ETAS (version 0.7.2)

rates: Declustering Probabilities, Background Seismicity Rate and Clustering Coefficient

Description

Functions to estimate the declustering probabilities, background seismicity rate and clustering (triggering) coefficient for a fitted ETAS model.

Usage

probs(fit)
  rates(fit, lat.range = NULL, long.range = NULL,
        dimyx=NULL, plot.it=TRUE)

Value

If plot.it=TRUE, the function produces plots of the background seismicity and total spatial rate, clustering coefficient and conditional intensity function at the end of study period.

If plot.it=FALSE, it returns a list with components

• bkgd the estimated background siesmicity rate

• total the estimated total spatial rate

• clust the estimated clustering coefficient

• lamb the estimated conditional intensity function at time \(t=t_{\mathrm{start}}\)

Arguments

fit

A fitted ETAS model. An object of class "etas".

lat.range

Latitude range of the rectangular grid. A numeric vector of length 2.

long.range

Longitude range of the rectangular grid. A numeric vector of length 2.

dimyx

Dimensions of the rectangular discretization grid for the geographical study region. A numeric vector of length 2.

plot.it

Logical flag indicating whether to plot the rates or return them as pixel images.

Author

Abdollah Jalilian

Details

The function probs returns estimates of the declustering probabilities $$p_j = 1 - \frac{\mu(x_j, y_j)}{lambda(t_j, x_j, y_j|H_{t_j})}$$ where \(1-p_j\) is the probability that event \(j\) is a background event.

The function rates returns kernel estimate of the background seismicity rate \(\mu(x,y)\) and the clustering (triggering) coefficient $$\omega(x,y)=1-\frac{\mu(x,y)}{\Lambda(x,y)}$$ where \(\Lambda(x,y)\) is the total spatial intensity function.

The argument dimyx determines the rectangular discretization grid dimensions. If it is given, then it must be a numeric vector of length 2 where the first component dimyx[1] is the number of subdivisions in the y-direction (latitude) and the second component dimyx[2] is the number of subdivisions in the x-direction (longitude).

References

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. tools:::Rd_expr_doi("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. tools:::Rd_expr_doi("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. tools:::Rd_expr_doi("10.5047/eps.2010.12.010").

See Also

etas

Examples

  # preparing the catalog
  iran.cat <- catalog(iran.quakes, time.begin="1973/01/01",
     study.start="1996/01/01", study.end="2016/01/01",
     lat.range=c(25, 42), long.range=c(42, 63), mag.threshold=4.5)

  print(iran.cat)
  if (FALSE) {
  plot(iran.cat)}

  # initial parameters values
  param01 <- c(0.46, 0.23, 0.022, 2.8, 1.12, 0.012, 2.4, 0.35)

  # fitting the model and
  if (FALSE) {
  iran.fit <- etas(iran.cat, param0=param01)}

  # estimating the declustering probabilities
  if (FALSE) {
  pr <- probs(iran.fit)
  plot(iran.cat$longlat.coord[,1:2], cex=2 * (1 - pr$prob))}

  # estimating the  background seismicity rate and clustering coefficient
  if (FALSE) {
  rates(iran.fit, dimyx=c(100, 125))
  iran.rates <- rates(iran.fit, dimyx=c(200, 250), plot.it=FALSE)
  summary(iran.rates$background)}