R/g_cfar.r
In liuxialu0328/CFAR:
Defines functions
g_cfar1
g_cfar2
g_cfar
g_cfar2h
Documented in
g_cfar1
#' CFAR program
#' based on the original codes of Xialu Liu of SDSU.
#' @export
g_cfar1 <- function(tmax=1001,rho=5,phi_func=NULL,grid=1000,sigma=1){
#Simulate CFAR(1) processes
#parameter setting
#rho parameter for error process epsilon_t, O-U process
#sigma is the standard deviation of OU-process
#tmax length of time
#iter number of replications in the simulation, iter=1
#grid the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func: User supplied convolution function phi(.). The following default is used if not specified.
if(is.null(phi_func)){
phi_func= function(x){
return(dnorm(x,mean=0,sd=0.1))
}
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
x_grid<- seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid<- seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid <- matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi <- exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b <- phi_func(b_grid) # phi(s) when s in [-1,1]
#the convolutional function values phi(s-u), for different s in [0,1]
phi_matrix <- matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix[i,]= phi_b[seq((i+grid),i,by=-1)]/(grid+1)
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(1) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt((1-fi^2))*rnorm(n,0,1)) ###error process
f_grid[1,]= eps_grid;
for (i in 2:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix%*%f_grid[i-1,]+ eps_grid;
}
g_cfar1 <- list(cfar1=f_grid*sigma,epsilon=eps_grid*sigma)
return(g_cfar1)
}
#' @export
g_cfar2 <- function(tmax=1001,rho=5,phi_func1=NULL, phi_func2=NULL,grid=1000,sigma=1){
#Simulate CFAR(2) processes
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
#parameter setting
# rho=5 parameter for error process epsilon_t, O-U process
#tmax= 1001; length of time
#iter= 100; number of replications in the simulation
#grid=1000; the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func1 and phi_func2: User specified convolution functions. The following defaults are used if not given
if(is.null(phi_func1)){
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
}
if(is.null(phi_func2)){
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
x_grid= seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid= seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid= matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi = exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b1= phi_func1(b_grid) # phi_1(s) when s in [-1,1]
phi_b2= phi_func2(b_grid) # phi_2(s) when s in [-1,1]
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix1= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix1[i,]= phi_b1[seq((i+grid),i,by=-1)]/(grid+1)
}
phi_matrix2= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix2[i,]= phi_b2[seq((i+grid),i,by=-1)]/(grid+1)
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(2) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid;
i=2
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,])+ eps_grid;
for (i in 3:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,])+ phi_matrix2%*%as.matrix(f_grid[i-2,])+ eps_grid;
}
# The last iteration of epsilon_t is returned.
g_cfar2 <- list(cfar2=f_grid*sigma,epsilon=eps_grid*sigma)
}
#' @export
g_cfar <- function(tmax=1001,rho=5,phi_list=NULL, grid=1000,sigma=1){
#Simulate CFAR(p) processes
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
# parameter setting
# rho=5 parameter for error process epsilon_t, O-U process
# tmax= 1001; # length of time
# iter= 100; # number of replications in the simulation
#grid=1000; # the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func User specified convolution functions. The following defaults are used if not given
if(is.null(phi_list)){
phi_func1= function(x){
return(dnorm(x,mean=0,sd=0.1))
}
phi_list=phi_func1
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
x_grid= seq(0, 1, by= 1/grid);
b_grid= seq(-1, 1, by=1/grid);
eps_grid= matrix(0, 1, grid+1);
fi = exp(-rho/grid);
p=length(phi_list)
if(is.list(phi_list)){
phi_b=sapply(phi_list,mapply,b_grid)
}else{
phi_b=matrix(phi_list(b_grid),2*grid+1,1)
}
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix= matrix(0, (grid+1), p*(grid+1));
for(j in 1:p){
for (i in 1:(grid+1)){
phi_matrix[i,((j-1)*(grid+1)+1):(j*(grid+1))]= phi_b[seq((i+grid),i,by=-1),j]/(grid+1)
}
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(2) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid;
if(p>1){
for(i in 2:p){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix[,1:((i-1)*(grid+1))]%*%matrix(t(f_grid[(i-1):1,]),(i-1)*(grid+1),1)+ eps_grid;
}
}
for (i in (p+1):tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix%*%matrix(t(f_grid[(i-p):(i-1),]),p*(grid+1),1)+ eps_grid;
