# rates: Declustering Probabilities, Background Seismicity Rate and... In ETAS: Modeling Earthquake Data Using 'ETAS' Model

## Description

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

## Usage

 ```1 2 3``` ``` 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 - 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-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` is the number of subdivisions in the y-direction (latitude) and the second component `dimyx` 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_{start} + T

## Author(s)

Abdollah Jalilian jalilian@razi.ac.ir

## 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.

`etas`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30``` ``` # 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) ```