Res)=c("F","G1","G2","G3","H1","H2","D11","D21","D22","D31","D32","D33") }else{Res=cbind(FGDres,Gres,Hres)colnames(Res)=c("F","G1","G2","G3","H1","H2")}return(Res)}#######First Order Approximation Method#Commented out for the CRAN CPU time#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="ZERO")#(EstRes=EstStep())#4sec#(CovRes=CovStep())#2sec#PostHocEta()#Using e$FinalPara from EstStep()#TabStep()########First Order Conditional Estimation with Interaction Method#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="COND")#(EstRes=EstStep())#2min#(CovRes=CovStep())#1min#get("EBE",envir=e)#TabStep()########Laplacian Approximation with Interacton Method#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="LAPL")#(EstRes=EstStep())#4min#(CovRes=CovStep())#1min#get("EBE",envir=e)#TabStep()AddCox Add a Covariate Column to an Existing NONMEM datasetDescriptionA new covariate column can be added to an existing NONMEM dataset.4CombDmExPcUsageAddCox(nmData,coxData,coxCol,dateCol="DATE",idCol="ID")ArgumentsnmData an existing NONMEM datasetcoxData a data table containing a covariate columncoxCol the covariate column name in the coxData tabledateCol date column name in the NONMEM dataset and the covariate data tableidCol ID column name in the NONMEM dataset and the covariate data tableDetailsItfirst carry forward for the missing data.If NA is remained,it carry backward.ValueA new NONMEM dataset containing the covariate columnAuthor(s)Kyun-Seop Bae<********>CombDmExPc Combine the demographics(DM),dosing(EX),and DV(PC)tables intoa new NONMEM datasetDescriptionA new NONMEM dataset can be created from the demographics,dosing,and DV tables.UsageCombDmExPc(dm,ex,pc)Argumentsdm A demographics table.It should contain a row per subject.ex An exposure table.Drug administration(dosing)history table.pc A DV(dependent variable)or PC(drug concentration)tableDetailsCombining a demographics,a dosing,and a concentration table can produce a new NONMEM dataset.CovStep5ValueA new NONMEM datasetAuthor(s)Kyun-Seop Bae<********>CovStep Covariance StepDescriptionIt calculates standard errors and various variance matrices with the e$FinalPara after estimation step.UsageCovStep()DetailsBecause EstStep uses nonlinear optimization,covariance step is separated from estimation step.It calculates variance-covariance matrix of estimates in the original scale.ValueTime consumed timeStandard Error standard error of the estimates in the order of theta,omega,and sigmaCovariance Matrix of Estimatescovariance matrix of estimates in the order of theta,omega,and sigma.This isinverse(R)x S x inverse(R)by default.Correlation Matrix of Estimatescorrelation matrix of estimates in the order of theta,omega,and sigma Inverse Covariance Matrix of Estimatesinverse covariance matrix of estimates in the order of theta,omega,and sigma Eigen Values eigen values of covariance matrixR Matrix R matrix of NONMEM,the second derivative of log likelihood function with respect to estimation parametersS Matrix S matrix of NONMEM,sum of individual cross-product of thefirst derivative of log likelihood function with respect to estimation parametersAuthor(s)Kyun-Seop Bae<********>6EstStepReferencesNONMEM Users GuideSee AlsoEstStep,InitStepExamples#Only after InitStep and EstStep#CovStep()EstStep Estimation StepDescriptionThis estimates upon the conditions with InitStep.UsageEstStep()DetailsIt does not have arguments.All necessary arguments are stored in the e environment.It assumes "INTERACTION"between eta and epsilon for"COND"and"LAPL"options.The output is basically same to NONMEM output.ValueInitial OFV initial value of the objective functionTime time consumed for this stepOptim the raw output from optim functionFinal Estimatesfinal estimates in the original scaleAuthor(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideSee AlsoInitStepExamples#Only After InitStep#EstStep()InitStep Initialization StepDescriptionIt receives parameters for the estimation and stores them into e environment.UsageInitStep(DataAll,THETAinit,OMinit,SGinit,LB,UB,Pred,METHOD)ArgumentsDataAll Data for all subjects.It should contain columns which Pred function uses.THETAinit Theta initial valuesOMinit Omega matrix initial valuesSGinit Sigma matrix initial valuesLB Lower bounds for theta vectorUB Upper bounds for theta vectorPred Prediction function nameMETHOD one of the estimation methods"ZERO","COND",or"LAPL"DetailsPrediction function should return not only prediction values(F or IPRED)but also G(first derivative with respect to etas)and H(first derivative of Y with respect to epsilon).For the"LAPL",prediction function should return second derivative with respect to eta also."INTERACTION"is TRUE for "COND"and"LAPL"option,and FALSE for"ZERO".Omega matrix should be full block one.Sigma matrix should be diagonal one.ValueThis does not return values,but stores necessary values into the environment e.Author(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideExamplesDataAll=Theophcolnames(DataAll)=c("ID","BWT","DOSE","TIME","DV")DataAll[,"ID"]=as.numeric(as.character(DataAll[,"ID"]))nTheta=3nEta=3nEps=2THETAinit=c(2,50,0.1)#Initial estimateOMinit=matrix(c(0.2,0.1,0.1,0.1,0.2,0.1,0.1,0.1,0.2),nrow=nEta,ncol=nEta)OMinitSGinit=diag(c(0.1,0.1))SGinitLB=rep(0,nTheta)#Lower boundUB=rep(1000000,nTheta)#Upper boundFGD=deriv(~DOSE/(TH2*exp(ETA2))*TH1*exp(ETA1)/(TH1*exp(ETA1)-TH3*exp(ETA3))*(exp(-TH3*exp(ETA3)*TIME)-exp(-TH1*exp(ETA1)*TIME)),c("ETA1","ETA2","ETA3"),function.arg=c("TH1","TH2","TH3","ETA1","ETA2","ETA3","DOSE","TIME"),func=TRUE,hessian=TRUE)H=deriv(~F+F*EPS1+EPS2,c("EPS1","EPS2"),function.arg=c("F","EPS1","EPS2"),func=TRUE) PRED=function(THETA,ETA,DATAi){FGDres=FGD(THETA[1],THETA[2],THETA[3],ETA[1],ETA[2],ETA[3],DOSE=320,DATAi[,"TIME"]) Gres=attr(FGDres,"gradient")Hres=attr(H(FGDres,0,0),"gradient")if(e$METHOD=="LAPL"){Dres=attr(FGDres,"hessian")Res=cbind(FGDres,Gres,Hres,Dres[,1,1],Dres[,2,1],Dres[,2,2],Dres[,3,])colnames(Res)=c("F","G1","G2","G3","H1","H2","D11","D21","D22","D31","D32","D33") }else{Res=cbind(FGDres,Gres,Hres)colnames(Res)=c("F","G1","G2","G3","H1","H2")}return(Res)}#########First Order Approximation MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB, Pred=PRED,METHOD="ZERO")#########First Order Conditional Estimation with Interaction MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB, Pred=PRED,METHOD="COND")#########Laplacian Approximation with Interacton MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,TabStep9 Pred=PRED,METHOD="LAPL")TabStep Table StepDescriptionThis produces standard table.UsageTabStep()DetailsIt does not have arguments.All necessary arguments are stored in the e environment.This is similar to other standard results table.ValueA table with ID,TIME,DV,PRED,RES,WRES,derivatives of G and H.If the estimation methodis other than’ZERO’(First-order approximation),it includes CWRES,CIPREDI(formerly IPRED), CIRESI(formerly IRES).Author(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideSee AlsoEstStepExamples#Only After EstStep#TabStep()10TrimOut TrimOut Trimming and beutifying NONMEM original OUTPUTfileDescriptionTrimOut removes unnecessary parts from NONMEM original OUTPUTfile.UsageTrimOut(inFile,outFile="PRINT.OUT")ArgumentsinFile NONMEM original untidy OUTPUTfile nameoutFile Outputfile name to be writtenDetailsNONMEM original OUTPUTfile contains unnecessary parts such as CONTROLfile content, Start/End Time,License Info,Print control characters such as"+","0","1".This function trims those.ValueoutFile will be written in the current working folder or designated folder.Ths returns TRUE if the process was smooth.Author(s)Kyun-Seop Bae<********>Index∗Covariance StepCovStep,5∗Data PreparationAddCox,3CombDmExPc,4∗Estimation StepEstStep,6∗Initialization StepInitStep,7∗NONMEM OUTPUTTrimOut,10∗Nonlinear Mixed Effects Modelingnmw-package,2∗Population Pharmacokineticsnmw-package,2∗Tabulation StepTabStep,9AddCox,3CombDmExPc,4CovStep,5EstStep,6,6,9InitStep,6,7nmw(nmw-package),2nmw-package,2TabStep,9TrimOut,1011。