}
# The last iteration of epsilon_t is returned.
g_cfar <- list(cfar=f_grid*sigma,epsilon=eps_grid*sigma)
}
#' @export
g_cfar2h <- function(tmax=1001,grid=1000,rho=1,min_obs=40, pois=5,phi_func1=NULL, phi_func2=NULL, weight=NULL){
#Simulate CFAR(2) processes with heteroscedasticity, irregular observation locations
# We need to have f_grid to record the functional time series
# We need to have x_pos_full to record the observation positions
# parameter setting
# rho=1 ### parameter for error process epsilon_t, O-U process
# tmax= 1001; ### length of time
# iter= 100; ### number of replications in the simulation
# grid=1000; ### the number of grid points used to construct the functional time series X_t and epsilon_t\
# min_obs=40 ### the minimal number of observations at each time
# phi_func1, phi_func2, weight: User specified convolution functions and weight function. The following defaults
# are used if not given
if(is.null(phi_func1)){
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
}
if(is.null(phi_func2)){
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
}
if(is.null(weight)){
###heteroscedasticity weight function
x_grid <- seq(0,1,by=1/grid)
weight0= function(w){
return((w<=0.6)*exp(-10*w)+(w>0.6)*(exp(-6)+0.2*(w-0.6))+0.1);
}
const=sum(weight0(x_grid)/(grid+1))
weight= function(w){
return(((w<=0.6)*exp(-10*w)+(w>0.6)*(exp(-6)+0.2*(w-0.6))+0.1)/const)
}
}
num_full=rpois(tmax,pois)+min_obs # number of observations at time t follows a Poisson distribution
x_grid= seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid= seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid= matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi = exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b1= phi_func1(b_grid) # phi_1(s) when s in [-1,1]
phi_b2= phi_func2(b_grid) # phi_2(s) when s in [-1,1]
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix1= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix1[i,]= phi_b1[seq((i+grid),i,by=-1)]/(grid+1)
}
phi_matrix2= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix2[i,]= phi_b2[seq((i+grid),i,by=-1)]/(grid+1)
}
n=max(num_full)
x_pos_full=matrix(0,tmax,n) # observation positions
for(k in 1:(tmax)){
x_pos_full[k,1:num_full[k]]=sort(runif(num_full[k]))
}
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid*weight(x_grid);
i=2
f_grid[i,]= phi_matrix1 %*% as.matrix(f_grid[i-1,])+ eps_grid*weight(x_grid);
for (i in 3:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,]) + phi_matrix2%*%as.matrix(f_grid[i-2,])+ eps_grid*weight(x_grid);
}
# Functional time series, number of observations at different time of periods,
# observation points at different time of periods, the last iteration of epsilon_t is returned.
g_cfar2h <- list(cfar2h=f_grid, num_obs=num_full, x_pos=x_pos_full,epsilon=eps_grid)
}
liuxialu0328/CFAR documentation built on May 21, 2019, 9:34 a.m.
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Defines functions g_cfar1 g_cfar2 g_cfar g_cfar2h
Documented in g_cfar1
#' CFAR program
#' based on the original codes of Xialu Liu of SDSU.
