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check bulk data library size: vary from few hundred(100-200) to few thousands(1600). Here
GSE50244bulkeset <- readRDS("~/misc/data/deconv/GSE50244bulkeset.rds")
GSE50244bulkesetExpressionSet (storageMode: lockedEnvironment)
assayData: 32581 features, 89 samples
element names: exprs
protocolData: none
phenoData
sampleNames: Sub1 Sub2 ... Sub89 (89 total)
varLabels: sampleID SubjectName ... tissue (7 total)
varMetadata: labelDescription
featureData: none
experimentData: use 'experimentData(object)'
Annotation: summary(as.numeric(apply(exprs(GSE50244bulkeset),2,mean))) Min. 1st Qu. Median Mean 3rd Qu. Max.
376.3 506.4 605.8 587.1 664.4 793.6 Mousebulkeset <- readRDS("~/misc/data/deconv/Mousebulkeset.rds")
summary(as.numeric(apply(exprs(Mousebulkeset),2,mean))) Min. 1st Qu. Median Mean 3rd Qu. Max.
1431 1546 1608 1625 1729 1805 We have scRNA-seq data from Segerstolpe et al. (2016) including the 1097 cells from 6 healthy subjects, taken from here. This dataset is non-UMI so the read counts are huge. In the following simulation study, 4 cell types - acinar, alpha, beta, and ductal cells are included.
EMTAB.eset = readRDS('data/deconv/EMTABesethealthy.rds')
EMTAB.esetExpressionSet (storageMode: lockedEnvironment)
assayData: 25453 features, 1097 samples
element names: exprs
protocolData: none
phenoData
sampleNames: AZ_A10 AZ_A11 ... HP1509101_P9 (1097 total)
varLabels: sampleID SubjectName cellTypeID cellType
varMetadata: labelDescription
featureData: none
experimentData: use 'experimentData(object)'
Annotation: table(EMTAB.eset$cellType)
acinar alpha beta
112 443 171
co-expression delta ductal
26 59 135
endothelial epsilon gamma
13 5 75
mast MHC class II PSC
4 1 23
unclassified unclassified endocrine
1 29 Now obtain the true gene relative expression in each cell type.
Preprocess the data: 1. remove genes appearing in less than 10 cells; 2. remove top \(5\%\) expressed genes
cell_types = c('acinar','alpha','beta','ductal')
###remove genes appeared in too few cells
###remove genes that are overly expressed
#rm.idx = which(rowSums((exprs(EMTAB.eset)[,which(EMTAB.eset$cellType%in%cell_types)])!=0)<10)
#rr = rowSums((exprs(EMTAB.eset)[,which(EMTAB.eset$cellType%in%cell_types)]))
###rm.idx = which(rr==0)
###rm.idx1 = which(rr<=quantile(rr[-rm.idx],0.01))
#rm.idx2 = which(rr>=quantile(rr[-rm.idx],0.99))
#rm.idx = unique(c(rm.idx,rm.idx2))
acinar = exprs(EMTAB.eset)[,which(EMTAB.eset$cellType=='acinar')]
alpha = exprs(EMTAB.eset)[,which(EMTAB.eset$cellType=='alpha')]
beta = exprs(EMTAB.eset)[,which(EMTAB.eset$cellType=='beta')]
ductal = exprs(EMTAB.eset)[,which(EMTAB.eset$cellType=='ductal')]
rm.idx1 = which(((rowSums(acinar!=0)<=10)|(rowSums(acinar)>=quantile(rowSums(acinar),0.95))))
rm.idx2 = which(((rowSums(alpha!=0)<=10)|(rowSums(alpha)>=quantile(rowSums(alpha),0.95))))
rm.idx3 = which(((rowSums(beta!=0)<=10)|(rowSums(beta)>=quantile(rowSums(beta),0.95))))
rm.idx4 = which(((rowSums(ductal!=0)<=10)|(rowSums(ductal)>=quantile(rowSums(ductal),0.95))))
rm.idx = unique(c(rm.idx1,rm.idx2,rm.idx3,rm.idx4))
acinar = acinar[-rm.idx,]
alpha = alpha[-rm.idx,]
beta = beta[-rm.idx,]
ductal = ductal[-rm.idx,]Check cell library size: cell library sizes are big
summary(colSums(acinar)) Min. 1st Qu. Median Mean 3rd Qu. Max.