二维传播算子DOA估计的改进算法刁鸣;陈超;杨丽丽【摘要】针对传播算子算法在低信噪比和小快拍数环境下进行二维DOA估计性能下降的实际问题,提出了一种改进的二维传播算子DOA估计方法.该方法继承了ESPRIT算法无需谱峰搜索的优点,并且利用线性运算代替特征分解求得旋转不变关系矩阵,再通过简单的除法运算实现方位角和俯仰角的快速配对,极大地降低了运算量.重新构造的协方差矩阵对接收数据进行了共轭重排再利用,使该算法在快拍数有限、信噪比较低的条件下,估计性能明显提高.该文从理论上论证了将数据共轭重构的思想引申到2-D传播算子算法的合理性,仿真实验证明了该方法的有效性.%In order to improve the accuracy of DOA estimation in a poor environment with low SNR and a limited number of snapshots, a new two-dimensional DOA estimation algorithm using a modified propagator method based on two parallel uniform linear arrays was proposed. In this method, two-dimensional spectral peak searching was avoided just as with the ESPRIT algorithm; Eigen-decomposition of a large matrix was replaced by a linear operator;parameter matching was automatically achieved by a simple division operation. The computational complexity was greatly reduced. The conjugate data was rearranged and re-used in the new covariance matrix, so that the performance of DOA estimation was obviously improved in the condition of low SNR and a limited number of snapshots.The rationality of introducing the conjugate data rearrangement into 2-D propagator method was demonstrated in theory. Simulation results show the superiority of this proposed method in precision and computation.【期刊名称】《哈尔滨工程大学学报》【年(卷),期】2011(032)001【总页数】5页(P98-102)【关键词】阵列信号处理;二维DOA估计;数据共轭重排;传播算子算法【作者】刁鸣;陈超;杨丽丽【作者单位】哈尔滨工程大学信息与通信工程学院,黑龙江哈尔滨,150001;哈尔滨工程大学信息与通信工程学院,黑龙江哈尔滨,150001;哈尔滨工程大学信息与通信工程学院,黑龙江哈尔滨,150001【正文语种】中文【中图分类】TN911.7目前，在阵列信号处理领域，对二维波达方向估计方法的研究已经涌现了很多的成果，传统的二维MUSIC算法需要在整个参数平面上进行谱峰搜索，计算量巨大，难以应用于工程实际，ESPRIT方法不需谱峰搜索，与MUSIC方法相比，具有较好的实时性，但是它存在一个参数配对的问题.文献［1］提出的DOA矩阵法可以避免谱峰搜索而直接计算出二维DOA参数，且参数能自动匹配，在一定程度上解决了上述问题，但存在相位模糊;文献［2］提出的基于L型阵列的波达方向估计算法，无相位模糊，解决了信号方位角和俯仰角的配对问题，但需计算多个相关矩阵后再对大矩阵进行特征分解，计算量仍很大;文献［3］基于双平行线阵列的特点，对子阵进行合并求解，使得协方差矩阵中不再含有冗余数据，但仍不可避免协方差矩阵特征分解带来的复杂计算.传播算子算法［4］利用线性运算代替特征分解，在解决计算量问题上有着巨大优势.文献［5］中Tayem等人提出的一种基于双L阵的二维DOA估计的传播算子算法，为了提高估计精度，需要进行2次方位角估计，文献［6］采用了空间三平行线阵列结构的传播算子算法，以增加一条均匀线阵来提高算法的估计精度.本文将数据共轭重排［7］的思想引入到传播算子算法，基于双平行线的阵列结构，提出一种改进的传播算子二维DOA估计方法，该方法在不增加计算量和阵元数的前提下，通过接收数据的共轭重排再利用，可以减少信源间的相关性，并且提高算法在快拍数有限及低信噪比条件下的估计性能.1 阵列结构和信号模型阵列结构由如图1所示的2个平行均匀线阵构成.其中，以位于原点的阵元为参考点，Y轴上共有M+1个阵元，前M个阵元组成子阵列L1，后M个阵元组成子阵列L2，X-Y平面上的M个阵元组成子阵列L3，且X轴方向和Y轴方向的阵元间距均为d，d的取值为1/2信号波长.图1 双平行线阵的阵列结构Fig.1 Structure of two parallel ULAs假设空间有D个同中心频率的远场窄带信号入射到天线阵上，第i个入射信号的方位角和俯仰角分别为θi和φi(i=1，2，…，D)，各阵元输出的噪声是统计独立、均值为0、方差为σ2的加性高斯白噪声，噪声与信号不相关.则L1、L2、L3接收到的信号分别表示为式中:式中:λ为波长，d为天线阵列中2个相邻阵元的距离，Xi(t)和Ni(t)分别为第i个子阵列在第t个快拍时刻的接收数据和接收到的噪声数据.这样Φ(θ，φ)和Ψ(θ，φ)分别反映了子阵列L2和子阵列L3相对于子阵列L1的旋转不变关系.2 改进算法的原理及性能分析不考虑噪声影响，将X1、X2、X3按式(4)所示构造矩阵X:将A和C分别进行分块处理:式中:假设C1为非奇异矩阵，即C1的D行互相独立，那么C2是C1的线性变换，有式中:P定义为传播算子，P的求解需要信源方位信息，可以由空间协方差矩阵求解传播算子的估计值.对Rx进行分块处理:Rx=［Rx1，Rx2］，其中，Rx1、Rx2的维数分别为3M×D 和3M×(3M－D).由式(7)易推得由于噪声的影响，使数据模型和实际情况有所偏差，因此，式(9)不能严格相等，可通过最小化求得传播算子的估计值.为进一步提高估计性能，将接收数据共轭重排的思想推广到传播算子算法的二维DOA估计中，提出传播算子算法的二维DOA估计改进算法.对式(4)接收的数据向量进行共轭重排，令，式中J为副对角线上元素为1，其余元素均为0的3M阶方阵，则的协方差矩阵为式中:JM代表副对角线上元素为1，其余元素均为0的M阶方阵.可见，Rx和Rx均可以划分成9个M×M维的子矩阵，将这些子矩阵按式(12)进行重排相加求平均:观察矩阵R中的第1个M×M维子矩阵:式中:对于独立的信号源，协方差矩阵RS应为实对角阵，即RS=，又根据Q*，R，QT3个对角阵相乘可交换顺序进行计算，并且Q*QT=I，可以推导出:同理可推得矩阵R中9个M×M维子矩阵的2个相加项分别相等，因此，利用Rx 估计传播算子和利用R估计传播算子应得到相同的.由于R将接收数据共轭重排再利用了一次，使协方差矩阵的估计更准确，当信噪比较低，快拍数较少时，使用R 进行估计可以获得比Rx更好的性能.得到传播算子的估计值后，对进行分块处理:式中:P1，P3，P5的维数与A2的维数相同，P2，P4的维数与A1的维数相同.根据式(5)～(7)可得由式(17)、(19)可知，Φ和Ψ的对角线元素分别对应于P2、P4的特征值.