COAP: simulation'
In COAP: High-Dimensional Covariate-Augmented Overdispersed Poisson Factor Model

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

This vignette introduces the usage of COAP for the analysis of high-dimensional count data with additional high-dimensional covariates, by comparison with other methods.

The package can be loaded with the command:

library(COAP)
library(GFM)

Generate the simulated data

First, we generate the data simulated data.

n <- 200; p <- 200; 
d= 50
rank0 <- 6;
q = 5;
datList <- gendata_simu(seed = 1, n=n, p=p, d= d, rank0 = rank0, q= q, rho=c(2, 2),
                        sigma2_eps = 1)
X_count <- datList$X; Z <- datList$Z
H0 <- datList$H0; B0 <- datList$B0
bbeta0 <- cbind( datList$mu0, datList$bbeta0)

Fit the COAP model using the function RR_COAP() in the R package COAP. Users can use ?RR_COAP to see the details about this function

hq <- 5; hr <- 6
system.time({
  tic <- proc.time()
  reslist <- RR_COAP(X_count, Z= Z, q=hq, rank_use= hr, epsELBO = 1e-6)
  toc <- proc.time()
  time_coap <- toc[3] - tic[3]
})

Check the increased property of the envidence lower bound function.

library(ggplot2)
dat_iter <- data.frame(iter=1:length(reslist$ELBO_seq), ELBO=reslist$ELBO_seq)
ggplot(data=dat_iter, aes(x=iter, y=ELBO)) + geom_line() + geom_point() + theme_bw(base_size = 20)

We calculate the metrics to measure the estimatioin accuracy, where the trace statistic is used to measure the estimation accuracy of loading matrix and prediction accuracy of factor matrix, which is evaluated by the function measurefun() in the R package GFM, and the root of mean square error is adopted to measure the estimation error of bbeta.

library(GFM)
metricList <- list()
metricList$COAP <- list()
metricList$COAP$Tr_H <- measurefun(reslist$H, H0)
metricList$COAP$Tr_B <- measurefun(reslist$B, B0)

norm_vec <- function(x) sqrt(sum(x^2/ length(x)))
metricList$COAP$err_bb <- norm_vec(reslist$bbeta-bbeta0)
metricList$COAP$err_bb1 <- norm_vec(reslist$bbeta[,1]-bbeta0[,1])
metricList$COAP$Time <- time_coap

Compare with other methods

We compare COAP with various prominent methods in the literature. They are (1) High-dimensional LFM (Bai and Ng 2002) implemented in the R package GFM; (2) PoissonPCA (Kenney et al. 2021) implemented in the R package PoissonPCA; (3) Zero-inflated Poisson factor model (ZIPFA, Xu et al. 2021) implemented in the R package ZIPFA; (4) Generalized factor model (Liu et al. 2023) implemented in the R package GFM; (5) PLNPCA (Chiquet et al. 2018) implemented in the R package PLNmodels; (6) Generalized Linear Latent Variable Models (GLLVM, Hui et al. 2017) implemented in the R package gllvm. (7) Poisson regression model for each $x_{ij}, (j = 1,··· ,p)$, implemented in stats R package; (8) Multi-response reduced-rank Poisson regression model (MMMR, Luo et al. 2018) implemented in rrpack R package.

(1). First, we implemented the linear factor model (LFM) and record the metrics that measure the estimation accuracy and computational cost.

metricList$LFM <- list()
tic <- proc.time()
fit_lfm <- Factorm(X_count, q=q)
toc <- proc.time()
time_lfm <- toc[3] - tic[3]

hbb1 <- colMeans(X_count)
metricList$LFM$Tr_H <- measurefun(fit_lfm$hH, H0)
metricList$LFM$Tr_B <- measurefun(fit_lfm$hB, B0)
metricList$LFM$err_bb1 <- norm_vec(hbb1- bbeta0[,1])
metricList$LFM$err_bb <- NA
metricList$LFM$Time <- time_lfm

(2). Then, we implemented PoissonPCA and recorded the metrics.

metricList$PoissonPCA <- list()
library(PoissonPCA)
tic <- proc.time()
fit_poispca <- Poisson_Corrected_PCA(X_count, k= hq) 
toc <- proc.time()
time_ppca <- toc[3] - tic[3]

hbb1 <- colMeans(X_count)
metricList$PoissonPCA$Tr_H <- measurefun(fit_poispca$scores, H0)
metricList$PoissonPCA$Tr_B <- measurefun(fit_poispca$loadings, B0)
metricList$PoissonPCA$err_bb1 <- norm_vec(log(1+fit_poispca$center)- bbeta0[,1])
metricList$PoissonPCA$err_bb <- NA
metricList$PoissonPCA$Time <- time_ppca

(3) Thirdly, we implemented the zero-inflated Poisson factor model:

