rETASp {stopp} | R Documentation |
Simulation of a spatio-temporal ETAS (Epidemic Type Aftershock Sequence) model
Description
This function simulates a spatio-temporal ETAS
(Epidemic Type Aftershock Sequence) process as a stpm
object.
It follows the generating scheme for simulating a pattern from an Epidemic Type Aftershocks-Sequences (ETAS) process (Ogata and Katsura 1988) with conditional intensity function (CIF) as in Adelfio and Chiodi (2020), adapted for the space location of events to be constrained.
See the 'Details' section.
Usage
rETASp(
pars = NULL,
betacov = 0.39,
m0 = 2.5,
b = 1.0789,
tmin = 0,
t.lag = 200,
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 1,
covsim = FALSE,
all.marks = FALSE
)
Arguments
pars |
A vector of parameters of the ETAS model to be simulated. See the 'Details' section. |
betacov |
Numerical array. Parameters of the ETAS model covariates. |
m0 |
Parameter for the background general intensity of the ETAS model. In the common seismic analyses it represents the threshold magnitude. |
b |
1.0789 |
tmin |
Minimum value of time. |
t.lag |
200 |
xmin |
Minimum of x coordinate range |
xmax |
Maximum of x coordinate range |
ymin |
Minimum of y coordinate range |
ymax |
Maximum of y coordinate range |
covsim |
Default |
all.marks |
Logical value indicating whether to store
all the simulation information as marks in the |
Details
The CIF of an ETAS process as in Adelfio and Chiodi (2020) can be written as
\lambda_{\theta}(t,\textbf{u}|\mathcal{H}_t)=\mu f(\textbf{u})+\sum_{t_j<t} \frac{\kappa_0 \exp(\eta_j)}{(t-t_j+c)^p} \{ (\textbf{u}-\textbf{u}_j)^2+d \}^{-q} ,
where
\mathcal{H}_t
is the past history of the process up to time
t
\mu
is the large-scale general intensity
f(\textbf{u})
is
the spatial density
\eta_j=\boldsymbol{\beta}' \textbf{Z}_j
is a linear predictor
\textbf{Z}_j
the external known covariate vector, including the
magnitude
\boldsymbol{\theta}= (\mu, \kappa_0, c, p, d, q, \boldsymbol{\beta})
are the parameters to be estimated
\kappa_0
is a
normalising constant
c
and p
are characteristic parameters of the
seismic activity of the given region,
and d
and q
are two parameters
related to the spatial influence of the mainshock
In the usual ETAS
model for seismic analyses, the only external covariate represents the magnitude,
\boldsymbol{\beta}=\alpha
, as
\eta_j = \boldsymbol{\beta}' \textbf{Z}_j = \alpha (m_j-m_0)
, where
m_j
is the magnitude of the j^{th}
event and m_0
the threshold
magnitude, that is, the lower bound for which earthquakes with higher
values of magnitude are surely recorded in the catalogue.
Value
A stpm
object
Author(s)
Nicoletta D'Angelo and Marcello Chiodi
References
Adelfio, G., and Chiodi, M. (2021). Including covariates in a space-time point process with application to seismicity. Statistical Methods & Applications, 30(3), 947-971.
Ogata, Y., and Katsura, K. (1988). Likelihood analysis of spatial inhomogeneity for marked point patterns. Annals of the Institute of Statistical Mathematics, 40(1), 29-39.
Examples
set.seed(95)
X <- rETASp(pars = c(0.1293688525, 0.003696, 0.013362, 1.2,0.424466, 1.164793),
betacov = 0.5,
xmin = 600, xmax = 2200, ymin = 4000, ymax = 5300)
plot(X)