DPMdencens之学解

DPMdencens {DPpackage} R Documentation Bayesian density estimation for interval-censored data using a DPM ofnormalsDescriptionThis function generates a posterior density sample for a Dirichlet proc ess mixture of normals model for interval-censored data区间删失数据.UsageDPMdencens(left,right,ngrid=100,grid=NULL,prior,mcmc,state,status) Argumentsleft a vector or matrix giving the lower limit for each response variable. Note that the responses are defined on the entire real line and that unknown limitsshould be indicated by NA.right a vector or matrix giving the upper limit for each response variable. Note that the responses are defined on the entire real line and that unknown limitsshould be indicated by NA.ngrid number of grid points where the density estimate is evaluated. The default value is 100.grid matrix of dimension ngrid*nvar of grid points where the density estimate is evaluated. The default value is NULL and the grid is chosen according to therange of the interval limits.prior a list giving the prior information. The list includes the following parameter: a0 and b0 giving the hyperparameters for prior distribution of theprecision parameter of the Dirichlet process prior, alpha giving the value ofthe precision parameter (it must be specified if a0 is missing, see detailsbelow), nu2 and psiinv2 giving the hyperparameters of the inverted Wishartprior distribution for the scale matrix, Psi1, of the inverted Wishart part ofthe baseline distribution, tau1 and tau2 giving the hyperparameters for thegamma prior distribution of the scale parameter k0 of the normal part of thebaseline distribution,m2 and s2 giving the mean and the covariance of thenormal prior for the mean, m1, of the normal component of the baselinedistribution, respectively, nu1 and psiinv1 (it must be specified if nu2 ismissing, see details below) giving the hyperparameters of the invertedWishart part of the baseline distribution and, m1giving the mean of the normal part of the baseline distribution (it must be specified if m2 is missing, seedetails below) and, k0 giving the scale parameter of the normal part of thebaseline distribution (it must be specified if tau1 is missing, see detailsbelow).mcmc a list giving the MCMC parameters. The list must include the following integers: nburn giving the number of burn-in scans, nskip giving the thinning interval,nsave giving the total number of scans to be saved,and ndisplay giving the number of saved scans to be displayed on screen(the function reports on the screen when every ndisplay iterations have beencarried out).state a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis.status a logical variable indicating whether this run is new (TRUE) or the continuation of a previous analysis (FALSE). In the latter case the currentvalue of the parameters must be specified in the object state.DetailsThis generic function fits a Dirichlet process mixture of normal model for density estimation (Escobar and West, 1995) based on interval-censored data:yij in [lij,uij),i=1,…,n, j=1,…,m,yi | mui, Sigmai ~ N(mui,Sigmai), i=1,…,n,(mui,Sigmai) | G ~ G,G | alpha, G0 ~ DP(alpha G0),where, yi=(yi1,…,yim), and