在实际情况中，对P2和P4的特征分解是分别进行的，因此不能保证其特征值是一一对应的，在此可以采用快速配对算法［8］来解决参数配对问题.对P2进行特征分解得到其D个特征值分别为［λ1 λ2 … λD］，相应的特征向量分别为W=［w1w2 … wD］，与P2的第i个特征值λi相对应的P4的特征值λi'应为式中:yik是矩阵Y第i个列向量yi的第k个元素.由此可得，Φ和Ψ的对角线上第i个元素分别为λi和λi'，结合Φ和Ψ的表达式，可以估计入射信号的方位角和俯仰角分别为同理，也可根据式(16)、(18)、(20)分别求和的特征值来获得入射信号方位角和俯仰角的估计.在MUSIC和ESPRIT算法中，对协方差矩阵进行特征分解的时间复杂度近似为O(M3)阶，而估计传播算子的时间复杂度为O(DM2)阶［4］.可见，传播算子算法与MUSIC和ESPRIT算法相比，有着计算量小的优势.本文所提出的改进算法，在PM(propagator method)算法的基础上仅增加了取共轭和子矩阵换位相加的运算，用很少的计算量换取了较高的估计性能.3 仿真实验为验证方法的正确性及有效性，采用如图1所示的阵列结构进行了计算机模拟仿真实验，各子阵阵元数M=4，信号的中心频率f=50 MHz，阵元间距d为1/2个信号波长.实验1:本例给出了方位角和俯仰角在0°～90°范围内变化的估计性能.实验中使俯仰角以5°为间隔，从0°变化到90°，对应每个俯仰角，方位角亦以5°为间隔在0°～90°变化，快拍数200，信噪比5dB，对应每个入射角度做300次独立的仿真实验.图2给出了仿真结果，其中均方根误差定义为图2 方位角和俯仰角在0°～90°之间变化的均方根误差Fig.2 RMSE ofestimation with a zimuth and elevation angle varying from 0°to 90°实验2:基于图1所示的双平行线阵，本例对常规传播算子算法和文中提出的修正传播算子算法的估计性能进行了对比.2个等功率的远场窄带信号入射到图1所示天线阵列，信号源相互独立，二维入射角度分别为［35°20°］和［45°70°］.图3所示为估计的均方根误差随信噪比变化的曲线，快拍数为250，信噪比从0 dB变化到25 dB，对应每个信噪比做1 000次独立的仿真实验.图4所示为均方根误差随快拍数变化的曲线，信噪比固定为5dB，快拍数从100变化到1 000，对应每个快拍数做1 000次独立的仿真实验.由图3及图4可以看出，在对非相干信源进行DOA估计时，修正的传播算子算法在信噪比较低和快拍数较少的情况下明显优于常规传播算子算法.可见，采用数据共轭重排的修正传播算子算法可以提高非相干信源的DOA估计性能.图3 本文算法与常规传播算子算法对比(均方根误差随信噪比变化曲线)Fig.3 Comparison between normal propagator method and thealgorithm proposed in thispaper (RMSE varying with SNR)图4 本文算法与常规传播算子算法对比(均方根误差随快拍数变化曲线)Fig.4 Comparison between normal propagator method and thealgorithm proposed in thispaper (RMSE varying with snapshots)4 结束语本文在基于双平行线阵的基础上，将数据共轭重排的思想成功的引入到传播算子算法的二维DOA估计中，理论分析和仿真结果表明，改进的算法相当于对协方差矩阵进行了一次前后向平滑，具有平均的意义，可以减少信源间的相关性，提高对非相干信源的估计能力.该方法在低信噪比，小快拍数的条件下仍能获得较好的估计性能，并且有着计算量小的明显优势，具有较好的实用性.参考文献:【相关文献】［1］殷勤业，邹理和，NEWCOMB R W.一种高分辨率二维信号参量估计方法:波达方向矩阵法［J］.通信学报，1991，12(4):1-7.YIN Qinye，ZOU Lihe，NEWCOMB R W.A high resolution approach to 2-D signal parameter estimation:DOA matrix method［J］.Journal of China Institute of Communications，1991，12(4):1-7.［2］董轶，吴云韬，廖桂生.一种二维到达方向估计的ESPRIT新方法［J］.西安电子科技大学学报，2003，30(5):569-573.DONG Yi，WU Yuntao，LIAO Guisheng.A novel method for estimating 2-D DOA ［J］.Journal of Xidian University，2003，30(5):569-573.［3］刁鸣，吴小强，张鹏.基于修正ESPRIT算法的二维DOA估计［J］.哈尔滨工程大学学报，2008，29(4):407-410.DIAO Ming，WU Xiaoqiang，ZHANG Peng.2-D DOA estimation based on a modified ESPRIT algorithm［J］.Journal of Harbin Engineering University，2008，29(4):407-410. ［4］MARCOS S，MARSAL A，BENIDIR M.The propagator method for source bearing estimation［J］.Signal Processing，1995，42(2):121-138.［5］TAYEM N，KWON H M.L-shape 2-dimensional arrival angle estimation with propagator method［J］.IEEE Transactions on Antennas and Propagation，2005，53(5):1622-1630.［6］苏淑靖，颜景龙，马维贤，曹东杰.基于传播算子的二维波达方向估计新算法［J］.北京理工大学学报，2008，28 (7):602-605.SU Shujing，YAN Jinglong，MA Weixian，CAO Dongjie.A new algorithm for 2-D DOA estimation based on propagator method［J］.Transactions of Beijing Institute of Technology，2008，28(7):602-605.［7］张光斌，廖桂生，吴云韬，王万林.基于数据共轭重排修正的传播算子DOA估计算法［J］.系统仿真学报，2004，16(8):1662-1664.ZHANG Guangbin，LIAO Guisheng，WU Yuntao，WANG Wanlin.A modified propagator algorithm for DOA estimation based on conjugate data rearrangement［J］.Journal of System Simulation，2004，16(8):1662-1664.［8］刁鸣，缪善林.二维ESPRIT算法参数的快速配对［J］.哈尔滨工程大学学报，2008，29(3):290-293.DIAO Ming，MIAO Shanlin.Fast parameter matching for 2-D ESPRIT algorithm［J］.Journal of Harbin Engineering University，2008，29(3):290-293.。

