---
title: 'Low Dimensional Example of MultiRFM'
author: "Wei Liu"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Low Dimensional Example of MultiRFM}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

This vignette introduces the usage of MultiRFM for the analysis of low-dimensional multi-study multivariate data with heavy tail, by comparison with other methods.


The package can be loaded with the command, and define some metric functions:
```{r  eval = FALSE}
library(MultiRFM)
trace_statistic_fun <- function(H, H0){

  tr_fun <- function(x) sum(diag(x))
  mat1 <- t(H0) %*% H %*% qr.solve(t(H) %*% H) %*% t(H) %*% H0

  tr_fun(mat1) / tr_fun(t(H0) %*% H0)

}
trace_list_fun <- function(Hlist, H0list){
  trvec <- rep(NA, length(Hlist))
  for(i in seq_along(trvec)){
    trvec[i] <- trace_statistic_fun(Hlist[[i]], H0list[[i]])
  }
  return(mean(trvec))
}
```

## Generate the simulated data
First, we generate the simulated data with heavy tail, where the error term follows from a multivariate t-distribution with degree of freedom 3.
```{r  eval = FALSE}
nu <- 3 # nu is set to
p <- 100
nvec <- c(150,200);  q <- 3;qs <- c(2,2); S <- length(nvec)
sigma2_eps <- 1
datList <- gendata_simu_multi(seed=1, nvec=nvec, p=p, q=q, qs=qs, rho=c(5,5), err.type='mvt', sigma2_eps = sigma2_eps, nu=nu)


```
Fit the MultiRFM model using the function `MultiRFM()` in the R package `MultiRFM`. Users can use `?MultiRFM` to see the details about this function. For two matrices $\widehat D$ and $D$, we use trace statistic to measure their similarity.  The trace statistic  ranges from 0 to 1,  with higher values indicating better performance.
```{r  eval = FALSE}
methodNames <- c("MultiRFM", "MSFA-CAVI", "MSFA-SVI")
metricMat <- matrix(NA, nrow=length(methodNames), ncol=5)
colnames(metricMat) <- c('A_tr', 'B_tr',  'F_tr', 'H_tr', 'Time')
row.names(metricMat) <- methodNames
XList <- datList$Xlist;

res <- MultiRFM(XList, q=q, qs=qs)
#str(res)
metricMat["MultiRFM",'Time'] <- res$time_use
metricMat["MultiRFM",'A_tr'] <- trace_statistic_fun(res$A, datList$A0)
metricMat["MultiRFM",'B_tr'] <- trace_list_fun(res$B, datList$Blist0)
metricMat["MultiRFM",'F_tr'] <- trace_list_fun(res$F, datList$Flist)
metricMat["MultiRFM",'H_tr'] <- trace_list_fun(res$H, datList$Hlist)


```

## Compare with other methods

We compare MultiRFM with two prominent methods: MSFA-CAVI and MSFA-SVI

First, we implement MSFA-CAVI:
```{r  eval = FALSE}
X_s <- lapply(XList, scale, scale=FALSE)
hmu <- sapply(XList, colMeans)
library(VIMSFA)
### MSFA-CAVI
print("MSFA-CAVI")
tic <- proc.time()
cavi_est <- cavi_msfa(X_s,  K=q, J_s=qs)
toc <- proc.time()
time_cavi <- toc[3] - tic[3]
hLam <- Reduce(cbind, cavi_est$mean_psi_s)
hF_cavi <- hH_cavi <- list()
for(s in 1:S){
  # s <- 1
  hF_cavi[[s]] <- t(Reduce(cbind, cavi_est$mean_f[[s]]))
  hH_cavi[[s]] <- t(Reduce(cbind, cavi_est$mean_l[[s]]))
}

metricMat["MSFA-CAVI",'Time']  <- time_cavi
metricMat["MSFA-CAVI",'A_tr']  <- trace_statistic_fun(cavi_est$mean_phi, datList$A0)
metricMat["MSFA-CAVI",'B_tr']  <- trace_list_fun(cavi_est$mean_lambda_s, datList$Blist0)
metricMat["MSFA-CAVI",'F_tr'] <- trace_list_fun(hF_cavi, datList$Flist)
metricMat["MSFA-CAVI",'H_tr'] <- trace_list_fun(hH_cavi, datList$Hlist)

