Description Usage Arguments Details Value Control Arguments Author(s) References Examples
The function generates n
random numbers of a spatial ARCH process for given parameters and weighting schemes.
1 | 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.,
Y = diag(h)^(1/2) ε ,
where ε is a spatial White Noise process. The definition of h depends on the chosen type
. The following types are available.
type = "spARCH"
- simulates ε from a truncated normal distribution on the interval [-a, a], such that h > 0 with
h = α + ρ W Y^(2) and a = 1 / (ρ^2||W^2||_1)^(1/4)
Note that the normal distribution is not trunctated (a = ∞), if W is a strictly triangular matrix, as it is ensured that h > 0. Generally, it is sufficient that if there exists a permutation such that 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 h = α + ρ W g_b(ε) .
For the log-spARCH process, the errors follow a standard normal distribution. The function g_b is given by
g_b(ε) = (ln|h(ε_1)|^b, ..., ln|h(ε_n)|^b)' .
type = "complex-spARCH"
- allows for complex solutions of h^(1/2) with
h = | α + ρ W Y^(2) |.
The errors follow a standard normal distribution.
The functions returns a vector 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
, W is a triangular matrix and there are no checks to verify this assumption (default FALSE
)
Philipp Otto potto@europa-uni.de
Philipp Otto, Wolfgang Schmid, Robert Garthoff (2018). Generalised Spatial and Spatiotemporal Autoregressive Conditional Heteroscedasticity. Spatial Statistics 26, pp. 125-145. arXiv:1609.00711
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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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