Step 1. Prepare simulation of time-series data
library(pompom)
library(ggplot2)
library(qgraph)
require(reshape2)
set.seed(1234)Set up a 3 node model
The 3-node network is a hypothetical example used for the purpose of illustration. As explained in the paper, the variables in time-series data are nodes and the temporal relations are edges, in the network terminology.
n.obs <- 200 # number of observation
p <- 3 # number of variables
noise.level <- 0.1
epsilon <- cbind(rnorm(n.obs,0, noise.level),
rnorm(n.obs,0, noise.level),
rnorm(n.obs,0, noise.level)) # 3 time-series of noise for 3 variables
true.beta <- matrix(c(0,0,0,0,0,0,
0,0,0,0,0,0,
0,0,0,0,0,0,
0.2,0,0.25,0,0,0.6,
0,0.3,0,-0.2,0,-0.6,
0,-0.2,0.3,0,0,0),
nrow = 2 * p,
ncol = 2 * p,
byrow = TRUE)
contemporaneous.relations <- matrix(true.beta[(p+1):(2*p),(p+1):(2*p)], nrow = p, ncol = p, byrow = F)
lag.1.relations <- matrix(true.beta[(p+1):(2*p),1:p], nrow = p, ncol = p, byrow = F)True model is:
true.beta## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.0 0.0 0.00 0.0 0 0.0
## [2,] 0.0 0.0 0.00 0.0 0 0.0
## [3,] 0.0 0.0 0.00 0.0 0 0.0
## [4,] 0.2 0.0 0.25 0.0 0 0.6
## [5,] 0.0 0.3 0.00 -0.2 0 -0.6
## [6,] 0.0 -0.2 0.30 0.0 0 0.0Network graph
To introduce the terminologies of networks, each of the 3 variables is depicted as a node (circle), and the temporal relations among the nodes are depicted as edges (arrows, and the arrow indicate directionality of the temporal relationship), with color indicating sign of relation (green = positive, red = negative), thickness indicating strength of relation, and line shape indicating temporality of the association (dashed = lag-1, solid = contemporaneous). Lag-1 relations mean the temporal relations between variables from measurement t -1 to the measurement t, and contemporaneous relations means the temporal relations between variables within the same measurement.
plot_network_graph(true.beta, p)
## NULLSimulate time-series
Time series was simulated based on the temporal relations and process noise. Our hypohetical 3-node newtork was using simulated data based on a pre-defined temporal relationship matrix and process noise. The temporal relationship is as follows, and process noise is at Mean = 0, SD = .1.
\(\eta(t) = (I-\ A \ )^{-1}\Phi\eta(t-1) + (I-\ A \ )^{-1}\epsilon(t)\)
where \(\ A\) is the block of contemporaneous relations (southeast block of the beta matrix),
\(\Phi\) is the block of lag-1 relations (southwest block of the beta matrix),
\(\epsilon\) is the process noise.
We simulated 200 occasions to represent 200 repeated measurements of the three variables in the real studies.
time.series <- matrix(rep(0, p * n.obs), nrow = n.obs, ncol = p)
time.series[1,] <- rnorm(p,0,1)
row <- p
for (row in 2:n.obs)
{
time.series[row,] <- solve(diag(1,p, p) - contemporaneous.relations) %*%
lag.1.relations %*% time.series[row-1,] +
solve(diag(1,p, p) - contemporaneous.relations) %*% epsilon[row, ]
}
time.series <- data.frame(time.series)
names(time.series) <- c("x", "y", "z")Plot the simulated time-series
time.series$time <- seq(1,length(time.series[,1]),1)
time.series.long <- melt(time.series, id="time") ## convert to long format
ggplot(data=time.series.long,
aes(x=time, y=value, colour=variable)) +
geom_line() +
facet_wrap( ~ variable , ncol = 1) +
scale_y_continuous(breaks = seq(0, 100, by = 50)) +
theme(
strip.background = element_blank(),
panel.background = element_blank(),
legend.title = element_blank(),
# strip.text.x = element_blank(),
# strip.text.y = element_blank(),
# axis.text.y = element_text(),
# axis.title.y = element_blank()
legend.key = element_blank(),
legend.position = "none",
# legend.title = element_blank(),
# panel.background = element_blank(),
axis.text.y=element_text(color="black",size=12),
axis.text.x=element_text(color="black",size=12),
axis.title.y=element_text(color="black",size=12),
axis.title.x=element_text(color="black",size=12),
axis.line = element_line(color = 'black')
)+
ylim(-1,1)+
xlab("Time") +
ylab("Value")## Scale for 'y' is already present. Adding another scale for 'y', which will
## replace the existing scale.## Warning: Removed 1 row(s) containing missing values (geom_path).
time.series$time <- NULL
