# Appendix

## 6.1 Implementaion of LTV-SOBI in R

The primary R-functions that support LTV-SOBI are listed as following,

• ltvsobi(x, lags = 12, quadratic = TRUE, fix_symmetry = TRUE, verbose = FALSE) returns list of class tvbss that is compatible with JADE pacakge’s bss class;

• ltvsobi2(x, lags = 12, quadratic = TRUE, fix_symmetry = TRUE, verbose = FALSE) implements the LTV-SOBI-alt in the same manner as ltvsobi

• SIR_all(bss_res, Omega, Epsilon, S) calculates all applicable performance metric, supporting both tvbss class and bss class;

• tvmix(z, Omega, Epsilon, x_only = TRUE) and tvunmix <- function(x, Omega_hat, Epsilon_hat) serve as TV-SOS utility that generates and restores time-varying mixture based on source signal and mixing parameters.

For major computer platforms, linear estimation is faster and more reliable compared with the inverse matrix. Consequently, inversion is minimized in LTV-SOBI. It is possible to use a vectorization version of ltvsobi, and it indeed reduces the CPU time in processing compared with looping. However, vectorization and Kronecker product significantly raise the dimension of matrices; therefore, the improvement in CPU time is at the cost of RAM usage, which becomes a problem if the dimension is too high and R crash might occur. It is advisable to use non-vectorized version for large $$p$$ and $$T$$ ($$p \geq 10,\ T \geq 50000$$).

## 6.2 Introduction to LTV-SOBI Performance Metric Explorer

As Section 5 stated the complexity in presents LTV-SOBI performance due to a large number of factors, an interactive dashboard is designed to enable customizable performance exploring. The explorer is available at http://bss.yan.fi. The simulation results are conducted in R, and the data is stored in structured database while the dashboard is prepared in an open platform. The explorer allows the user to view and compare performance metrics from different perspectives.

There are 5 dynamic pages designed to aid performance evaluation of multiple algorithms, involving a trend of performance change over the number of observations. The “Simple View” page allows users to select one single cohort and aggregate over 1 or more lags, and the “View All Corhots” will display all 4 plots that representing each cohort on the same page. “View All Lags” shows 3 plots of different lags given user selection of 1 or more cohorts. Control on all possible parameters is available in the “View All Options” page. In addition, the “Method Relationships” attempts to overview the hierarchies of all methods discussed, where the size of blocks shall indicate the number of valid simulations done.

## 6.3R Code for Simulation Study

path <- paste0(getwd(), "/sim/")
library(tidyverse)
source("rfun.R")  # contains algorithms and utilities of LTV-SOBI
source("rsim.R")  # contains functions that simulate various sources
source("rlab_x.R")# an implementation of Yeredors' TV-SOBI

# simulation sampling interval --------------------------------------------

do_it_once <- function(x, z, lll = 6, id = "ID", Omega, Epsilon){

for (i in 1:8) {
flag <- TRUE
tryCatch({
if(i == 1) bss_res <- JADE::SOBI(x, k = lll)
if(i == 2) bss_res <- tvsobi  (x, lag.max = lll, TRUE)
if(i == 3) bss_res <- tvsobi  (x, lag.max = lll, FALSE)
if(i == 4) bss_res <- ltvsobi (x, lags = lll,
fix_symmetry = TRUE)
if(i == 5) bss_res <- ltvsobi (x, lags = lll,
fix_symmetry = FALSE)
if(i == 6) bss_res <- ltvsobi (x, lags = lll,
fix_symmetry = TRUE)
if(i == 7) bss_res <- ltvsobi (x, lags = lll,
fix_symmetry = FALSE)
if(i == 8) bss_res <- ltvsobi2(x, lags = lll)
}, error = function(e) {
flag <<- FALSE
})

if(flag) {
#save_estimator(bss_res, id) #save_restored(bss_res, id)
benchmarks <- SIR_all(bss_res, Omega, Epsilon, z)
remove(bss_res)
save_eval(benchmarks, id)
}
}
}

save_eval <- function(benchmarks, id){
df <- NULL
for(i in 2:length(benchmarks)){
df <- data.frame(criteria = attributes(benchmarks)$names[i], value = benchmarks[[i]]) %>% rbind(df, .) } df$detail <- benchmarks$method df$method <- word(benchmarks$method) df$id     <- id
df$desc <- str_remove(benchmarks$method, " NearestSPD")
df$N <- benchmarks$N
df$p <- benchmarks$p

df <- df %>% filter(criteria != "N" & criteria != "p")

Sys.sleep(runif(1))
fname <- paste0(path, "benchmarks-", id, ".rds")
if (file.exists(fname)) saveRDS(rbind(readRDS(fname), df), file = fname)
else  saveRDS(df, file = fname)
}

# function to run the repeated simulation----------------------------------

multido <- function(E, N, sn){
for(i in 1:100){
Omega <- matrix(c(2, -9, -4, -6, 5, 6, 0.5, 3, 8), ncol =3)
Epsilon <- 10^(-E) * matrix(c(-3, - 4, 9  ,
6, 2.5, 2.1,
-6, 6  , 7  ), ncol = 3)
zall <- sim_good_sources(N = 1e4, 3)
xall <- tvmix(zall, Omega, Epsilon)

# loop for freqs
freq_list <- 2^(0:10)
for(freq in freq_list){
for(l in c(3,6,12,1)){
ids  <- seq(from = 1, to = nrow(xall), by = freq)
x <- xall[ids,]
z <- zall[ids,]
do_it_once(x, z, lll = l,
id = paste0("seq", sn, "_fixed_freq_E",
E, "N", N, "_Boot_lag", l), Omega, Epsilon)
}
}
}
}

# submitting the job using multi-core--------------------------------------

library(parallel) # enable parallel processing
mclapply(
as.list(1:500),
function(seq) {
multido(5,5,seq)
multido(5,4,seq);
multido(4,5,seq);
multido(4,4,seq);
},
mc.cores = detectCores()
)

## 6.4 Supplementary Simulation Results

Figure 6.3 contains the identical configuration as Figure 5.4 with an extra comparison with the original SOBI algorithm. Further views can be found in the interactive performance metric explorer (Appendix 6.2).