SPSS 专业技术词汇、短语的中英文对照索引% of cases 各类别所占百分比1-tailed 单尾的2 Independent Samples 两个独立样本的检验2 Related Samples 两个相关样本检验2-tailed 双尾的3-D (=dimensional) 三维-->三维散点图AAbove 高于Absolute 绝对的-->绝对值Add 加，添加Add Cases 合并个案Add cases from... 从……加个案Add Variables 合并变量Add variables from... 从……加变量Adj.(=adjusted) standardized 调整后的标准化残差Aggregate 汇总-->分类汇总Aggregate Data 对数据进行分类汇总Aggregate Function 汇总函数Aggregate Variable 需要分类汇总的变量Agreement 协议Align 对齐-->对齐方式Alignment 对齐-->对齐方式All 全部，所有的All cases 所有个案All categories equal 所有类别相等All other values 所有其他值All requested variables entered 所要求变量全部引入Alphabetic 按字母顺序的-->按字母顺序列表Alternative 另外的，备选的Analysis by groups is off 分组分析未开启Analyze 分析-->统计分析Analyze all cases, do not create groups 分析全部个案，不建立分组Annotation 注释ANOVA Table ANOVA表ANOVA table and eta (对分组变量)进行单因素方差分析并计算其 η 值Apply 应用Apply Data Dictionary 应用数据字典Apply Dictionary 应用数据字典Approximately 大约Approximately X% of all cases 从所有个案中随机选择约 X%的个案Approximation 近似估计Area 面积Ascend 上升Ascending counts 按频数的升序排列Ascending means 按均值升值排序Ascending values 按变量值的升序排列Assign 指定，分配Assign Rank 1 to 把秩值 1 分配给Assume 假定Asymp. Sig.(=Asymptotic Significance) (2-sided) 双尾渐近显著性检验Automatic 自动的Automatic Recode 自动重新编码Axis 轴Axis Title 横轴名称BBack 返回-->上一步Backward 向后-->向后剔除法Bar chart 条形图Bar Spacing 条形间距Bars Represent 条的长度所代表的统计量Based on negative ranks 基于负秩Based on time or case range 在给定范围内随机选择个案Below 低于Between Groups (Combined) 组间值Binomial 二项分布Bivariate 双变量的-->双变量相关分析Bivariate Correlations 双变量相关分析Both files provide cases 是由外部文件和当前文件二者都提供个案Bottom 底部-->下边框Break Variable 分类变量Brown-Forsythe Brown-Forsythe检验Browse 浏览CCache 贮存Cache Data 数据暂存Calculated from data 根据数据计算Cancel 取消Case 个案Case Label 个案标签Case Processing Summary 个案处理概要Casewise 以个案为单位Casewise diagnostics 个案诊断表Categories 分组数，分类变量，分类模式Categorize Variables 变量分类Category 类别-->单元格数据类型Category Axis 分类轴Cell 单元格Cell Display 单元格显示Cell Percentages 在每个单元格中输出百分比Cell Properties 单元格属性Cell Statistics 单元格中的统计量Center 中心-->居中，中间对齐Central Tendency 集中趋势Change 变化，更改Change Summary 改变汇总函数Chart 图表，图形Chart Type 统计图类型Chart values 作图数据选项Charts 确定生成的图形选项Chi-square test 卡方检验Chi-square（χ2）卡方Choose 选择Choose Destination Location 选择安装路径Classify 分类Clear 清除Close 关闭Cluster 群，组，分组Clustered 分组的-->分组条形图Clustered bar charts 分组条形图Cochran's and Mantel-Haenszel statistics ：Cochran 统计量与Mantel-Haenszel 统计量Cochran's Q Cochran Q 检验Code 编码Coefficient 系数Coefficient statistics 系数统计Collinearity 共线性Collinearity diagnostics 共线性诊断Column 列-->单元格中个案数占列总数的百分比Columns 列数-->显示宽度Comma 逗号-->带逗号的数值型变量Common 共同的Compact 最低要求安装Compare 比较，对比Compare groups 分组对比Compare Means 比较平均数Compare variables 比较变量Complete 完成Compute 计算Concordance 和谐Condition 条件Confidence Interval 置信区间Contingency coefficient 列联相关的 C 系数Continue 继续Contrast 对比Control 控制Convert 转换Convert numeric strings to numbers 将数值型字符转换为数字Copy 拷贝Copy Objects 拷贝对象Copy old value 拷贝旧变量Correlate 与……相关-->作相关分析Correlation 相关，关联Correlation Coefficient 相关系数Correlations 皮尔逊（Pearson）相关系数Count 个案数，次数，频数Counted value 用于计数的值Covariance 协方差Covariance Matrix 协方差矩阵Covariance ratio 协方差比率Creat new data file 创建新的数据文件Create Time Series 生成时间序列Criteria 标准（复数）Criterion 标准（单数）Cross-product 叉积Cross-product deviation 叉积离差Cross-product deviations and covariances 叉积离差与协方差Crosstab (=Cross-tabulation) 交叉列表Crosstabs 交叉列表（列联表）分析Cum.(=Cumulative) % of cases 累积百分比Cum.(=Cumulative) N of cases 累积频数Cumulative Percent 累计百分比Cumulative Sum 累积和Currency 货币-->货币型变量Current 目前的Current Selections 当前选择Current Settings 目前设置Current Status 目前状态Curve 曲线Curve Estimation 曲线估算Custom 自定义安装Custom Currency 自定义货币记法Cut 剪切Cut point 切分点Cut points for N equal groups 将数据分为N个个案数相同的组的切分点DData 数据-->数据文件的建立与编辑Data in Chart Are 图中数据为（用于统计量模式的选项）Data View 数据(编辑)窗口Database 数据库，数据文件Date 日期-->日期型变量Decimal 小数-->按小数点对齐Decimals 小数位数Define 定义Define Clusters by 以……定义分组（确定分组变量）Define Dates 定义日期Define Dichotomy 确定二分值Define Groups 确定分组Define Ranges 定义范围Define Sets 定义多选变量集Define Simple Bar 定义简单条形图Define Slices by 以……定义扇形块（确定分块变量）Define Variable Ranges 定义变量范围Degree 度，程度Degrees of freedom 自由度Delete 删除Deleted 删除掉-->删除未选个案，将残差删除Dependent 非独立的，依赖的-->因变量Dependent List 因变量/分析变量列表Derive 推导Derived 推导出的Derived axis 转换轴Descend 下降Descending counts 按频数的降序排列Descending means 按均值的降序列表Descending values 按变量值的降序排列Descriptive 描述性的Descriptive Statistics 描述统计Descriptives 描述统计Destination 目标Deviation 偏差，离差df (=degrees of freedom) 自由度Diagnostics 诊断Dialog 对话（框）Dichotomies 二分变量，二分模式Dichotomy 二分法，二分值，二分变量Dictionary 字典Directory 目录Discrete 离散的Discrete missing values 离散缺失值Dispersion 离散趋势Display 显示Display axis line 显示轴线Display chart with case labels 在图中显示个案标签Display clustered bar charts 显示聚类条图Display Data Info 显示数据的基本信息Display derived axis 显示转换轴Display frequencies tables （在输出结果中）显示频数表Display groups defined by missing values 对由缺失值定义的组也加以显示Display labels 显示标签Display legend 显示图例Display normal curve 显示正态曲线Display order 输出结果列表顺序Display summary tables 显示提要表Display the ReadMe file now? 