2686 40757 75941 129023 186640 553802 summary(colSums(alpha)) Min. 1st Qu. Median Mean 3rd Qu. Max.
3523 40465 83358 113020 159609 690603 summary(colSums(beta)) Min. 1st Qu. Median Mean 3rd Qu. Max.
2970 47179 100240 124888 190024 537763 summary(colSums(ductal)) Min. 1st Qu. Median Mean 3rd Qu. Max.
6405 38504 86323 135121 191832 739986 # cell type specific gene relative expression
Theta = cbind(rowSums(acinar)/sum(acinar),
rowSums(alpha)/sum(alpha),
rowSums(beta)/sum(beta),
rowSums(ductal)/sum(ductal))
cor(Theta) [,1] [,2] [,3] [,4]
[1,] 1.0000000 0.6653738 0.5972479 0.8288408
[2,] 0.6653738 1.0000000 0.8156478 0.6819325
[3,] 0.5972479 0.8156478 1.0000000 0.6363127
[4,] 0.8288408 0.6819325 0.6363127 1.0000000kappa(t(Theta)%*%Theta)[1] 34.74529# cell library size
S = c(sum(acinar)/ncol(acinar),
sum(alpha)/ncol(alpha),
sum(beta)/ncol(beta),
sum(ductal)/ncol(ductal))
#S=S/100 Generate reference data: 1. obtain \(\tilde{\theta}_{gk}\) by summing up read counts and normalize; 2. obtain \(y_{k}^{r,+}\) by multiple cell library size and number of cells(set to 100); 3. Generate reference data using the Poisson model
Generate bulk data: 1. simulate library size \(y^{b,+}\) from Poisson\((300G)\); 2. generate \(x^b_g\) by multiplying \(N=400\), \(\beta = (0.1,0.2,0.3,0.4)\), and \(S\) cell library size then normalize to \(\theta^b_g\). 3. generate bulk data using Poisson distribution.
G = nrow(Theta)
K = ncol(Theta)
set.seed(12345)
# bulk data library size: 300*number of genes.
# bulk_ls = rpois(1,300*nrow(Theta))
# total number of cells in bulk data
bulk_ncell = 400
# cell proportions
bulk_beta = c(1,2,3,4)
bulk_beta = bulk_beta/sum(bulk_beta)
bulk_p = (bulk_beta * (S))/sum(bulk_beta * (S))
# bulk data gene relative expression.
bulk_X = bulk_ncell*Theta%*%diag(S)%*%bulk_beta
bulk_theta = bulk_X/sum(bulk_X)
ref_ncell = 100
ref_X = Theta%*%diag(S)*ref_ncell
w = rep(1,G)
p_tilde_est = c()
p_tilde_var = c()
p_est = c()
p_var = c()
nrep = 100
for(rep in 1:nrep){
if(rep%%200==0){print(rep)}
bulk_ls = rpois(1,300*nrow(Theta))
Z = matrix(rpois(G*K,ref_X),ncol=K)
y = rpois(G,bulk_ls*bulk_theta)
#W = diag(w)
#beta_hat = solve(t(Z)%*%W%*%Z - diag(colSums(W%*%Z)))%*%t(Z)%*%W%*%y
U = diag(colSums(Z))
Zu = Z%*%(solve(U))
Zuu = Zu%*%solve(U)
p_tilde_hat = solve(t(Zu)%*%Zu - diag(colSums(Zuu)))%*%t(Zu)%*%y
p_tilde_est = rbind(p_tilde_est,c(p_tilde_hat))
#beta_hat = solve(t(Z)%*%Z)%*%t(Z)%*%y
#beta_hat
#beta_hat/sum(beta_hat)
Q = 0
Sigma=0
deltas = c()
for(i in 1:length(y)){
ag = Zu[i,]%*%t(Zu[i,])-diag(Zuu[i,])
Q = Q + ag
Delta = (ag%*%p_tilde_hat-y[i]*Zu[i,])
Sigma = Sigma + Delta%*%t(Delta)
}
Q = Q/G
Sigma = Sigma/G
J = matrix(nrow = K,ncol=K)
for(i in 1:K){
for(j in 1:K){
if(i==j){
J[i,j] = sum(p_tilde_hat)-p_tilde_hat[i]
}else{
J[i,j] = -p_tilde_hat[i]
}
}
}
J = J/sum(p_tilde_hat)^2
asyV = (t(J)%*%solve(Q)%*%Sigma%*%solve(Q)%*%J)/G
p_hat = p_tilde_hat/sum(p_tilde_hat)
p_est = rbind(p_est,c(p_hat))
p_tilde_var = rbind(p_tilde_var,diag(solve(Q)%*%Sigma%*%solve(Q)))
p_var = rbind(p_var,diag(asyV))
}
#### coverage before delta method:
#ci_l = p_tilde_est - qnorm(0.975)*sqrt(p_tilde_var)
#ci_r = p_tilde_est + qnorm(0.975)*sqrt(p_tilde_var)
#coverage
#coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
#apply(coverage,2,mean)
#### converage after delta method
ci_l = p_est - qnorm(0.975)*sqrt(p_var)
ci_r = p_est + qnorm(0.975)*sqrt(p_var)
#coverage
coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
apply(coverage,2,mean)[1] 0.98 0.98 0.94 0.84#for(i in 1:4){
# plot(ci_l[,i],type='l',ylim=range(c(ci_l[,i],ci_r[,i])))
# lines(ci_r[,i],col='grey80')
# lines(rep(bulk_p[i],100),lty = 2)
#}how about decrease bulk library size to \(50G\)?