#' @export
g_cfar1 <- function(tmax=1001,rho=5,phi_func=NULL,grid=1000,sigma=1){
#Simulate CFAR(1) processes
#parameter setting
#rho parameter for error process epsilon_t, O-U process
#sigma is the standard deviation of OU-process
#tmax length of time
#iter number of replications in the simulation, iter=1
#grid the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func: User supplied convolution function phi(.). The following default is used if not specified.
if(is.null(phi_func)){
phi_func= function(x){
return(dnorm(x,mean=0,sd=0.1))
}
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
x_grid<- seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid<- seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid <- matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi <- exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b <- phi_func(b_grid) # phi(s) when s in [-1,1]
#the convolutional function values phi(s-u), for different s in [0,1]
phi_matrix <- matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix[i,]= phi_b[seq((i+grid),i,by=-1)]/(grid+1)
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(1) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt((1-fi^2))*rnorm(n,0,1)) ###error process
f_grid[1,]= eps_grid;
for (i in 2:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix%*%f_grid[i-1,]+ eps_grid;
}
g_cfar1 <- list(cfar1=f_grid*sigma,epsilon=eps_grid*sigma)
return(g_cfar1)
}
#' @export
g_cfar2 <- function(tmax=1001,rho=5,phi_func1=NULL, phi_func2=NULL,grid=1000,sigma=1){
#Simulate CFAR(2) processes
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
#parameter setting
# rho=5 parameter for error process epsilon_t, O-U process
#tmax= 1001; length of time
#iter= 100; number of replications in the simulation
#grid=1000; the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func1 and phi_func2: User specified convolution functions. The following defaults are used if not given
if(is.null(phi_func1)){
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
}
if(is.null(phi_func2)){
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
x_grid= seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid= seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid= matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi = exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b1= phi_func1(b_grid) # phi_1(s) when s in [-1,1]
phi_b2= phi_func2(b_grid) # phi_2(s) when s in [-1,1]
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix1= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix1[i,]= phi_b1[seq((i+grid),i,by=-1)]/(grid+1)
}
phi_matrix2= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix2[i,]= phi_b2[seq((i+grid),i,by=-1)]/(grid+1)
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(2) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid;
i=2
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,])+ eps_grid;
for (i in 3:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,])+ phi_matrix2%*%as.matrix(f_grid[i-2,])+ eps_grid;
}
# The last iteration of epsilon_t is returned.
g_cfar2 <- list(cfar2=f_grid*sigma,epsilon=eps_grid*sigma)
}
#' @export
g_cfar <- function(tmax=1001,rho=5,phi_list=NULL, grid=1000,sigma=1){
#Simulate CFAR(p) processes
# OUTPUTS
# A matrix f_grid is generated, with (tmax) rows and (grid+1) columns.
# parameter setting
# rho=5 parameter for error process epsilon_t, O-U process
# tmax= 1001; # length of time
# iter= 100; # number of replications in the simulation
#grid=1000; # the number of grid points used to construct the functional time series X_t and epsilon_t
# phi_func User specified convolution functions. The following defaults are used if not given
if(is.null(phi_list)){
phi_func1= function(x){
return(dnorm(x,mean=0,sd=0.1))
}
phi_list=phi_func1
}
if(is.null(grid)){
grid=1000
}
if(is.null(sigma)){
sigma=1
}
x_grid= seq(0, 1, by= 1/grid);
b_grid= seq(-1, 1, by=1/grid);
eps_grid= matrix(0, 1, grid+1);
fi = exp(-rho/grid);
p=length(phi_list)
if(is.list(phi_list)){
phi_b=sapply(phi_list,mapply,b_grid)
}else{
phi_b=matrix(phi_list(b_grid),2*grid+1,1)
}
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix= matrix(0, (grid+1), p*(grid+1));
for(j in 1:p){
for (i in 1:(grid+1)){
phi_matrix[i,((j-1)*(grid+1)+1):(j*(grid+1))]= phi_b[seq((i+grid),i,by=-1),j]/(grid+1)