是否现在显示软件说明文件Distribution 分布Division 分，除法Do not filter cases 不过滤个案Dollar 元，美元-->带美元符号的数值型变量Dot 圆点，句号-->带圆点的数值型变量Draft output 文本输出文件Drop-line 垂线图Durbin-Watson 系列相关检验，Durbin-Watson 系数EEdit 编辑-->文件编辑Enter 进入-->强行进入法Entry 进入Equal 等于Equal Variances Assumed 等方差假定Equal Variances Not Assumed 不假定等方差Equality 相等Estimate 估计-->估计值Eta η系数（用于测量一个定类变量与一个定比/定距变量之间的相关比率）Eta Squared η2Every 每一个Every N labels 每 N个标签Exactly 精确地Exactly M cases from the first N cases 在前N个个案中随机选择M个个案Exclude 排除，拒绝Exclude cases analysis by analysis 剔除分析变量为缺失值的个案Exclude cases listwise 剔除任何含有缺失值的个案Exclude cases listwise within categories 排除分类变量中的缺失值Exclude cases listwise within dichotomies 排除二分变量中的缺失值Exclude cases pairwise 剔除参与相关系数计算的两个变量中有缺失值的个案Exclude cases test-by-test 排除对比中的缺失值Excluded Variables 被拒绝变量Existing 现有的Exit 退出Expect 期望Expected 期望的-->期望频次Expected Range 理论分布范围Expected Value 期望值，理论值Exponential 指数的-->指数分布Export model information to XML file 将模型的信息输出到 XML 文件External file is keyed table 外部文件为关键表Extreme 极端FF F检验的值Factor 因素，因子-->影响因素变量File 文件-->文件操作File is already sorted 数据文件已排序Filter 过滤-->过滤变量Filter variable 过滤变量Filtered 过滤掉的-->生成过滤变量Find 查找Finish 完成First Case 第一个个案（最小值）First value 第一个观测值Fixed and random effects 确定性影响因素和随机影响因素Flag 旗帜，标记；做标记Flag significant correlations 标出达到显著性水平的相关系数Font 字体Footnote 脚注Format 格式Fraction 比率Frame 框，框架Freedom 自由Frequencies 频数分析，按频数作图Frequency 频率，频数Friedman Friedman检验Function 函数GGallery 美术馆-->图形转换功能Gamma γ等级相关系数General 一般General Linear Model 一般线性模型Get from data 由样本观测值确定Go to Case 查找个案Graph 图形Graphs 统计图Grid 网格Grid line 格线Grid lines 用竖线作刻度标志Group 群，组，分组Group Based on 根据……分组Group Statistics 分组统计分析Grouped Median 分组中位值Grouping Variable 分组变量HHelp 帮助Hidden 隐藏High 最大值Histogram 直方图Homogeneity 同质性-->齐次性Homogeneity of variance 方差齐次性Homogeneity-of-variance 方差齐次性检验Horizontal 水平的，横向的Horizontal Alignment 横向对齐方式Hypothesis 假设IIf condition is satisfied 如果满足一定条件Include 包含，包括Include constant in equation 在方程中包含常数项Increment 增量Independent 独立的-->自变量Independent List 自变量/分组变量列表Independent-Samples T Test 独立样本的 T 检验Independent-Samples Test 独立样本检验Indicate 指明，标明Indicate case source as variable 生成一个表明个案来源的变量Individual 单个，个体Influence 影响Influence Statistics 影响因素统计量Info (=Information) 信息Information 信息Inner 里面的Inner Frame 内框-->给图形增加内框Input Variable 输入变量Insert 插入Inside 在（某范围）之内Install (动) 安装Installation (名) 安装Into Different Variables 用重新编码的变量生成一个新变量Into Same Variables 用重新编码的变量取代原变量JJustification 对齐方式KK Independent Samples K个独立样本检验Kappa κ系数Kendall's Coefficient of Concordance 肯德尔和谐系数Kendall's tau-b 肯得尔等级相关τ-b 系数Kendall's tau-c 肯得尔等级相关τ-c 系数Kendall's W Kendall W 检验Key 关键的Key Variables 关键变量Kolmogorov-Smirnov Z 柯尔莫哥洛夫-斯米诺夫 Z 检验Kruskal-Wallis H Kruskal-Wallis H检验（用秩的平方和检验）Kurtosis 峰度，峰度系数LLabel 标签-->变量名标签Label Cases by 以……为个案标签Label Text 标签文字Lambda λ系数Largest value 最大值Last Case 最后一个个案（最大值）Last value 最后一个观测值Launch 开始Launch tutorial now? 现在开始看使用辅导Layer 层-->层变量Left 左-->居左，左对齐Left/bottom 左下Legend 图例Legend Title 图例标题Less 较少，较小Less than 小于Levene StatisticLevene’s Test for Equality of Variances 方差齐性检验License 许可，授权Likelihood 似然性Likelihood Ratio 似然比卡方Line 行，线-->线形图Line Represents 单线条所代表的统计量Line Style 线型Linear 线性Linear model 线性模型Linear Regression 线性回归分析，线性回归模型Lines Represent 多线条所代表的统计量List 列表Listwise 整个数据表Location 位置Log 对数-->对数尺度Lose 丢失Lost 丢失掉Low 最小值Lower 下限Lower Bound 下限Lowest through X 从最小值到 XLSD (=Least-significance difference) 能达到显著性水平的最小差异MMajor 主要的Major Divisions 大分度Mann-Whitney U 曼-维特尼 U检验（用秩和检验的方法进行）Margin 边距Margins 边距设置Marker 标记，标志Match 匹配Match cases on key variables in sorted files 按排序的关键变量匹配个案Match variables across response sets 输出与多选变量进行交叉分析的匹配变量的选择数和以选择总数为基础计算的边缘频率分布Matrix 矩阵-->矩阵散点图Maximum 最小值McNemar 麦克讷马Mean 均值Mean Difference 均值差Mean of Values 变量值的平均数Mean Rank 平均秩和Mean Square 平均平方和Means 平均数分析Means and standard deviations 均值与标准差Means plot 均值分布图Measure 