rates {ETAS}

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)

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.

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).

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}}

Author(s)

Abdollah Jalilian [e-mail]

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. 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. 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. 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)
  ## Not run: 
  plot(iran.cat)
## End(Not run)

  # 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
  ## Not run: 
  iran.fit <- etas(iran.cat, param0=param01)
## End(Not run)

  # estimating the declustering probabilities
  ## Not run: 
  pr <- probs(iran.fit)
  plot(iran.cat$longlat.coord[,1:2], cex=2 * (1 - pr$prob))
## End(Not run)

  # estimating the  background seismicity rate and clustering coefficient
  ## Not run: 
  rates(iran.fit, dimyx=c(100, 125))
  iran.rates <- rates(iran.fit, dimyx=c(200, 250), plot.it=FALSE)
  summary(iran.rates$background)
## End(Not run)