## ZIPFA runs very slowly, so we do not run it here.
library(ZIPFA)
metricList$ZIPFA <- list()
system.time(
  tic <- proc.time()
  fit_zipfa <- ZIPFA(X_count, k=hq, display = FALSE)
  toc <- proc.time()
  time_zipfa <- toc[3] - tic[3]
)



idx_max_like <- which.max(fit_zipfa$Likelihood) 
hbb1 <- colMeans(X_count)
metricList$ZIPFA$Tr_H <- measurefun(fit_zipfa$Ufit[[idx_max_like]], H0)
metricList$ZIPFA$Tr_B <- measurefun(fit_zipfa$Vfit[[idx_max_like]], B0)
metricList$PoissonPCA$Time <- time_zipfa

(4) Fourthly, we also applied the generalized factor model to estimate the loading matrix and factor matrix.

metricList$GFM <- list()
tic <- proc.time()
fit_gfm <- gfm(list(X_count),  type='poisson', q= q, verbose = F)
toc <- proc.time()
time_gfm <- toc[3] - tic[3]
metricList$GFM$Tr_H <- measurefun(fit_gfm$hH, H0)
metricList$GFM$Tr_B <- measurefun(fit_gfm$hB, B0)
metricList$GFM$err_bb1 <- norm_vec(fit_gfm$hmu- bbeta0[,1])
metricList$GFM$err_bb <- NA
metricList$GFM$Time <- time_gfm

(5) Fifthly, we implemented PLNPCA:

PLNPCA_run <- function(X_count, covariates, q,  Offset=rep(1, nrow(X_count))){
  require(PLNmodels)

  if(!is.character(Offset)){
    dat_plnpca <- prepare_data(X_count, covariates)
    dat_plnpca$Offset <- Offset
  }else{
    dat_plnpca <- prepare_data(X_count, covariates, offset = Offset)
  }

  d <- ncol(covariates)
  #  offset(log(Offset))+
  formu <- paste0("Abundance ~ 1 + offset(log(Offset))+",paste(paste0("V",1:d), collapse = '+'))


  myPCA <- PLNPCA(as.formula(formu), data = dat_plnpca, ranks = q)

  myPCA1 <- getBestModel(myPCA)
  myPCA1$scores

  res_plnpca <- list(PCs= myPCA1$scores, bbeta= myPCA1$model_par$B, 
                     loadings=myPCA1$model_par$C)

  return(res_plnpca)
}



  tic <- proc.time()
  fit_plnpca <- PLNPCA_run(X_count,  covariates = Z[,-1], q= q)
  toc <- proc.time()
  time_plnpca <- toc[3] - tic[3]
message(time_plnpca, " seconds")

metricList$PLNPCA$Tr_H <- measurefun(fit_plnpca$PCs, H0)
metricList$PLNPCA$Tr_B <- measurefun(fit_plnpca$loadings, B0)
metricList$PLNPCA$err_bb1 <- norm_vec(fit_plnpca$bbeta[,1]- bbeta0[,1])
metricList$PLNPCA$err_bb <- norm_vec(as.vector(fit_plnpca$bbeta) - as.vector(bbeta0)) 
metricList$PLNPCA$Time <- time_plnpca

(6) Sixthly, we implement the generalized linear latent variable models (GLLVM, Hui et al. 2017).

## GLLVM runs very slowly, so we do not run it here.

library(gllvm)
colnames(Z) <- c(paste0("V",1: ncol(Z)))
tic <- proc.time()
fit <- gllvm(y=X_count, X=Z, family=poisson(), num.lv= q, control = list(trace=T))
toc <- proc.time()
time_gllvm <- toc[3] - tic[3]

metricList$GLLVM <- list()
metricList$GLLVM$Tr_H <- measurefun(fit$lvs, H0)
metricList$GLLVM$Tr_B <- measurefun(fit$params$theta, B0)
metricList$GLLVM$err_bb1 <- norm_vec(fit$params$beta0- bbeta0[,1])
metricList$GLLVM$err_bb <- norm_vec(as.vector(cbind(fit$params$beta0,fit$params$Xcoef)) - as.vector(bbeta0)) 
metricList$GLLVM$Time <- time_gllvm
}

(7) Seventhly, Poisson regression model for each variable was implemented.

PoisReg <- function(X_count, covariates){
     library(stats)
     hbbeta <- apply(X_count, 2, function(x){
       glm1 <- glm(x~covariates+0, family = "poisson")
       coef(glm1)
     } )
     return(t(hbbeta))
}
tic <- proc.time()
hbbeta_poisreg <- PoisReg(X_count, Z)
toc <- proc.time()
time_poisreg <- toc[3] - tic[3]
metricList$GLM <- list()
metricList$GLM$Tr_H <- NA
metricList$GLM$Tr_B <- NA
metricList$GLM$err_bb1 <- norm_vec(hbbeta_poisreg[,1]- bbeta0[,1])
metricList$GLM$err_bb <- norm_vec(as.vector(hbbeta_poisreg) - as.vector(bbeta0)) 
metricList$GLM$Time <- time_poisreg

(8) Eightly, we implemented the first version of multi-response reduced-rank Poisson regression model (MMMR, Luo et al. 2018) implemented in rrpack R package (MRRR-Z), that did not consider the latent factor structure but only the covariates.