测量-->测量层次Measurement level 测量层次Measures of AssociationMedian 中位数-->中位数检验法Median of Values 变量值的中位数Merge 合并Merge Files 合并(数据)文件Method 方法Minimum 最大值Minor 次要的Minor Divisions 小分度Missing 缺失，变量的缺失值Missing value 缺失值Missing Value Analysis 缺失值分析Mix 混合Mixed 混合的-->混合对齐Mode 众数Mode of Values 变量值的众数Model 模型Model fit 模型配置Model Summary 模型概要More 更多（选项）Moses 莫西Moses extreme reactions 莫西极值反应Most Extreme 最极值Mult (=Multiple) Response Sets 多选变量集Multiple 多个-->多线图Multiple Response 多选变量分析Multiple Variables 多变量选项NN (=Number) 个案数N of cases 各类别的频数Name 名称-->变量名Name & Label 名称与标签Name Variable 名称变量Negative 负Negative Difference 负差Negative Rank 负秩Network 网络New 新的-->新建(各种文件)New Value 新值New Working Data File 新的当前数据文件Next 下一个-->下一步No 否，无No missing values 无缺失值Nominal 定类变量Nominal by Interval 定类变量与定距变量间的相关系数None 无Nonlinear 非线性的-->非线性模型Nonparametric 非参数的Nonparametric Tests 非参数检验Normal 正态的-->正态分布Normal curve 正态曲线Normal probability plot 残差的正态概率图Notation 记号，记法Note 便条Notes 说明文字Number 数字，编号-->数字型变量Number Above 大于设定值的个案数Number Below 小于设定值的个案数Number of Cases 个案数目，频数Number of Runs 游程数Numeric 数值的-->标准数值型变量OObject 对象Observation 观察，观测-->样本观测值Observed 观测到的-->观测到的频次Observed Prop. (=proportion) 观察到的比例Odds 几率Off 关闭，未开启Offset from 距离OK 行了-->执行Old and New Values 新旧变量值的转换Old Value 旧值Omit 省略，略去On 开启One Sample T Test 单样本 T 检验One-sample K-S(Kolmogorov-Smirnov) test 单样柯尔莫哥洛夫-斯米诺夫检验One-Sample Statistics 单样本统计量One-Sample Test 单样本检验One-tailed 单尾的-->单尾检验One-Way ANOVA 简单方差分析Open 打开Open an existing data source 打开现存的数据文件Open another type of file 打开另一种类型的数据文件Open File 打开文档Option 选择，先项Options 先项Order 顺序Order by 排序选项Ordinal 定序变量Organization 组织Organize 组织（动）Organize output by groups 按分组变量组织输出Organize output by variables 各变量单独输出Orientation 定向Other summary function 其它汇总函数Outer 外面的Outer Frame 外框-->给图形增加外框Outlier 远离中心者-->远离均值的值Outliers outside X standard deviations 离均值 X个标准差之外的值Output 输出-->输出文件Output file specification 规定输出文件Output Variable 输出变量Output variables are strings 输出变量是字符型变量Outside 在（某范围）之外Overlay 重叠-->重叠散点图PPackage 包Pair 对，一对Paired 已配对的Paired Sample Correlations 配对样本相关系数Paired Sample Statistics 配对样本统计分析Paired Sample Test 配对样本检验Paired Variables 已配对样本Paired-Samples T Test 配对样本的 t 检验Pairwise 成对地Parameter 参数Parametric 参数的Part and partial correlations 部分相关与偏相关系数Part correlation 部分相关系数Partial 部分，偏Partial correlation 偏相关系数Paste 粘贴PC (=personal computer) 个人电脑Pct (=Percent) 百分比Pearson 皮尔逊Pearson Chi-Square 皮尔逊卡方值Pearson Correlation 皮尔逊相关系数Percentage 百分比Percentage Above 大于设定值的个案的百分比Percentage Below 小于设定值的个案的百分比Percentages 按百分比作图Percentages Based on 百分比基于……Percentile 百分位数Percentile Values 百分位数选项栏Percents and totals based on respondents 百分比与总数基于回答者人数Personal 个人的-->个人安装Personal or Shared Installation 个人或共享安装Phi and Cramér's V 列联相关的 V系数Pie chart 圆饼图，圆形图Pivot 支点，枢轴，在枢轴上的旋转运动-->表格转置（行列互换）Plot 绘图，图形Point 点Poisson 泊桑-->泊桑分布Positive 正Positive Difference 正差Positive Rank 正秩Post Hoc 发生于其后者-->确定均值有差别后的多重比较Predict 预测Predicted 预测的Predicted Values 预测值Prediction Intervals 预测区间Predictor 预测变量Preview 预览Print 打印Print Preview 打印预览Probability 概率Process 处理Processor 处理器Produce 产生，生成Provide 提供pt.(=point) 点-->象素点QQuartile 四分位数RR squared change R2 变化Random 随机的Random Number 随机数Random Number Seed 随机数初始值Random sample of cases 随机选择个案样本Range 范围，全距，极差Range of Grouping Variable 分组变量值的范围Range of missing values 缺失值范围Rank 秩Rank Assigned to Ties 对同值的秩的分配方法选项Rank Cases 对个案排秩Rank Types 秩变量类型Ratio 比率Read 读Read File 读取文件Read Text Data 读入文本格式的数据ReadMe 软件说明文件Ready 准备好了Ready to Install Files 准备安装界面Recent 最近Recently Used Data 最近用过的数据（文件）Recently Used Files 最近用过的文件Recode 重新编码Redo 恢复-->恢复上一步被撤销的操作Reference 参考，参照Reference Line 参照线Refresh 刷新Regression 回归Regression Standardized Residual 回归分析的标准化残差Remote 远程Remote server 远程服务器Removal 剔除Remove 取消-->删除Rename 重命名Replace 替换Replace Missing Values 替换缺失值Replace with mean 用平均值代替（缺失值）Replace working data file 替代当前的数据文件Report 报告Represent 代表Request 要求Reset 重设Residual 残余的，残余物-->残差Residual Statistics 残差统计表Response 回答Right 右-->居右，右对齐Right/top 右上Row 行-->单元格中个案数占行总数的百分比Row Order 行序（行的排列顺序）Run 执行Run Pending Transforms 运行等待中的转换Runs 游程Runs Test 游程检验SS.E. mean 均值的标准误S.E. of mean predictions 对均值标准误的预测Sample 样本Sample size 样本大小Satisfy 满足Save 保存Save As 另存为Save number of case in break group as variable 存储各分组变量的个案数Save standardized values as variables 将标准分存为新变量Save to New File 存为新文档Scale 尺度-->定距/定比变量Scatter 驱散-->散点Scatterplot 