G = nrow(Theta)
K = ncol(Theta)
set.seed(12345)
# bulk data library size: 300*number of genes.
# bulk_ls = rpois(1,300*nrow(Theta))
# total number of cells in bulk data
bulk_ncell = 400
# cell proportions
bulk_beta = c(1,2,3,4)
bulk_beta = bulk_beta/sum(bulk_beta)
bulk_p = (bulk_beta * (S))/sum(bulk_beta * (S))
# bulk data gene relative expression.
bulk_X = bulk_ncell*Theta%*%diag(S)%*%bulk_beta
bulk_theta = bulk_X/sum(bulk_X)
ref_ncell = 100
ref_X = Theta%*%diag(S)*ref_ncell
w = rep(1,G)
p_tilde_est = c()
p_tilde_var = c()
p_est = c()
p_var = c()
nrep = 100
for(rep in 1:nrep){
if(rep%%200==0){print(rep)}
bulk_ls = rpois(1,50*nrow(Theta))
Z = matrix(rpois(G*K,ref_X),ncol=K)
y = rpois(G,bulk_ls*bulk_theta)
#W = diag(w)
#beta_hat = solve(t(Z)%*%W%*%Z - diag(colSums(W%*%Z)))%*%t(Z)%*%W%*%y
U = diag(colSums(Z))
Zu = Z%*%(solve(U))
Zuu = Zu%*%solve(U)
p_tilde_hat = solve(t(Zu)%*%Zu - diag(colSums(Zuu)))%*%t(Zu)%*%y
p_tilde_est = rbind(p_tilde_est,c(p_tilde_hat))
#beta_hat = solve(t(Z)%*%Z)%*%t(Z)%*%y
#beta_hat
#beta_hat/sum(beta_hat)
Q = 0
Sigma=0
deltas = c()
for(i in 1:length(y)){
ag = Zu[i,]%*%t(Zu[i,])-diag(Zuu[i,])
Q = Q + ag
Delta = (ag%*%p_tilde_hat-y[i]*Zu[i,])
Sigma = Sigma + Delta%*%t(Delta)
}
Q = Q/G
Sigma = Sigma/G
J = matrix(nrow = K,ncol=K)
for(i in 1:K){
for(j in 1:K){
if(i==j){
J[i,j] = sum(p_tilde_hat)-p_tilde_hat[i]
}else{
J[i,j] = -p_tilde_hat[i]
}
}
}
J = J/sum(p_tilde_hat)^2
asyV = (t(J)%*%solve(Q)%*%Sigma%*%solve(Q)%*%J)/G
p_hat = p_tilde_hat/sum(p_tilde_hat)
p_est = rbind(p_est,c(p_hat))
p_tilde_var = rbind(p_tilde_var,diag(solve(Q)%*%Sigma%*%solve(Q)))
p_var = rbind(p_var,diag(asyV))
}
ci_l = p_est - qnorm(0.975)*sqrt(p_var)
ci_r = p_est + qnorm(0.975)*sqrt(p_var)
#coverage
coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
apply(coverage,2,mean)[1] 0.96 0.98 0.95 0.77#for(i in 1:4){
# plot(ci_l[,i],type='l',ylim=range(c(ci_l[,i],ci_r[,i])))
# lines(ci_r[,i],col='grey80')
# lines(rep(bulk_p[i],100),lty = 2)
#}how about change \(beta\) to be \((0.4,0.3,0.2,0.1)\)?
