rETASp: 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
)

Value

A stpm object

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 FALSE
all.marks: Logical value indicating whether to store all the simulation information as marks in the stpm object. If FALSE (default option) only the magnitude is returned.

Author

Nicoletta D'Angelo and Marcello Chiodi

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.

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

Run this code


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)

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