mrrr_run <- function(Y, X, rank0, q=NULL, family=list(poisson()), familygroup=rep(1,ncol(Y))){


  require(rrpack)

  n <- nrow(Y); p <- ncol(Y)

  if(!is.null(q)){
    rank0 <- rank0+q
    X <- cbind(X, diag(n))
  }

  svdX0d1 <- svd(X)$d[1]
  init1 = list(kappaC0 = svdX0d1 * 5) ## this setting follows the example that authors provided.

  fit.mrrr <- mrrr(Y=Y, X=X[,-1], family = family, familygroup = familygroup,
                   penstr = list(penaltySVD = "rankCon", lambdaSVD = 0.1),
                   init = init1, maxrank = rank0)
  hbbeta_mrrr <-t(fit.mrrr$coef[1:ncol(Z), ])
  if(!is.null(q)){
    Theta_hb <- (fit.mrrr$coef[(ncol(Z)+1): (nrow(Z)+ncol(Z)), ])
    svdTheta <- svd(Theta_hb, nu=q, nv=q)
    return(list(hbbeta=hbbeta_mrrr, factor=svdTheta$u, loading=svdTheta$v))
  }else{
    return(list(hbbeta=hbbeta_mrrr))
  }


}
tic <- proc.time()

res_mrrrz <- mrrr_run(X_count, Z, rank0)
toc <- proc.time()
time_mrrrz <- toc[3] - tic[3]

metricList$MRRR_Z <- list()
metricList$MRRR_Z$Tr_H <- NA
metricList$MRRR_Z$Tr_B <-NA
metricList$MRRR_Z$err_bb1 <- norm_vec(res_mrrrz$hbbeta[,1]- bbeta0[,1])
metricList$MRRR_Z$err_bb <- norm_vec(as.vector(res_mrrrz$hbbeta) - as.vector(bbeta0)) 
metricList$MRRR_Z$Time <- time_mrrrz

(9) Lastly, we implemented the second version of MRRR (MRRR-F) that considered both covariates and the latent factor structure.

tic <- proc.time()
res_mrrrf <- mrrr_run(X_count, Z, rank0, q=q)
toc <- proc.time()
time_mrrrf <- toc[3] - tic[3]
metricList$MRRR_F <- list()
metricList$MRRR_F$Tr_H <- measurefun(res_mrrrf$factor, H0)
metricList$MRRR_F$Tr_B <- measurefun(res_mrrrf$loading, B0)
metricList$MRRR_F$err_bb1 <- norm_vec(res_mrrrf$hbbeta[,1]- bbeta0[,1])
metricList$MRRR_F$err_bb <- norm_vec(as.vector(res_mrrrf$hbbeta) - as.vector(bbeta0)) 
metricList$MRRR_F$Time <- time_mrrrf

Visualize the comparison of performance

Next, we summarized the metrics for COAP and other compared methods in a dataframe object.

list2vec <- function(xlist){
  nn <- length(xlist)
  me <- rep(NA, nn)
  idx_noNA <- which(sapply(xlist, function(x) !is.null(x)))
  for(r in idx_noNA) me[r] <- xlist[[r]]
  return(me)
}

dat_metric <- data.frame(Tr_H = sapply(metricList, function(x) x$Tr_H), 
                         Tr_B = sapply(metricList, function(x) x$Tr_B),
                         err_bb1 =sapply(metricList, function(x) x$err_bb1),
                         err_bb = list2vec(lapply(metricList, function(x) x[['err_bb']])),
                         Method = names(metricList))
dat_metric

Plot the results for COAP and other methods, which suggests that COAP achieves better estimation accuracy for the quantiites of interest.

library(cowplot)
p1 <- ggplot(data=subset(dat_metric, !is.na(Tr_B)), aes(x= Method, y=Tr_B, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL) + theme_bw(base_size = 16)
p2 <- ggplot(data=subset(dat_metric, !is.na(Tr_H)), aes(x= Method, y=Tr_H, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL)+ theme_bw(base_size = 16)
p3 <- ggplot(data=subset(dat_metric, !is.na(err_bb1)), aes(x= Method, y=err_bb1, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL)+ theme_bw(base_size = 16)
p4 <- ggplot(data=subset(dat_metric, !is.na(err_bb)), aes(x= Method, y=err_bb, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL)+ theme_bw(base_size = 16)
plot_grid(p1,p2,p3, p4, nrow=2, ncol=2)

Select the parameters

We applied the singular value ratio based method to select the number of factors and the rank of coefficient matrix. The results showed that the SVR method has the potential to identify the true values.

datList <- gendata_simu(seed = 1, n=n, p=p, d= d, rank0 = rank0, q= q, rho=c(3, 6),
                        sigma2_eps = 1)
X_count <- datList$X; Z <- datList$Z
res1 <- selectParams(X_count=datList$X, Z=datList$Z, verbose=F)

print(c(q_true=q, q_est=res1['hq']))
print(c(r_true=rank0, r_est=res1['hr']))