G = nrow(Theta)
K = ncol(Theta)
set.seed(12345)
# bulk data library size: 300*number of genes.
# bulk_ls = rpois(1,300*nrow(Theta))
# total number of cells in bulk data
bulk_ncell = 400
# cell proportions
bulk_beta = c(4,3,2,1)
bulk_beta = bulk_beta/sum(bulk_beta)
bulk_p = (bulk_beta * (S))/sum(bulk_beta * (S))
# bulk data gene relative expression.
bulk_X = bulk_ncell*Theta%*%diag(S)%*%bulk_beta
bulk_theta = bulk_X/sum(bulk_X)
ref_ncell = 100
ref_X = Theta%*%diag(S)*ref_ncell
w = rep(1,G)
p_tilde_est = c()
p_tilde_var = c()
p_est = c()
p_var = c()
nrep = 100
for(rep in 1:nrep){
if(rep%%200==0){print(rep)}
bulk_ls = rpois(1,300*nrow(Theta))
Z = matrix(rpois(G*K,ref_X),ncol=K)
y = rpois(G,bulk_ls*bulk_theta)
#W = diag(w)
#beta_hat = solve(t(Z)%*%W%*%Z - diag(colSums(W%*%Z)))%*%t(Z)%*%W%*%y
U = diag(colSums(Z))
Zu = Z%*%(solve(U))
Zuu = Zu%*%solve(U)
p_tilde_hat = solve(t(Zu)%*%Zu - diag(colSums(Zuu)))%*%t(Zu)%*%y
p_tilde_est = rbind(p_tilde_est,c(p_tilde_hat))
#beta_hat = solve(t(Z)%*%Z)%*%t(Z)%*%y
#beta_hat
#beta_hat/sum(beta_hat)
Q = 0
Sigma=0
deltas = c()
for(i in 1:length(y)){
ag = Zu[i,]%*%t(Zu[i,])-diag(Zuu[i,])
Q = Q + ag
Delta = (ag%*%p_tilde_hat-y[i]*Zu[i,])
Sigma = Sigma + Delta%*%t(Delta)
}
Q = Q/G
Sigma = Sigma/G
J = matrix(nrow = K,ncol=K)
for(i in 1:K){
for(j in 1:K){
if(i==j){
J[i,j] = sum(p_tilde_hat)-p_tilde_hat[i]
}else{
J[i,j] = -p_tilde_hat[i]
}
}
}
J = J/sum(p_tilde_hat)^2
asyV = (t(J)%*%solve(Q)%*%Sigma%*%solve(Q)%*%J)/G
p_hat = p_tilde_hat/sum(p_tilde_hat)
p_est = rbind(p_est,c(p_hat))
p_tilde_var = rbind(p_tilde_var,diag(solve(Q)%*%Sigma%*%solve(Q)))
p_var = rbind(p_var,diag(asyV))
}
ci_l = p_est - qnorm(0.975)*sqrt(p_var)
ci_r = p_est + qnorm(0.975)*sqrt(p_var)
#coverage
coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
apply(coverage,2,mean)[1] 0.88 0.96 0.95 1.00#for(i in 1:4){
# plot(ci_l[,i],type='l',ylim=range(c(ci_l[,i],ci_r[,i])))
# lines(ci_r[,i],col='grey80')
# lines(rep(bulk_p[i],100),lty = 2)
#}The bigger the beta, the lower the coverage??
