spARCHsim: Simulation of spatial ARCH models
In spGARCH: Spatial ARCH and GARCH Models (spGARCH)

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 `length(n)` > 1, the length is taken to be the number required. Default `dim(W)[1]`
`rho`	spatial dependence parameter rho
`alpha`	unconditional variance level alpha
`W`	`n` times `n` spatial weight matrix
`b`	parameter `b` for logarithmic spatial ARCH (only needed if `type = "log-spARCH"`). Default 2.
`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

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]))