etas: Fit the space-time ETAS model to data

Description

A function to fit the space-time version of the Epidemic Type Aftershock Sequence (ETAS) model to a catalog of earthquakes (a spatio-temporal point pattern) and perform a stochastic declustering method.

Usage

etas(object, param0 = NULL, bwd = NULL, nnp = 5, bwm = 0.05,
      verbose = TRUE, plot.it = FALSE, ndiv = 1000, no.itr = 11,
      rel.tol=1e-03, eps = 1e-06, cxxcode = TRUE, nthreads = 1)

Arguments

object

An object of class "catalog" containing an earthquake catalog dataset.

param0

Initial guess for model parameters. A numeric vector of appropriate length (currently 8). See details.

bwd

Optional. Bandwidths for smoothness and integration on the geographical region win. A numeric vector which has the length of the number of events. If not supplied, the following arguments nnp and bwm determine bandwidths.

nnp

Number of nearest neighbors for bandwidth calculations. An integer.

bwm

Minimum bandwidth. A positive numeric value.

verbose

Logical flag indicating whether to print progress reports.

plot.it

Logical flag indicating whether plot probabilities of each event being a background event on a map.

ndiv

An integer indicating the number of knots on each side of the geographical region for integral approximation.

no.itr

An integer indicating the number of iterations for convergence of the iterative approach of simultaneous estimation and declustering algorithm. See details.

rel.tol

Relative iteration convergence tolerance of the iterative estimation approach.

eps

Optimization convergence tolerance in the Davidon-Fletch-Powell algorithm

cxxcode

Logical flag indicating whether to use the C++ code. The C++ code is slightly faster and allows parallel computing.

nthreads

An integer indicating number of threads in the parallel region of the C++ code

Value

A list with components

param:: The ML estimates of model parameters.
bk:: An estimate of the $u(x, y)$.
pb:: The probabilities of being background event.
opt:: The results of optimization: the value of the log-likelihood function at the optimum point, its gradient at the optimum point and AIC of the model.
rates:: Pixel images of the estimated total intensity, background intensity, clustering intensity and conditional intensity.

Details

Ogata (1988) introduced the epidemic type aftershock sequence (ETAS) model based on Gutenberg-Richter law and modified Omori law. In its space-time representation (Ogata, 1998), the ETAS model is a temporal marked point process model, and a special case of marked Hawkes process, with conditional intensity function $$ \lambda(t, x, y | H_t) = \mu(x,y) + \sum_{t_i < t} k(m_i)g(t - t_i)f(x - x_i, y - y_i|m_i) $$ where

$H_t$:: is the observational history up to time t, but not including t; that is $$H_t=\{(t_i, x_i, y_i, m_i): t_i < t\}$$
$\mu(x,y)$:: is the background intensity. Currently it is assumed to take the semi-parametric form $$\mu(x,y)=\mu u(x,y)$$ where $\mu$ is an unknown constant and $u(x,y)$ is an unknown function.
$k(m)$:: is the expected number of events triggered from an event of magnitude $m$ given by $$k(m) = A\exp(\alpha(m - m_0))$$
$g(t)$:: is the p.d.f of the occurrence times of the triggered events, taking the form $$g(t) = \frac{p-1}{c}(1 + \frac{t}{c})^{-p}$$
$f(x,y|m)$:: is the p.d.f of the locations of the triggered events, considered to be either the long tail inverse power density $$ f(x, y|m) = \frac{q-1}{\pi \sigma(m))} (1 + \frac{x^2 + y^2}{\sigma(m)})^{-q} $$ or the light tail Gaussian density (currently not implemented) $$ f(x,y|m)= \frac{1}{2\pi \sigma(m)}\exp(-\frac{x^2 + y^2}{2\sigma(m)}) $$ with $$ \sigma(m) = D\exp(\gamma(m - m_0)) $$

The ETAS models classify seismicity into two components, background seismicity $\mu(x, y)$ and clustering seismicity $\lambda(t, x, y|H_t) - \mu(x, y)$, where each earthquake event, whether it is a background event or generated by another event, produces its own offspring according to the branching rules controlled by $k(m)$, $g(m)$ and $f(x, y|m)$.

Background seismicity rate $u(x, y)$ and the model parameters $$\theta=(\mu, A, c, \alpha, p, D, q, \gamma)$$ are estimated simultaneously using an iterative approach proposed in Zhuang et al. (2002). First, for an initial $u_0(x, y)$, the parameter vector $\theta$ is estimated by maximizing the log-likelihood function $$l(\theta)=\sum_{i} \lambda(t_i, x_i, y_i|H_{t_i}) - \int \lambda(t, x, y|H_t) dx dy dt.$$ Then the procedure calculates the probability of being a background event for each event in the catalog by $$ \phi_i = \frac{\mu(x_i, y_i)}{\lambda(t_i, x_i, y_i|H_{t_i})}. $$ Using these probabilities and kernel smoothing method with Gaussian kernel and appropriate choice of bandwidth (determined by bwd or nnp and bwm arguments), the background rate $u_0(x, y)$ is updated. These steps are repeated until the estimates converge (stabilize).

