| ETAS_Boots {ETASbootstrap} | R Documentation |
Compute bootstrap confidence intervals
Description
A number (1000 by default) of earthquake data catalogs are
simulated by bootstrap and recorded. A 2-D spatial and temporal ETAS model is
fitted to each bootstrap-simulated
earthquake data catalog, and the corresponding parameter estimates are
recorded, which provides as many distributions of estimates.
For a given confidence level 1-\alpha (0.95 by default), bootstrap
confidence intervals are built from the empirical \alpha/2 (0.025) and
1 -\alpha/2 (0.975) quantiles
of the distributions of estimates for the parameters
(A,c,\alpha,p,D,q,\gamma) of the ETAS model.
Usage
ETAS_Boots(
earthquake_data,
longitude_boundaries = NULL,
latitude_boundaries = NULL,
longitude_study = NULL,
latitude_study = NULL,
time_begin = NULL,
study_start = NULL,
study_end = NULL,
magnitude_threshold = NULL,
time_zone = "GMT",
parameters_0 = NULL,
number_simulations = 1000,
confidence_level = 0.95,
output_datasets = FALSE,
output_estimates = FALSE
)
Arguments
earthquake_data |
An object of class "data.frame" containing the following 5 columns:
See VCI_earthquakes for an example. Note that West longitude and South latitude values should be negative, whereas the East longitude and North latitude values are positive. |
longitude_boundaries |
A numerical vector of length 2 (long_min, long_max) with the longitude boundaries of a rectangular space window, for which the earthquake catalog data are contained in earthquake_data. If NULL (at the beginning of the execution of the program), long_min and long_max will be set (by the program) to the minimum and maximum values of the longitudes of earthquakes in earthquake_data. |
latitude_boundaries |
A numerical vector of length 2 (lat_min, lat_max) with the latitude boundaries of a rectangular space window, for which the earthquake catalog data are contained in earthquake_data. If NULL, lat_min and lat_max will be set to the minimum and maximum values of the latitudes of earthquakes in earthquake_data. |
longitude_study |
A numerical vector of length 2 (long_study_min, long_study_max) with the longitude boundaries of a rectangular "study space window" |
latitude_study |
A numerical vector of length 2 (lat_study_min, lat_study_max) with the latitude boundaries of the study space window. Note: The study space window should be narrower than the rectangular space window defined by longitude_boundaries and latitude_boundaries. If NULL, longitude_study and latitude_study will be filled with boundaries defining a rectangular space window 20% narrower than the space window built from the longitude_boundaries and latitude_boundaries, while keeping the same center. |
time_begin |
A character string, in the date-time format (yyyy-mm-dd hh:mm:ss), which identifies the beginning of the time span in earthquake_data. If NULL, time_begin will be set to the date-time of the first event in earthquake_data. |
study_start |
A character string, in the date-time format, which specifies the start of the study period. If NULL, study_start will be set to the date-time corresponding to that of time_begin plus 20% of the length of the time span in earthquake_data. |
study_end |
A character string, in the date-time format, which specifies the end of the study period. If NULL, it will be set to the date-time of the last event in earthquake_data. Note: study_end coincides with the end of the time span in earthquake_data. |
magnitude_threshold |
A decimal number, which specifies the threshold to be used for the magnitudes of earthquakes. Only earthquakes with a magnitude greater or equal to magnitude_threshold will be considered, while the model is fitting. is used consistently and thoroughly. If NULL, the minimum magnitude calculated from the events in earthquake_data will be used for magnitude_threshold. |
time_zone |
A character string specifying the time zone in which the occurrence times of earthquakes were recorded. The default "GMT"is the UTC (Universal Time Coordinates). |
parameters_0 |
A decimal vector of size 8 |
number_simulations |
A positive integer, which stands for the number of requested bootstrap simulations. The default value is 1000. |
confidence_level |
A decimal number in (0, 1), which specifies the confidence level associated with the bootstrap confidence intervals that are built for the ETAS model parameters, and saved as outputs. It is set to 0.95 by default. |
output_datasets |
A logical flag indicating whether or not the bootstrap-simulated earthquake data catalogs must be written in CSV files. The default setting is FALSE. |
output_estimates |
A logical flag indicating whether or not the maximum likelihood estimates of parameters from each bootstrap-simulated earthquake data catalog must be written in a CSV file. The default setting is FALSE. |
Details
Ogata (1998) proposed the 2-D spatial and temporal ETAS model, which is
now widely used to decluster earthquake catalogs and, to a lesser extent,
make short-term forecasts. In the 2-D spatial and temporal ETAS model, the
behavior of the point process for which
\{(t_i,x_i,y_i,m_i),i=1,\dots,n\} is a partial realization is
characterized by the conditional intensity function
\lambda_{\beta,\mathbf{\theta}}(t,x,y,m
\mid H_t) = s_{\beta}(m)\lambda_{\mathbf{\theta}}(t,x,y \mid H_t),
where \beta and \mathbf{\theta} = (\nu,A,\alpha,c,p,q,D,\gamma)
are the model parameters. s_\beta is the probability density function
(pdf) associated with the distribution of earthquake magnitudes. It is
assumed that the distribution of the magnitude of earthquakes is independent
of the distributions of the occurrence time of earthquakes and the 2-D
spatial location of their epicenters. It can be expressed, for arbitrary
\beta \in (0, \infty) as
s_{\beta}(m) = \beta \exp \{
-\beta(m-m_0)\},
where m and m_0 represent the magnitude of the
earthquake and the magnitude threshold, respectively.
