spec_density_covlagh_operator: Calculate the lag-h autocovariance operator for FTS defined...

In tomasrubin/specsimfts: Spectral domain simulation methods for functional time series

Description

Numerically calculate the lag-h covariance operators for functional time series dynamics defined directly by its spectral density operator. The calculation is done by numerically integrating the inverse formula, i.e. the spectral density multiplied by exp(-1i*lag*omega)

Usage

spec_density_covlagh_operator(spec_density, lag, n_grid, n_grid_freq = 2000)

Arguments

spec_density	The spectral density operator defined as an integral operator through with the given kernel. Function of three variables, omega, x, y, that assigns the value of the spectral density kernel at point (x,y) in [0,1]^2 at frequency omega in [0,pi]. Must be well defined for frequencies (0,pi]. The interval [pi,2pi) is not used and is calculated by mirroring of (0,pi].
lag	The lag of the autocovariance to evaluate.
n_grid	Number of grid points (spatial resolution) of the discretisation of [0,1]^2 for the operator kernel to evaluate.
n_grid_freq	The grid points for the spectral density to evaluate at. Partition of [0,pi].

Value

lag-h autocovariance operator, matrix of size (n_grid,n_grid)

References

Rubin, Panaretos. Simulation of stationary functional time series with given spectral density. arXiv, 2020

See Also

CKL_simulate

Examples

# Define the spectral density operator as an integral operator with kernel
k_bbridge <- function(x,y) { pmin(x,y)-x*y }
spec_density <- function( omega, x,y ){ 1/(1-0.9 *cos(omega)) * k_bbridge( (x-omega/pi)%%1, (y-omega/pi)%%1  ) }

# evaluation setting
lag <- 1 # change here to evaluate different lag-h autocovariance operator. put "lag <- 0" for lag-0 covariance operator
n_grid <- 101

# calculate the lag-h autocovariance operator
covlagh <- spec_density_covlagh_operator(spec_density, lag, n_grid)

# visualise as a surface plot
persp(covlagh)

tomasrubin/specsimfts documentation built on March 26, 2021, 1:37 p.m.