```

Next, we implement MSFA-SVI:
```{r  eval = FALSE}
print("MSFA-SVI")
tic <- proc.time()
svi_est <- svi_msfa(X_s, K=q, J_s=qs, verbose = 0)
toc <- proc.time()
time_svi <- toc[3] - tic[3]
hLam <- Reduce(cbind, svi_est$mean_psi_s)
hF_cavi <- hH_cavi <- list()
for(s in 1:S){
  # s <- 1
  hF_cavi[[s]] <- t(Reduce(cbind, svi_est$mean_f[[s]]))
  hH_cavi[[s]] <- t(Reduce(cbind, svi_est$mean_l[[s]]))
}

metricMat["MSFA-SVI",'Time']  <- time_cavi
metricMat["MSFA-SVI",'A_tr']  <- trace_statistic_fun(svi_est$mean_phi, datList$A0)
metricMat["MSFA-SVI",'B_tr']  <- trace_list_fun(svi_est$mean_lambda_s, datList$Blist0)
metricMat["MSFA-SVI",'F_tr'] <- trace_list_fun(hF_cavi, datList$Flist)
metricMat["MSFA-SVI",'H_tr'] <- trace_list_fun(hH_cavi, datList$Hlist)

```


## Visualize the comparison of performance

Next, we summarized the metrics for MultiRFM and other compared methods in a data.frame object.
```{r  eval = FALSE}


dat_metric <- data.frame(metricMat)
dat_metric$Method <- factor(row.names(dat_metric), levels=row.names(dat_metric))
```


Plot the results for MultiRFM and other methods, which suggests that MultiRFM achieves better estimation accuracy for the study-shared loading matrix A, study-specified loading matrix B and factor matrix H.  MultiRFM significantly outperforms the compared methods in terms of estimation accuracy of B and H, as well as computational efficiency. 
```{r  eval = FALSE, fig.width=13, fig.height=5}
library(cowplot)
library(ggplot2)
p1 <- ggplot(data=subset(dat_metric, !is.na(A_tr)), aes(x= Method, y=A_tr, 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(F_tr)), aes(x= Method, y=F_tr, 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(B_tr)), aes(x= Method, y=B_tr, 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(H_tr)), aes(x= Method, y=H_tr, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL)+ theme_bw(base_size = 16)
p5 <- ggplot(data=subset(dat_metric, !is.na(Time)), aes(x= Method, y=Time, fill=Method)) + geom_bar(stat="identity") + xlab(NULL) + scale_x_discrete(breaks=NULL)+ theme_bw(base_size = 16)
plot_grid(p1,p2,p3, p4,  p5, nrow=2, ncol=3)
```


## Select the parameters

We applied the proposed 'CUP' method to select the number of factors. The results showed that  the CUP method has the potential to identify the true values.
```{r  eval = FALSE}

datList <- gendata_simu_multi(seed=1, nvec=nvec, p=p, q=q, qs=qs, rho=c(5,5), err.type='mvt', sigma2_eps =
                                sigma2_eps, nu=3)
XList <- datList$Xlist;
q_max <- 6; qs_max <- 4
hq.list <- selectFac.MultiRFM(XList,  q_max=q_max, qs_max=qs_max, verbose = FALSE)
message("hq = ", hq.list$q, " VS true q = ", q)
message("hqs.vec = ", paste(hq.list$qs, collapse =", "), " VS true qs.vec = ", paste(qs, collapse =", "))
```


<details>
  <summary>**Session Info**</summary>
```{r}
sessionInfo()
```
</details>