\lambda_{\mathbf{\theta}}(t,x,y \mid H_t) represents the rate of
observation of earthquakes in time and space, given the information on
earthquakes prior to time t. This rate is expressed as the sum of two
terms and is defined as
\lambda_{\mathbf{\theta}}(t,x,y \mid H_t) = \mu(x,y) + \sum_{i:t_i<t}k(m_i)g(t-t_i)f(x-x_i,y-y_i \mid m_i)
with
\mu(x,y) = \nu u(x,y),
where \nu \in (0, \infty).
The term \mu(x,y) is usually called “background seismicity rate" and represents the rate at which earthquakes independently occur around longitude x and latitude y.
The ith term of the summation in \lambda_{\theta}, namely
k(m_i)g(t-t_i)f(x-x_i,y-y_i \mid m_i)
represents the effect of the ith earthquake before time t on the occurrence rate of earthquakes that would occur at time t, with an epicenter around (x,y). Thus,
\sum_{i:t_i<t}k(m_i)g(t-t_i)f(x-x_i,y-y_i \mid m_i)
describes the total effect of all the earthquakes that occurred prior to time t, on the rate at which earthquakes would occur with an epicenter around (x, y) at time t.
The expressions of k, g, and f are discussed individually as follows. First,
k(m) = Ae^{\alpha(m-m_0)},\quad m \geq m_0 ,
can be interpreted as the expected number of earthquakes triggered by a previous earthquake with magnitude m, where A \in (0, \infty) and \alpha \in (0, \infty). Second, for all t \in (t_i, \infty),
g(t-t_i) = \frac{p-1}{c} \, \left (1+\frac{t-t_i}{c} \right )^{-p},
is the pdf for the occurrence time of an earthquake triggered by the ith earthquake in the catalog, which occurred at time t_i, where c \in (0, \infty) and p \in (1, \infty). Third,
f(x-x_i,y-y_i \mid m_i) = \frac{q-1}{\pi De^{\gamma(m_i-m_0)}} \, \left\{ 1+\frac{(x-x_i)^2+(y-y_i)^2}{De^{\gamma(m_i-m_0)}} \right\}^{-q}
is the pdf for the occurrence location (epicenter) of an earthquake triggered by the ith earthquake in the catalog, which occurred with magnitude m_i and an epicenter at (x_i, y_i), where D \in (0, \infty), \gamma \in (0, \infty), and q \in (1, \infty).
Value
A list of three components:
MLE: A numerical vector recording the maximum likelihood estimates of the ETAS model parameters
(A,c,\alpha,p,D,q,\gamma).ASE: A numerical vector recording the corresponding asymptotic standard errors.
BootstrapCI: A matrix recording the corresponding bootstrap confidence intervals for the confidence_level entered as input and all the other input arguments of the ETAS_Boots function starting with earthquake_data.
When output_datasets=TRUE, the simulated earthquake data catalogs are written in "Boot_N.csv", where "N" denotes the number of bootstrap simulation runs.
When output_estimates=TRUE, the maximum likelihood estimates of parameters from each simulated earthquake data catalog are written in "estimates.csv".
References
Jalilian, A. (2019). ETAS: An R package for fitting the space-time ETAS model to earthquake data. Journal of Statistical Software 88, 1–39.
Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics 50(2), 379–402.
Zhuang, J., Y. Ogata, and D. Vere-Jones (2002). Stochastic declustering of space-time earthquake occurrences. Journal of the American Statistical Association 97(458), 369–380.
Zhuang, J., Y. Ogata, and D. Vere-Jones (2004). Analyzing earthquake clustering features by using stochastic reconstruction. Journal of Geophysical Research: Solid Earth 109(B05301).
Examples
set.seed(23)
ETAS_Boots(earthquake_data = VCI_earthquakes,
longitude_boundaries = c(-131, -126.25),
latitude_boundaries = c(48, 50),
longitude_study = c(-130.5,-126.75),
latitude_study = c(48.25,49.75),
time_begin = "2000/01/01 00:00:00",
study_start = "2008/04/27 00:00:00",
study_end = "2018/04/27 00:00:00",
magnitude_threshold = 4,
time_zone = "GMT",
parameters_0 = c(0.65, 0.24, 0.0068, 0.97, 1.22, 0.0033, 2.48, 0.17),
number_simulations = 4,
confidence_level = 0.95,
output_datasets = FALSE,
output_estimates = FALSE)