散点图Scientific 科学的Scientific Notation 科学计数法Script 程序Seed 种子-->初始值Select 选择Selected Label 选定的标签Selection Variable 选择变量Sequential 序列的Sequential ranks to unique values 相同值的秩次取第一个出现的秩次值Serial 系列的Serial Number 系列号Series 系列Series Displayed as 图形转换选项Server 服务器Set 集，集合Set Definition 多选变量集的定义Set Markers by 以……为标志Setting 设置Setup 安装Setup Complete 完成安装界面Setup Type 安装类型Shade 阴影Shading 阴影设置Share 共享Show 显示Sig. (=significance) 显著性，显著性水平Sign 符号-->符号检验法Significance level 显著性水平Simple 简单的-->简单条形图，单线图，简单散点图Skewness 偏度，偏度系数Slice 片，块-->扇形块Smallest value 最小值Social Science 社会科学Software 软件Software License Agreement 软件授权使用协议Somers' d Somers 等级相关 d 系数Sort 分类，排序Sort Cases 对个案进行归类、排序Sort order 排序规则Sort the file by grouping variables 按分组变量对数据文件进行排序Source 来源Specific 专门的，特定的Specific Values 专门值Specification 规定，规格Split 分割Split Files 分割文件SPSS (Statistical Package for Social Sciences) 社会科学统计软件包SPSS Data Editor 数据窗口编辑界面SPSS/PC(for DOS) DOS 版个人电脑用 SPSSStacked 分段条的-->分段条形图Standard deviation 标准差Standard Error of the Estimate 估计值的标准误Standardized 标准化的-->标准化残差Standardized Coefficient 标准化(回归)系数Standardized Residual Plots 标准化残差图Standardized value 标准分Statistic 统计量Statistical 统计的Statistics 统计学，统计量(复数)Statistics for First Layer 第一层分组的统计量Status 状态Std. Error Difference 标准误之差Std.(=Standard) Deviation 标准差Step 步Stepping 分步引入或剔除变量Stepping Method Criteria 变量引入模型或从模型中剔除的判断标准Stepwise 逐步-->逐步进入法Stop 停止，中断String 字符串-->字符型变量Studentized 将残差学生化Studentized deleted 将残差学生化并删除Style 风格，形式-->线型Subtitle 副标题Sum 总和Sum of Ranks 秩和Sum of Squares 总平方和Sum of Squares and Cross-products 离差平方和与叉积和Sum of Values 变量值的和Summaries for groups of cases 个案组概要（变量值模式）Summaries of separate variables 不同变量的概要（变量模式）Summary 概要-->统计概要Summary Function for Selected Va r i abl e s 选定变量的概要函数Suppress 压制，禁止Suppress table 隐藏表格Suppress tables with more than N categories 分组数大于N时禁止频数表在结果中输出Switch 切换Syntax 语句-->语句文件System 系统System-missing 系统缺失值System-or User-missing 系统或用户缺失值Tt t 统计量T test T 检验Table 表格Tables for 为（某变量）列表-->多选变量分析中要分析的变量Tail 尾Template 模板Tendency 趋势Test 检验Test distribution is Normal 待检分布为正态分布Test for linearity 检验线性相关Test Homogeneity of Variances 方差齐次性检验Test of Significance 显著性检验Test Pair List 待检变量对列表Test Prop.(=proportion) 待检比例Test Proportion 待检概率Test Type 检验类型，检验方法Test Value 待检值Test Variable 分析变量，待检变量Test Variable List 待检变量列表Tests for Several Independent Samples 多个独立样本检验Tick 标记Tick mark 标记Tick marks 用点作刻度标志Tick marks for skipped labels 给被越过的标签作标记Tie 同值-->相等的秩，无差（相等）Time 时间-->时间型变量Time Series 时间系列Title 标题Title Justification 标题对齐方式，标题位置Top 顶部-->上边框Total 总数-->单元格中个案数占个案总数的百分比Transform 转换-->数据转换Transpose 转置Transpose Rows and Columns 行列互换Transposition 转置（名）t-test for Equality of Means 均值相等的 t 检验Tutorial 辅导-->使用辅导Two Independent Samples 两个独立样本的检验Two-Related-Samples Tests 两个相关样本检验Two-tailed 双尾的-->双尾检验Type 类型-->变量类型Typical 典型安装UUncertainty 不确定性Uncertainty coefficient 不定系数Undo 撤销-->撤销上一步操作Uniform 均匀的-->均匀分布Unique 唯一的Unpaired 不匹配Unpaired variable 不匹配变量Unselected 未被选择的Unselected Case Are 未被选中的个案的处理方法选项Unstandardized 非标准化的-->非标准化残差Unstandardized Coefficient 非标准化(回归)系数Untransposed 未被转置的Unweighted 不加权的-->不加权的个案数Unweighted missing 不加权的缺失值数Upper 上限Upper Bound 上限Use chart specifications from 用……的图形规格Use F value 以 F值为标准Use filter variable 使用过滤变量Use probability of F 以 F的概率为标准Use specified range 使用特定的范围Use specified values 使用给定的变量值User 用户User Information 用户信息界面Utilities 实用程序Utility 利用，用途VValid 有效的Valid Cases 有效个案Valid Percent 有效百分比Va lue 值Value label 变量值标签Va lue s 变量值标签Values are group midpoints 取值为各组中点Values are grouped midpoints 变量值为各组中点Values of individual cases 个案值（观测值模式）Va r i abl e 变量Variable label 变量名标签Variable list 变量列表，按变量顺序列表Variable Name and Label 变量名称与标签Variable View 变量(编辑)窗口Variables Are Coded As 变量编码方式Variables Entered/Removed 引入或剔除的变量Variables in Set （多选变量）集中的变量Variance 方差Vertical 垂直的，纵向的Vertical Alignment 纵向对齐方式View 观看-->窗口外观控制Viewer 浏览器WWeigh Cases 给变量加权Weight 重量，轻重-->线的粗细Weight Estimation 权重估计Weighted 加权的-->加权的个案数Weighted missing 加权的缺失值数Welch Welch 检验Welcome 欢迎(界面)Width 宽度Wilcoxon Wilcoxon 检验(用符号秩检验)Wilcoxon Signed Rank Test Wilcoxon符号秩检验Wilcoxon W 威尔科克松 W 检验Window 窗口-->窗口控制Windows 视窗With normal curve 加上正态曲线Within Groups 组内值Working 当前正在用的Working Data File is keyed table 当前数据文件为关键表Working Date File 当前数据文件XX Axis X 轴X through highest 从 X到最大值X through Y 从 X到 YYY Axis Y轴Yes 是ZZ Z 检验值Z Approximation Z 近似估计(整理自网络)。

