sim.spARCH | R Documentation |
The function generates n
random numbers of a spatial ARCH process for given parameters and weighting schemes.
sim.spARCH(n = dim(W)[1], rho, alpha, W, b = 2, type = "spARCH", control = list())
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) |
The function simulates n
observations Y = (Y_1, ..., Y_n)'
of a spatial ARCH process, i.e.,
\boldsymbol{Y} = diag(\boldsymbol{h})^{1/2} \boldsymbol{\varepsilon} \, ,
where \boldsymbol{\varepsilon}
is a spatial White Noise process. The definition of \boldsymbol{h}
depends on the chosen type
. The following types are available.
type = "spARCH"
- simulates \boldsymbol{\varepsilon}
from a truncated normal distribution on the interval [-a, a]
, such that \boldsymbol{h} > 0
with
\boldsymbol{h} = \alpha + \rho \mathbf{W} \boldsymbol{Y}^{(2)} \; \mbox{and} \; a = 1 / ||\rho^2\mathbf{W}^2||_1^{1/4}.
Note that the normal distribution is not trunctated (a = \infty
), if \mathbf{W}
is a strictly triangular matrix, as it is ensured that \boldsymbol{h} > \boldsymbol{0}
. Generally, it is sufficient that if there exists a permutation such that \mathbf{W}
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.,
\ln\boldsymbol{h} = \alpha + \rho \mathbf{W} g(\boldsymbol{\varepsilon}) \, .
For the log-spARCH process, the errors follow a standard normal distribution. The function g_b
is given by
g_b(\boldsymbol{\varepsilon}) = (\ln|\varepsilon(\boldsymbol{s}_1)|^{b}, \ldots, \ln|\varepsilon(\boldsymbol{s}_n)|^{b})' \, .
type = "complex-spARCH"
- allows for complex solutions of \boldsymbol{h}^{1/2}
with
\boldsymbol{h} = \alpha + \rho \mathbf{W} \boldsymbol{Y}^{(2)} \, .
The errors follow a standard normal distribution.
The functions returns a vector \boldsymbol{y}
.
seed
- positive integer to initialize the random number generator (RNG), default value is a random integer in [1, 10^6]
silent
- if FALSE
, current random seed is reported
triangular
- if TRUE
, \mathbf{W}
is a triangular matrix and there are no checks to verify this assumption (default FALSE
)
Philipp Otto philipp.otto@glasgow.ac.uk
Philipp Otto, Wolfgang Schmid, Robert Garthoff (2018). Generalised Spatial and Spatiotemporal Autoregressive Conditional Heteroscedasticity. Spatial Statistics 26, pp. 125-145. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spasta.2018.07.005")}, arXiv: \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.1609.00711")}
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]))
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