sim.spARCH {spGARCH} | R Documentation |
Simulation of spatial ARCH models
Description
The function generates n
random numbers of a spatial ARCH process for given parameters and weighting schemes.
Usage
sim.spARCH(n = dim(W)[1], rho, alpha, W, b = 2, type = "spARCH", control = list())
Arguments
n |
number of observations. If |
rho |
spatial dependence parameter rho |
alpha |
unconditional variance level alpha |
W |
|
b |
parameter |
type |
type of simulated spARCH process (see details) |
control |
list of control arguments (see below) |
Details
The function simulates n
observations of a spatial ARCH process, i.e.,
where is a spatial White Noise process. The definition of
depends on the chosen
type
. The following types are available.
-
type = "spARCH"
- simulatesfrom a truncated normal distribution on the interval
, such that
with
Note that the normal distribution is not trunctated (
), if
is a strictly triangular matrix, as it is ensured that
. Generally, it is sufficient that if there exists a permutation such that
is strictly triangular. In this case, the process is called oriented spARCH process.
-
type = "log-spARCH"
- simulates a logarithmic spARCH process (log-spARCH), i.e.,For the log-spARCH process, the errors follow a standard normal distribution. The function
is given by
-
type = "complex-spARCH"
- allows for complex solutions ofwith
The errors follow a standard normal distribution.
Value
The functions returns a vector .
Control Arguments
-
seed
- positive integer to initialize the random number generator (RNG), default value is a random integer in -
silent
- ifFALSE
, current random seed is reported -
triangular
- ifTRUE
,is a triangular matrix and there are no checks to verify this assumption (default
FALSE
)
Author(s)
Philipp Otto potto@europa-uni.de
References
Philipp Otto, Wolfgang Schmid, Robert Garthoff (2018). Generalised Spatial and Spatiotemporal Autoregressive Conditional Heteroscedasticity. Spatial Statistics 26, pp. 125-145. arXiv:1609.00711
Examples
require("spdep")
# 1st example
##############
# parameters
rho <- 0.5
alpha <- 1
d <- 2
nblist <- cell2nb(d, d, type = "queen")
W <- nb2mat(nblist)
# simulation
Y <- sim.spARCH(rho = rho, alpha = alpha, W = W, type = "log-spARCH")
# visualization
image(1:d, 1:d, array(Y, dim = c(d,d)), xlab = expression(s[1]), ylab = expression(s[2]))
# 2nd example
##############
# two spatial weighting matrices W_1 and W_2
# h = alpha + rho_1 W_1 Y^2 + rho_2 W_2 Y^2
W_1 <- W
nblist <- cell2nb(d, d, type = "rook")
W_2 <- nb2mat(nblist)
rho_1 <- 0.3
rho_2 <- 0.7
W <- rho_1 * W_1 + rho_2 * W_2
rho <- 1
Y <- sim.spARCH(n = d^2, rho = rho, alpha = alpha, W = W, type = "log-spARCH")
image(1:d, 1:d, array(Y, dim = c(d,d)), xlab = expression(s[1]), ylab = expression(s[2]))