beyondWhittle (version 1.3.0)

psd_tvarma12: time-varying spectral density function of the tvARMA(1,2) processes for illustrations

Description

time-varying spectral density function of the tvARMA(1,2) processes for illustrations

Usage

psd_tvarma12(
  rescaled_time,
  freq,
  dgp = NULL,
  a1 = function(u) {
     rep(0, length(u))
 },
  b1 = function(u) {
     rep(0, length(u))
 },
  b2 = function(u) {
     rep(0, length(u))
 }
)

Value

a matrix of dimension length(rescaled_time) by length(freq).

Arguments

rescaled_time, freq

numeric vectors forming a rectangular grid on which the tv-PSD is evaluated.

dgp

optional: the tv-ARMA models demonstrated in section 4.2 of Tang et al. (2023). Should be chosen from "LS1", "LS2" and "LS3". See section Details.

a1, b1, b2

If dgp is not supplied, these arguments can be used to specify customized tv-ARMA process (up to order(1,2)). See Details. rescaled_time must be in \([0,1]\) and freq must be in \([0,\pi]\).

Details

See sim_tvarma12 for the precise definition of a tvARMA(1,2) process. The time-varying spectral density function of this process is defined as

$f(u,\lambda )\; =\; (2\pi )-1(1+b$₁²(u)+b₂²(u)+2b₁(u)(b₂(u)+1)cos(λ)+2b₂(u)cos(2λ))/(1+a₁²(u)-2a₁(u)cos(λ)), (u,λ)∈[0,1]×[0,π],

where \(u\) is called rescaled time and \(\lambda\) is called frequency.

For dgp = "LS1", it is a tvMA(2) process (MA order is 2) with

$a$₁(u)=0, b₁(u)=1.122(1-1.178sin(π/2 u)), b₂(u)=-0.81.

For dgp = "LS2", it is a tvMA(1) process (MA order is 1) with

$a$₁(u)=0, b₁(u)=1.1cos(1.5-cos(4π u)), b₂(u)=0.

For dgp = "LS3", it is a tvAR(1) process (MA order is 0) with

$a$₁(u)=1.2u-0.6, b₁(u)=0, b₂(u)=0.

References

Tang et al. (2023) Bayesian nonparametric spectral analysis of locally stationary processes ArXiv preprint <arXiv:2303.11561>

Examples

if (FALSE) {
res_time <- seq(0, 1, by = 0.005); freq <- pi*seq(0, 1, by = 0.01)
true_tvPSD <- psd_tvarma12(rescaled_time = res_time, freq = freq, dgp = "LS2")
plot(true_tvPSD)
}