load("~/misc/data/scde/scCD14.RData")
load("~/misc/data/scde/scCD4.RData")
load("~/misc/data/scde/scCD8.RData")
#load("~/misc/data/scde/scCDT.RData")
load("~/misc/data/scde/scMB.RData")
# remove genes
CD14 = as.matrix(CD14)
CD4 = as.matrix(CD4)
CD8 = as.matrix(CD8)
#CDT = as.matrix(CDT)
MB = as.matrix(MB)
rm.idx1 = which(((rowSums(CD14!=0)<=10)|(rowSums(CD14)>=quantile(rowSums(CD14),0.95))))
rm.idx2 = which(((rowSums(CD4!=0)<=10)|(rowSums(CD4)>=quantile(rowSums(CD4),0.95))))
rm.idx3 = which(((rowSums(CD8!=0)<=10)|(rowSums(CD8)>=quantile(rowSums(CD8),0.95))))
#rm.idx4 = which(((rowSums(CDT!=0)<=10)|(rowSums(CDT)>=quantile(rowSums(CDT),0.95))))
rm.idx5 = which(((rowSums(MB!=0)<=10)|(rowSums(MB)>=quantile(rowSums(MB),0.95))))
rm.idx = unique(c(rm.idx1,rm.idx2,rm.idx3,rm.idx5))
CD14 = CD14[-rm.idx,]
CD4 = CD4[-rm.idx,]
CD8 = CD8[-rm.idx,]
#CDT = CDT[-rm.idx,]
MB = MB[-rm.idx,]
Theta = cbind(rowSums(CD14)/sum(CD14),
rowSums(CD4)/sum(CD4),
rowSums(CD8)/sum(CD8),
#rowSums(CDT)/sum(CDT),
rowSums(MB)/sum(MB))
cor(Theta) [,1] [,2] [,3] [,4]
[1,] 1.0000000 0.8534940 0.7874808 0.6553133
[2,] 0.8534940 1.0000000 0.7390032 0.6612358
[3,] 0.7874808 0.7390032 1.0000000 0.5953219
[4,] 0.6553133 0.6612358 0.5953219 1.0000000kappa(t(Theta)%*%Theta)[1] 57.73301# cell library size
S = c(sum(CD14)/ncol(CD14),
sum(CD4)/ncol(CD4),
sum(CD8)/ncol(CD8),
sum(MB)/ncol(MB))G = nrow(Theta)
K = ncol(Theta)
set.seed(12345)
# bulk data library size: 300*number of genes.
# bulk_ls = rpois(1,300*nrow(Theta))
# total number of cells in bulk data
bulk_ncell = 400
# cell proportions
bulk_beta = c(1,2,3,4)
bulk_beta = bulk_beta/sum(bulk_beta)
bulk_p = (bulk_beta * (S))/sum(bulk_beta * (S))
# bulk data gene relative expression.
bulk_X = bulk_ncell*Theta%*%diag(S)%*%bulk_beta
bulk_theta = bulk_X/sum(bulk_X)
ref_ncell = 100
ref_X = Theta%*%diag(S)*ref_ncell
w = rep(1,G)
p_tilde_est = c()
p_tilde_var = c()
p_est = c()
p_var = c()
nrep = 100
for(rep in 1:nrep){
if(rep%%200==0){print(rep)}
bulk_ls = rpois(1,300*nrow(Theta))
Z = matrix(rpois(G*K,ref_X),ncol=K)
y = rpois(G,bulk_ls*bulk_theta)
#W = diag(w)
#beta_hat = solve(t(Z)%*%W%*%Z - diag(colSums(W%*%Z)))%*%t(Z)%*%W%*%y
U = diag(colSums(Z))
Zu = Z%*%(solve(U))
Zuu = Zu%*%solve(U)
p_tilde_hat = solve(t(Zu)%*%Zu - diag(colSums(Zuu)))%*%t(Zu)%*%y
p_tilde_est = rbind(p_tilde_est,c(p_tilde_hat))
#beta_hat = solve(t(Z)%*%Z)%*%t(Z)%*%y
#beta_hat
#beta_hat/sum(beta_hat)
Q = 0
Sigma=0
deltas = c()
for(i in 1:length(y)){
ag = Zu[i,]%*%t(Zu[i,])-diag(Zuu[i,])
Q = Q + ag
Delta = (ag%*%p_tilde_hat-y[i]*Zu[i,])
Sigma = Sigma + Delta%*%t(Delta)
}
Q = Q/G
Sigma = Sigma/G
J = matrix(nrow = K,ncol=K)
for(i in 1:K){
for(j in 1:K){
if(i==j){
J[i,j] = sum(p_tilde_hat)-p_tilde_hat[i]
}else{
J[i,j] = -p_tilde_hat[i]
}
}
}
J = J/sum(p_tilde_hat)^2
asyV = (t(J)%*%solve(Q)%*%Sigma%*%solve(Q)%*%J)/G
p_hat = p_tilde_hat/sum(p_tilde_hat)
p_est = rbind(p_est,c(p_hat))
p_tilde_var = rbind(p_tilde_var,diag(solve(Q)%*%Sigma%*%solve(Q)))
p_var = rbind(p_var,diag(asyV))
}
ci_l = p_est - qnorm(0.975)*sqrt(p_var)
ci_r = p_est + qnorm(0.975)*sqrt(p_var)
#coverage
coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
apply(coverage,2,mean)[1] 0.99 0.98 0.91 0.95how about change \(beta\) to be \((0.4,0.3,0.2,0.1)\)?
