rsfar {Rsfar}

Simulation of a Seasonal Functional Autoregressive SFAR(1) process.

Description

Simulation of a SFAR(1) process on a Hilbert space of L2[0,1].

Usage

rsfar(phi, seasonal, Z)

Arguments

phi

a kernel function corresponding to the seasonal autoregressive operator.

seasonal

a positive integer variable specifying the seasonal period.

Z

the functional noise object of the class 'fd'.

Value

A sample of functional time series from a SFAR(1) model of the class 'fd'.

Examples

# Set up Brownian motion noise process
N <- 300 # the length of the series
n <- 200 # the sample rate that each function will be sampled
u <- seq(0, 1, length.out = n) # argvalues of the functions
d <- 15 # the number of basis functions
basis <- create.fourier.basis(c(0, 1), d) # the basis system
sigma <- 0.05 # the stdev of noise norm
Z0 <- matrix(rnorm(N * n, 0, sigma), nr = n, nc = N)
Z0[, 1] <- 0
Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion
Z <- smooth.basis(u, Z_mat, basis)$fd

# Compute the standardized constant of a kernel function with respect to a given HS norm.
gamma0 <- function(norm, kr) {
  f <- function(x) {
    g <- function(y) {
      kr(x, y)^2
    }
    return(integrate(g, 0, 1)$value)
  }
  f <- Vectorize(f)
  A <- integrate(f, 0, 1)$value
  return(norm / A)
}
# Definition of parabolic integral kernel:
norm <- 0.99
kr <- function(x, y) {
  2 - (2 * x - 1)^2 - (2 * y - 1)^2
}
c0 <- gamma0(norm, kr)
phi <- function(x, y) {
  c0 * kr(x, y)
}

# Simulating a path from an SFAR(1) process
s <- 5 # the period number
X <- rsfar(phi, s, Z)
plot(X)