The no.itr argument specifies the maximum number of iterations in the iterative simultaneous estimation and declustering algorithm. The estimates often converge in less than ten iterations. The relative iteration convergence tolerance and the optimization convergence tolerance are, respectively, determined by rel.tol and eps arguments. The progress of the computations can be traced by setting the verbose and plot.it arguments to be TRUE.

If cxxcode = TRUE, then the internal function etasfit uses the C++ code implemented using the Rcpp package, which allows multi-thread parallel computing on multi-core processors with OpenMP. The argument nthreads in this case determines the number of threads in the parallel region of the code. If nthreads = 1 (the default case), then a serial version of the C++ code carries out the computations.

This version of the ETAS model assumes that the earthquake catalog is complete and the data are stationary in time. If the catalog is incomplete or there is non-stationarity (e.g. increasing or cyclic trend) in the time of events, then the results of this function are not reliable.

References

Ogata Y (1988). Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes. Journal of the American Statistical Association, 83(401), 9--27. 10.2307/2288914.

Ogata Y (1998). Space-time Point-process Models for Earthquake Occurrences. Annals of the Institute of Statistical Mathematics, 50(2), 379--402. 10.1023/a:1003403601725.

Zhuang J, Ogata Y, Vere-Jones D (2002). Stochastic Declustering of Space-Time Earthquake Occurrences. Journal of the American Statistical Association, 97(458), 369--380. 10.1198/016214502760046925.

Zhuang J, Ogata Y, Vere-Jones D (2006). Diagnostic Analysis of Space-Time Branching Processes for Earthquakes. In Case Studies in Spatial Point Process Modeling, pp. 275--292. Springer Nature. 10.1007/0-387-31144-0_15.

Zhuang J (2011). Next-day Earthquake Forecasts for the Japan Region Generated by the ETAS Model. Earth, Planets and Space, 63(3), 207--216. 10.5047/eps.2010.12.010.

Examples

Run this code

# NOT RUN {
  # fitting the ETAS model to an Iranian catalog
  # preparing the catalog
  iran.cat <- catalog(iran.quakes, time.begin="1973/01/01",
     study.start="1986/01/01", study.end="2016/01/01",
     lat.range=c(26, 40), long.range=c(44, 63), mag.threshold=5)
  print(iran.cat)
  
# }
# NOT RUN {
  plot(iran.cat)
# }
# NOT RUN {
  # setting initial parameter values
  param0 <- c(0.46, 0.23, 0.022, 2.8, 1.12, 0.012, 2.4, 0.35)

  # fitting the model
  
# }
# NOT RUN {
  iran.fit <- etas(iran.cat, param0=param0)
# }
# NOT RUN {

  # fitting the ETAS model to an Italian catalog
  # preparing the catalog
  italy.cat <- catalog(italy.quakes, dist.unit="km")
  
# }
# NOT RUN {
  plot(italy.cat)
# }
# NOT RUN {
  # setting initial parameter values
  mu <- 1
  k0 <- 0.005
  c <- 0.005
  alpha <- 1.05
  p <- 1.01
  D <- 1.1
  q <- 1.52
  gamma <- 0.6
  # reparametrization: transform k0 to A
  A <- pi * k0 / ((p - 1) * c^(p - 1) * (q - 1) * D^(q - 1))
  param0 <- c(mu, A, c, alpha, p, D, q, gamma)

  # fitting the model
  
# }
# NOT RUN {
  nthreads <- parallel::detectCores()
  italy.fit <- etas(italy.cat, param0, nthreads=nthreads)
# }
# NOT RUN {

  # fitting the ETAS model to a Japanese catalog
  # setting the target polygonal study region
  jpoly <- list(long=c(134.0, 137.9, 143.1, 144.9, 147.8,
      137.8, 137.4, 135.1, 130.6), lat=c(31.9, 33.0, 33.2,
      35.2, 41.3, 44.2, 40.2, 38.0, 35.4))
  # preparing the catalog
  japan.cat <- catalog(japan.quakes, study.start="1953-05-26",
      study.end="1990-01-08", region.poly=jpoly, mag.threshold=4.5)
  
# }
# NOT RUN {
  plot(japan.cat)
# }
# NOT RUN {
  # setting initial parameter values
  param0 <- c(0.592844590, 0.204288231, 0.022692883, 1.495169224,
  1.109752319, 0.001175925, 1.860044210, 1.041549634)

  # fitting the model
  
# }
# NOT RUN {
  nthreads <- parallel::detectCores()
  japan.fit <- etas(japan.cat, param0, nthreads=nthreads)
# }

Run the code above in your browser using DataLab