FARFIMA_test_stationarity: Check stationarity of the FARFIMA(p,d,q) process

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

Description

Numerically verify if the FARFIMA(p,d,q) process is stationary. The stationarity depends solely on the autoregressive part. The method construct the order-1 autoregressive process in the state space and calculate the 1000-th power of the state-space autoregressive operator. If its norm is less than one, the process is pronnounced stationary.

Usage

FARFIMA_test_stationarity(FARFIMA_pars, n_grid = 101)

Arguments

FARFIMA_pars

The list of the parameters for the FARFIMA(p,d,q) process. Must contain fields: (i) fractional_d, a real number in the open interval (-0.5,0.5) controling the fractional integration degree. fractional_d being positive corresponds to long-rande dependence behaviour. (ii) operators_ar, the list of length 'p' the order of the autoregressive part. The autoregressive operators are considered to be integral operators defined through their kernels which are saved as the elements of the list operators_ar as functions of two variables, x and y, returning the value of the kernel at point (x,y). In case of degenerate autoregressive part define operators_ar as an empty list. (iii) operators_ma, the list of length 'q', the order of the moving average part. Just like operators_ar its a liks of functions - the kernels of the moving average operators. (iv) The covariance opperator of the stochastic innovation process can be defined either through (iv-a) its kernel, (iv-b) finite rank eigendecomposition, (iv-c) infinite rank decomposition. In the case (iv-a), define sigma as a function of two variables x and y, returning the value of the covariance kernel at point (x,y). In the case (iv-b), define the elements sigma_eigenvalues as a vector of finitely many eigenvalues and sigma_eigenfunctions as a list of the same length as sigma_eigenvalues with each element being a function of variable x returning the value of that eigenfunction at point x. In the case (iv-c), define the elements sigma_eigenvalues as a function of the variable n returning the n-th eigenvalue and the element sigma_eigenfunctions as a function of two variables, n and x, returning the value of the n-th eigenfunctions at point x. See the example bellow for some examples on how to set up FARFIMA_pars.

n_grid

Number of grid points (spatial resolution) of the discretisation of [0,1]^2. The method checks if the process is stationary on the discretization level. Assuming smoothness, it shoudn't be dependent on the grid resolution unless very coarse.

Value

Returns TRUE or FALSE if stationary or not.

References

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

See Also

FARFIMA_simulate, FARFIMA_test_stationarity

Examples

# (i) fractional integration
fractional_d <- 0 # in the open interval (-0.5, 0.5), positive number means long-range dependence

# (ii) autoregressive operators
operators_ar <- list(
function(x,y){ 0.3*sin(x-y) },
function(x,y){ 0.3*cos(x-y) },
function(x,y){ 0.3*sin(2*x) },
function(x,y){ 0.3*cos(y) }
)
# operators_ar <- list() # use empty list for degenerate AR part

# (iii) moving average kernels
# you can put here arbitrary long list of operators
operators_ma <- list(
function(x,y){ x+y },
function(x,y){ x },
function(x,y){ y }
)
# operators_ma <- list() # use empty list for degenerate MA part

# (iv-b) covariance of the inovation defined through eigenvalues and eigenfunctions
# you can put here arbitrary long lists but their lenghts should match
sigma_eigenvalues <- c(1, 0.6, 0.3, 0.1, 0.1, 0.1, 0.05, 0.05, 0.05, 0.05)
sigma_eigenfunctions <- list(
function(x){ sin(2*pi*x) },
function(x){ cos(2*pi*x) },
function(x){ sin(4*pi*x) },
function(x){ cos(4*pi*x) },
function(x){ sin(6*pi*x) },
function(x){ cos(6*pi*x) },
function(x){ sin(8*pi*x) },
function(x){ cos(8*pi*x) },
function(x){ sin(10*pi*x) },
function(x){ cos(10*pi*x) }
)

# # (iv-c) innovation covariance operator (Brownian motion)
# sigma_eigenvalues <- function(n) { 1/((n-0.5)*pi)^2 }
# sigma_eigenfunctions <- function(n,x) { sqrt(2)*sin((n-0.5)*pi*x) }

# put the parameters into one list
FARFIMA_pars <- list(fractional_d=fractional_d, operators_ar=operators_ar, operators_ma=operators_ma, sigma_eigenvalues=sigma_eigenvalues,sigma_eigenfunctions=sigma_eigenfunctions)

# # (iv-a) Alternatively, define the kernel of the white noise innovation.
# sigma <- function(x,y) { pmin(x,y) } # Brownian motion
# FARFIMA_pars <- list(fractional_d=fractional_d,operators_ar=operators_ar,operators_ma=operators_ma, sigma=sigma)

# test stationarity
print(FARFIMA_test_stationarity(FARFIMA_pars))
 

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