the baseline distribution is the conjugate normal-inverted-Wishart distribution,G0 = N(mu| m1, (1/k0) Sigma) IW (Sigma | nu1, psi1)To complete the model specification, independent hyperpriors are assumed (optional),alpha | a0, b0 ~ Gamma(a0,b0)m1 | m2, s2 ~ N(m2,s2)k0 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)psi1 | nu2, psi2 ~ IW(nu2,psi2)Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).To let part of the baseline distribution fixed at a particular value, set the corresponding hyperparameters of the prior distributions to NULL in the hyperprior specification of the model.Although the baseline distribution, G0, is a conjugate prior in this model specification, an algorithm based on auxiliary parameters is adopted. Specifically, the algorithm 8 with m=1 of Neal (2000) is considered inthe DPMdencens function.Finally, note that this function can be used to fit the DPM of normals model for ordinal data proposed by Kottas, Mueller and Quintana (2005). In this case, the arbitrary cut-off points must be specified in left and right. Samples from the predictive distribution contained in the (last columns) of the object randsave (please see below) can be used to obtain an estimate of the cell probabilities.ValueAn object of class DPMdencens representing the DP mixture of normals model fit. Generic functions such as print, summary, and plot have methods to show the results of the fit. The results include the baseline parameters, alpha, and the number of clusters.The function DPrandom can be used to extract the posterior mean of the subject-specific means and covariance matrices.The MCMC samples of the parameters and the errors in the model are stored in the object thetasave and randsave, respectively. Both objects are included in the list save.state and are matrices which can be analyzed directly by functions provided by the coda package.The list state in the output object contains the current value of the parameters necessary to restart the analysis. If you want to specify different starting values to run multiple chains set status=TRUE and create the list state based on this starting values. In this case the list state must include the following objects:ncluster an integer giving the number of clusters.muclus a matrix of dimension (nobservations+100)*(nvariables) giving the means of the clusters (only the first ncluster are considered to start the chain). sigmaclus a matrix of dimension(nobservations+100)*( (nvariables)*((nvariables)+1)/2) giving the lowermatrix of the covariance matrix of the clusters (only the first ncluster areconsidered to start the chain).ss an interger vector defining to which of the ncluster clusters each observation belongs.alpha giving the value of the precision parameter.m1giving the mean of the normal components of the baseline distribution.k0giving the scale parameter of the normal part of the baseline distribution. psi1giving the scale matrix of the inverted-Wishart part of the baseline distribution.y giving the matrix of imputed data points.Author(s)Alejandro Jara <atjara@uc.cl>ReferencesEscobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.Kottas, A., Mueller, P., Quintana, F. (2005). Nonparametric Bayesian modeling for multivariate ordinal data. Journal of Computational and Graphical Statistics, 14: 610-625.Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.See AlsoDPrandom, DPdensityExamples## Not run:##################################### Bivariate example:# Censored data is artificially# created####################################data(airquality)缺失数据为NAattach(airquality)将数据中变量能直接读取ozone <- Ozone**(1/3)开立方根radiation <- Solar.R重新赋名y <- na.omit(cbind(radiation,ozone))删除带有na的行,并给出行号# create censored-dataxxlim <- seq(0,300,50)yylim <- seq(1.5,5.5,1)生成两个数列left <- matrix(0,nrow=nrow(y),ncol=2)right <- matrix(0,nrow=nrow(y),ncol=2)生成与变量y同行,列为2的全0阵for(i in 1:nrow(y)){left[i,1] <- NAright[i,1] <- NAif(y[i,1] < xxlim[1]) right[i,1] <- xxlim[1]for(j in 1:length(xxlim)){if(y[i,1] >= xxlim[j]) left[i,1] <- xxlim[j]if(y[i,1] >= xxlim[j]) right[i,1] <- xxlim[j+1] }left[i,2] <- NAright[i,2] <- NAif(y[i,2] < yylim[1]) right[i,2] <- yylim[1]for(j in 1:length(yylim)){if(y[i,2] >= yylim[j]) left[i,2] <- yylim[j]if(y[i,2] >= yylim[j]) right[i,2] <- yylim[j+1] }}> left[,1] [,2][1,] 150 2.5[2,] 100 2.5[3,] 100 1.5[4,] 300 2.5[5,] 250 2.5[6,] 50 2.5[7,] 0 1.5[8,] 250 2.5[9,] 250 1.5[10,] 250 1.5[11,] 50 2.5[12,] 300 1.5[13,] 300 2.5[14,] 50 1.5[15,] 300 2.5[16,] 0 1.5[17,] 0 NA[18,] 300 1.5[19,] 0 1.5[20,] 50 2.5[21,] 0 2.5[22,] 250 3.5[23,] 200 4.5[24,] 250 2.5[25,] 100 2.5[26,] 250 3.5[28,] 100 2.5 [29,] 150 2.5 [30,] 250 2.5 [31,] 0 2.5 [32,] 100 1.5 [33,] 100 1.5 [34,] 250 4.5 [35,] 200 3.5 [36,] 200 2.5 [37,] 150 3.5 [38,] 300 2.5 [39,] 250 3.5 [40,] 250 4.5 [41,] 250 4.5 [42,] 150 3.5 [43,] 250 1.5 [44,] 150 2.5 [45,] 0 1.5 [46,] 250 3.5 [47,] 250 2.5 [48,] 250 3.5 [49,] 150 3.5 [50,] 200 3.5 [51,] 0 2.5 [52,] 250 3.5 [53,] 200 4.5 [54,] 50 2.5 [55,] 50 3.5 [56,] 200 3.5 [57,] 250 3.5 [58,] 250 3.5 [59,] 250 3.5 [60,] 50 2.5 [61,] 0 1.5 [62,] 50 2.5 [63,] 250 4.5 [64,] 200 3.5 [65,] 200 4.5 [66,] 150 3.5 [67,] 250 2.5 [68,] 150 3.5 [69,] 50 2.5 [70,] 50 3.5[72,] 200 2.5 [73,] 150 3.5 [74,] 250 2.5 [75,] 0 1.5 [76,] 200 3.5 [77,] 200 5.5 [78,] 200 3.5 [79,] 200 3.5 [80,] 200 4.5 [81,] 200 3.5 [82,] 150 3.5 [83,] 150 4.5 [84,] 150 3.5 [85,] 150 3.5 [86,] 150 3.5 [87,] 50 3.5 [88,] 50 2.5 [89,] 250 2.5 [90,] 200 2.5 [91,] 200 2.5 [92,] 250 2.5 [93,] 200 3.5 [94,] 250 2.5 [95,] 200 2.5 [96,] 0 1.5 [97,] 100 1.5 [98,] 200 3.5 [99,] 200 2.5 [100,] 0 1.5 [101,] 200 2.5 [102,] 200 2.5 [103,] 200 1.5 [104,] 0 2.5 [105,] 100 2.5 [106,] 0 1.5 [107,] 0 1.5 [108,] 150 2.5 [109,] 150 1.5 [110,] 100 2.5 [111,] 200 2.5 > right[,1] [,2] [1,] 200 3.5[3,] 150 2.5 [4,] NA 3.5 [5,] 300 3.5 [6,] 100 3.5 [7,] 50 2.5 [8,] 300 3.5 [9,] 300 2.5 [10,] 300 2.5 [11,] 100 3.5 [12,] NA 2.5 [13,] NA 3.5 [14,] 100 2.5 [15,] NA 3.5 [16,] 50 2.5 [17,] 50 1.5 [18,] NA 2.5 [19,] 50 2.5 [20,] 100 3.5 [21,] 50 3.5 [22,] 300 4.5 [23,] 250 5.5 [24,] 300 3.5 [25,] 150 3.5 [26,] 300 4.5 [27,] NA 3.5 [28,] 150 3.5 [29,] 200 3.5 [30,] 300 3.5 [31,] 50 3.5 [32,] 150 2.5 [33,] 150 2.5 [34,] 300 5.5 [35,] 250 4.5 [36,] 250 3.5 [37,] 200 4.5 [38,] NA 3.5 [39,] 300 4.5 [40,] 300 5.5 [41,] 300 5.5 [42,] 200 