卡尔曼滤波矩阵名字
卡尔曼滤波（Kalman Filter）涉及到一系列矩阵，其中一些常见的矩阵有特定的名字，以便在卡尔曼滤波算法的文献和讨论中更容易识别。

以下是一些常见的卡尔曼滤波中使用的矩阵及其一般名称：
1. 状态转移矩阵 (State Transition Matrix)：
•通常表示为 A。

2. 观测矩阵（Observation Matrix）：
•通常表示为 H。

3. 控制输入矩阵（Control Input Matrix）：
•通常表示为 B。

4. 过程噪声协方差矩阵（Process Noise Covariance Matrix）：
•通常表示为 Q。

5. 测量噪声协方差矩阵（Measurement Noise Covariance Matrix）：
•通常表示为 R。

6. 估计误差协方差矩阵（Estimation Error Covariance Matrix）：
•通常表示为 P。

7. 卡尔曼增益矩阵（Kalman Gain Matrix）：
•通常表示为 K。

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这些矩阵在卡尔曼滤波算法中扮演着重要的角色，用于描述系统的动态和测量模型，以及噪声的影响。

它们通过卡尔曼滤波算法中的状态估计更新和预测步骤进行迭代计算，用于估计系统的状态。

在具体的卡尔曼滤波应用中，这些矩阵的具体值和尺寸可能会有所不同，取决于具体的问题和系统。

2/ 2。

中文图书分类号:O212.7密级:公开UDC:510学校代码:10005论文题目:部分线性变系数空间面板回归模型的统计推断论文作者:黄建杰学科:统计学指导教师:谢田法副教授论文提交日期:2018年5月UDC:510学校代码:10005中文图书分类号:O212.7学号:S201506084密级:公开北京工业大学理学硕士学位论文题目:部分线性变系数空间面板回归模型的统计推断英文题目:STATISTICAL INFERENCE OF PARTIALLY LINEARV ARYING-COEFFICIENT SPATIAL PANEL REGRESSION MODEL论文作者:黄建杰学科专业:统计学研究方向:应用统计申请学位:理学硕士指导老师:谢田法副教授所在单位:应用数理学院答辩日期:2018年5月授予学位单位:北京工业大学独创性声明本人声明所呈交的论文是我个人在导师指导下进行的研究工作及取得的研究成果。

尽我所知，除了文中特别加以标注和致谢的地方外，论文中不包含其他人已经发表或撰写过的研究成果，也不包含为获得北京工业大学或其它教育机构的学位或证书而使用过的材料。

与我一同工作的同志对本研究所做的任何贡献均已在论文中作了明确的说明并表示了谢意。

签名:黄建杰日期:2018年5月25日关于论文使用授权的说明本人完全了解北京工业大学有关保留、使用学位论文的规定，即：学校有权保留送交论文的复印件，允许论文被查阅和借阅；学校可以公布论文的全部或部分内容，可以采用影印、缩印或其他复制手段保存论文。

（保密的论文在解密后应遵守此规定）签名:黄建杰日期:2018年5月25日导师签名:谢田法日期:2018年5月25日摘要面板数据同时包含截面数据和时间序列，是近年来计量经济学和统计学的研究热点之一。

部分线性变系数回归模型结合了参数模型和非参数模型的特点，具有灵活、容易解释的优点，较经典模型有更好的拟合效果，在统计学和计量经济等领域有广泛的讨论和应用。