}
}
# Generate data
# functional time series f_grid is X_t in the paper
# A matrix f_grid is generated, with (tmax*iter) rows and (grid+1) columns.
# It contains iter CFAR(2) processes. The i-th process is in rows (i-1)*tmax+1:tmax.
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid;
if(p>1){
for(i in 2:p){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix[,1:((i-1)*(grid+1))]%*%matrix(t(f_grid[(i-1):1,]),(i-1)*(grid+1),1)+ eps_grid;
}
}
for (i in (p+1):tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix%*%matrix(t(f_grid[(i-p):(i-1),]),p*(grid+1),1)+ eps_grid;
}
# The last iteration of epsilon_t is returned.
g_cfar <- list(cfar=f_grid*sigma,epsilon=eps_grid*sigma)
}
#' @export
g_cfar2h <- function(tmax=1001,grid=1000,rho=1,min_obs=40, pois=5,phi_func1=NULL, phi_func2=NULL, weight=NULL){
#Simulate CFAR(2) processes with heteroscedasticity, irregular observation locations
# We need to have f_grid to record the functional time series
# We need to have x_pos_full to record the observation positions
# parameter setting
# rho=1 ### parameter for error process epsilon_t, O-U process
# tmax= 1001; ### length of time
# iter= 100; ### number of replications in the simulation
# grid=1000; ### the number of grid points used to construct the functional time series X_t and epsilon_t\
# min_obs=40 ### the minimal number of observations at each time
# phi_func1, phi_func2, weight: User specified convolution functions and weight function. The following defaults
# are used if not given
if(is.null(phi_func1)){
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
}
if(is.null(phi_func2)){
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
}
if(is.null(weight)){
###heteroscedasticity weight function
x_grid <- seq(0,1,by=1/grid)
weight0= function(w){
return((w<=0.6)*exp(-10*w)+(w>0.6)*(exp(-6)+0.2*(w-0.6))+0.1);
}
const=sum(weight0(x_grid)/(grid+1))
weight= function(w){
return(((w<=0.6)*exp(-10*w)+(w>0.6)*(exp(-6)+0.2*(w-0.6))+0.1)/const)
}
}
num_full=rpois(tmax,pois)+min_obs # number of observations at time t follows a Poisson distribution
x_grid= seq(0, 1, by= 1/grid); # the grid points for input variables of functional time series in [0,1]
b_grid= seq(-1, 1, by=1/grid); # the grid points for input variables of b-spline functions in [-1,1]
eps_grid= matrix(0, 1, grid+1); # the grid points for input variables of error process in [0,1]
fi = exp(-rho/grid); # the correlation of two adjacent points for an O-U process
phi_b1= phi_func1(b_grid) # phi_1(s) when s in [-1,1]
phi_b2= phi_func2(b_grid) # phi_2(s) when s in [-1,1]
# the convolutional function values phi_1(s-u) and phi_2(s-u), for different s in [0,1]
phi_matrix1= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix1[i,]= phi_b1[seq((i+grid),i,by=-1)]/(grid+1)
}
phi_matrix2= matrix(0, (grid+1), (grid+1));
for (i in 1:(grid+1)){
phi_matrix2[i,]= phi_b2[seq((i+grid),i,by=-1)]/(grid+1)
}
n=max(num_full)
x_pos_full=matrix(0,tmax,n) # observation positions
for(k in 1:(tmax)){
x_pos_full[k,1:num_full[k]]=sort(runif(num_full[k]))
}
f_grid=matrix(0,tmax,grid+1)
i=1
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[1,]= eps_grid*weight(x_grid);
i=2
f_grid[i,]= phi_matrix1 %*% as.matrix(f_grid[i-1,])+ eps_grid*weight(x_grid);
for (i in 3:tmax){
eps_grid= arima.sim(n= grid+1, model=list(ar=fi),rand.gen= function(n)sqrt(1-fi^2)*rnorm(n,0,1))
f_grid[i,]= phi_matrix1%*%as.matrix(f_grid[i-1,]) + phi_matrix2%*%as.matrix(f_grid[i-2,])+ eps_grid*weight(x_grid);
}
# Functional time series, number of observations at different time of periods,
# observation points at different time of periods, the last iteration of epsilon_t is returned.
g_cfar2h <- list(cfar2h=f_grid, num_obs=num_full, x_pos=x_pos_full,epsilon=eps_grid)
}
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