G = nrow(Theta)
K = ncol(Theta)
set.seed(12345)
# bulk data library size: 300*number of genes.
# bulk_ls = rpois(1,300*nrow(Theta))
# total number of cells in bulk data
bulk_ncell = 400
# cell proportions
bulk_beta = c(4,3,2,1)
bulk_beta = bulk_beta/sum(bulk_beta)
bulk_p = (bulk_beta * (S))/sum(bulk_beta * (S))
# bulk data gene relative expression.
bulk_X = bulk_ncell*Theta%*%diag(S)%*%bulk_beta
bulk_theta = bulk_X/sum(bulk_X)
ref_ncell = 100
ref_X = Theta%*%diag(S)*ref_ncell
w = rep(1,G)
p_tilde_est = c()
p_tilde_var = c()
p_est = c()
p_var = c()
nrep = 100
for(rep in 1:nrep){
if(rep%%200==0){print(rep)}
bulk_ls = rpois(1,300*nrow(Theta))
Z = matrix(rpois(G*K,ref_X),ncol=K)
y = rpois(G,bulk_ls*bulk_theta)
#W = diag(w)
#beta_hat = solve(t(Z)%*%W%*%Z - diag(colSums(W%*%Z)))%*%t(Z)%*%W%*%y
U = diag(colSums(Z))
Zu = Z%*%(solve(U))
Zuu = Zu%*%solve(U)
p_tilde_hat = solve(t(Zu)%*%Zu - diag(colSums(Zuu)))%*%t(Zu)%*%y
p_tilde_est = rbind(p_tilde_est,c(p_tilde_hat))
#beta_hat = solve(t(Z)%*%Z)%*%t(Z)%*%y
#beta_hat
#beta_hat/sum(beta_hat)
Q = 0
Sigma=0
deltas = c()
for(i in 1:length(y)){
ag = Zu[i,]%*%t(Zu[i,])-diag(Zuu[i,])
Q = Q + ag
Delta = (ag%*%p_tilde_hat-y[i]*Zu[i,])
Sigma = Sigma + Delta%*%t(Delta)
}
Q = Q/G
Sigma = Sigma/G
J = matrix(nrow = K,ncol=K)
for(i in 1:K){
for(j in 1:K){
if(i==j){
J[i,j] = sum(p_tilde_hat)-p_tilde_hat[i]
}else{
J[i,j] = -p_tilde_hat[i]
}
}
}
J = J/sum(p_tilde_hat)^2
asyV = (t(J)%*%solve(Q)%*%Sigma%*%solve(Q)%*%J)/G
p_hat = p_tilde_hat/sum(p_tilde_hat)
p_est = rbind(p_est,c(p_hat))
p_tilde_var = rbind(p_tilde_var,diag(solve(Q)%*%Sigma%*%solve(Q)))
p_var = rbind(p_var,diag(asyV))
}
ci_l = p_est - qnorm(0.975)*sqrt(p_var)
ci_r = p_est + qnorm(0.975)*sqrt(p_var)
#coverage
coverage = ((rep(1,100)%*%t(bulk_p))>=ci_l) & ((rep(1,100)%*%t(bulk_p))<=ci_r)
apply(coverage,2,mean)[1] 0.90 0.98 0.97 0.98
sessionInfo()R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)
Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] parallel stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] Matrix_1.2-15 Biobase_2.42.0 BiocGenerics_0.28.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.2 knitr_1.20 whisker_0.3-2 magrittr_1.5
[5] workflowr_1.6.0 lattice_0.20-38 R6_2.3.0 stringr_1.3.1
[9] highr_0.7 tools_3.5.1 grid_3.5.1 git2r_0.26.1
[13] htmltools_0.3.6 yaml_2.2.0 rprojroot_1.3-2 digest_0.6.18
[17] later_0.7.5 promises_1.0.1 fs_1.3.1 glue_1.3.0
[21] evaluate_0.12 rmarkdown_1.10 stringi_1.2.4 compiler_3.5.1
[25] backports_1.1.2 httpuv_1.4.5