4.5 [43,] 300 2.5 [44,] 200 3.5 [45,] 50 2.5[47,] 300 3.5 [48,] 300 4.5 [49,] 200 4.5 [50,] 250 4.5 [51,] 50 3.5 [52,] 300 4.5 [53,] 250 5.5 [54,] 100 3.5 [55,] 100 4.5 [56,] 250 4.5 [57,] 300 4.5 [58,] 300 4.5 [59,] 300 4.5 [60,] 100 3.5 [61,] 50 2.5 [62,] 100 3.5 [63,] 300 5.5 [64,] 250 4.5 [65,] 250 5.5 [66,] 200 4.5 [67,] 300 3.5 [68,] 200 4.5 [69,] 100 3.5 [70,] 100 4.5 [71,] 150 3.5 [72,] 250 3.5 [73,] 200 4.5 [74,] 300 3.5 [75,] 50 2.5 [76,] 250 4.5 [77,] 250 NA [78,] 250 4.5 [79,] 250 4.5 [80,] 250 5.5 [81,] 250 4.5 [82,] 200 4.5 [83,] 200 5.5 [84,] 200 4.5 [85,] 200 4.5 [86,] 200 4.5 [87,] 100 4.5 [88,] 100 3.5 [89,] 300 3.5[90,] 250 3.5[91,] 250 3.5[92,] 300 3.5[93,] 250 4.5[94,] 300 3.5[95,] 250 3.5[96,] 50 2.5[97,] 150 2.5[98,] 250 4.5[99,] 250 3.5[100,] 50 2.5[101,] 250 3.5[102,] 250 3.5[103,] 250 2.5[104,] 50 3.5[105,] 150 3.5[106,] 50 2.5[107,] 50 2.5[108,] 200 3.5[109,] 200 2.5[110,] 150 3.5[111,] 250 3.5# Prior informations2 <- matrix(c(10000,0,0,1),ncol=2)> s2[,1] [,2][1,] 10000 0[2,] 0 1m2 <- c(180,3)psiinv2 <- diag(c(1/10000,1),2)prior <- list(alpha=1,nu1=4,nu2=4,s2=s2,m2=m2,psiinv2=psiinv2,tau1=0.01,tau2=0.01)# Initial statestate <- NULL# MCMC parametersnburn <- 5000nsave <- 5000nskip <- 3ndisplay <- 1000mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)# Fitting the modelfit1 <- DPMdencens(left=left,right=right,ngrid=100,prior=prior,mcmc=mcmc,state=state,status=TRUE)fit1DPM model for interval-censored dataCall:DPMdencens.default(left = left, right = right, ngrid = 100, prior = prior, mcmc = mcmc, state = state, status = TRUE)Posterior Inference of Parameters:m1-var1 m1-var2 k0 psi1-var1:var1162.672129 3.184599 0.107712 0.000535psi1-var1:var2 psi1-var2:var2 ncluster alpha-0.009391 1.021222 4.219400 1.000000Number of Observations: 111Number of Variables: 2summary(fit1)DPM model for interval-censored dataCall:DPMdencens.default(left = left, right = right, ngrid = 100, prior = prior, mcmc = mcmc, state = state, status = TRUE)Baseline distribution:Mean Median Std. Dev. Naive Std.Errorm1-var1 1.627e+02 1.609e+02 6.431e+01 9.094e-01m1-var2 3.185e+00 3.190e+00 7.995e-01 1.131e-02k0 1.077e-01 5.657e-02 1.595e-01 2.256e-03psi1-var1:var1 5.350e-04 3.130e-04 6.453e-04 9.126e-06psi1-var1:var2 -9.391e-03 -4.588e-03 1.651e-02 2.334e-04psi1-var2:var2 1.021e+00 8.431e-01 6.868e-01 9.713e-0395%HPD-Low 95%HPD-Uppm1-var1 4.102e+01 2.914e+02m1-var2 1.586e+00 4.738e+00k0 1.785e-04 3.740e-01psi1-var1:var1 2.886e-05 1.664e-03psi1-var1:var2 -4.324e-02 1.130e-02psi1-var2:var2 2.344e-01 2.238e+00Precision parameter:Mean Median Std. Dev. Naive Std.Error 95%HPD-Low ncluster 4.21940 4.00000 1.32711 0.01877 2.00000 95%HPD-Uppncluster 7.00000Number of Observations: 111Number of Variables: 2# Plot the estimated densityplot(fit1)# Extracting the univariate density estimatescbind(fit1$grid[,1],fit1$funi[[1]])cbind(fit1$grid[,2],fit1$funi[[2]])# Extracting the bivariate density estimatesfit1$grid[,1]fit1$grid[,2]fit1$fbiv[[1]]# Plot of the estimated density along with the# true data points and censoring limitscontour(fit1$grid[,1],fit1$grid[,2],fit1$fbiv[[1]])points(y)abline(v=xxlim) abline(h=yylim)## End(Not run)。