simulateETAS: Simulates synthetic data from the ETAS model
In bayesianETAS: Bayesian Estimation of the ETAS Model for Earthquake Occurrences

Description Usage Arguments Value Author(s) References Examples

View source: R/simulateETAS.R

This function simulates sample data from the ETAS model over a particular interval [0,T]. The Epidemic Type Aftershock Sequence (ETAS) model is widely used to quantify the degree of seismic activity in a geographical region, and to forecast the occurrence of future mainshocks and aftershocks (Ross 2016). The temporal ETAS model is a point process where the probability of an earthquake occurring at time t depends on the previous seismicity Ht, and is defined by the conditional intensity function:

λ(t|Ht) = μ + ∑ κ(m[i]|K,α) h(t[i]|c,p)

where

κ(m[i]|K,α) = K * exp(α(m[i]-M0))

and

h(t[i]|c,p) = (p-1) * c^(p-1) * (t-t[i]+c)^(-p)

where the summation is over all previous earthquakes that occurred in the region, with the i'th such earthquake occurring at time t[i] and having magnitude m[i]. The quantity M0 denotes the magnitude of completeness of the catalog, so that m[i] ≥ M0 for all i. The temporal ETAS model has 5 parameters: μ controls the background rate of seismicity, K and α determine the productivity (average number of aftershocks) of an earthquake with magnitude m, and c and p are the parameters of the Modified Omori Law (which has here been normalized to integrate to 1) and represent the speed at which the aftershock rate decays over time. Each earthquake is assumed to have a magnitude which is an independent draw from the Gutenberg-Richter law p(m) = β * exp(β(m-M0).

This function simulates sample data from the ETAS model over a particular interval [0,T].

1	simulateETAS(mu, K, alpha, c, p, beta, M0, T, displayOutput = TRUE)

`mu`	Parameter of the ETAS model as described above.
`K`	Parameter of the ETAS model as described above.
`alpha`	Parameter of the ETAS model as described above.
`c`	Parameter of the ETAS model as described above.
`p`	Parameter of the ETAS model as described above.
`beta`	Parameter of the Gutenberg-Richter law used to generate earthquake magnitudes.
`M0`	Magnitude of completeness.
`T`	Length of the time window [0,T] to simulate the catalog over.
`displayOutput`	If TRUE then prints the number of earthquakes simulated so far.

A list consisting of

`ts`	The simulated earthquake times
`magnitudes`	The simulated earthquake magnitudes
`branching`	The simulated branching structure, where branching[i] is the index of the earthquake that triggered earthquake i, or 0 if earthquake i is a background event

Gordon J Ross

Gordon J. Ross - Bayesian Estimation of the ETAS Model for Earthquake Occurrences (2016), available from http://www.gordonjross.co.uk/bayesianetas.pdf

## Not run: 
beta <- 2.4; M0 <- 3
simulateETAS(0.2, 0.2, 1.5, 0.5, 2, beta, M0, T=500, displayOutput=FALSE)

## End(Not run)

$ts
  [1]   3.981303  11.224080  11.675037  28.904273  28.917568  35.187015
  [7]  35.286515  39.952947  40.399966  49.346380  62.918886  68.615387
 [13]  72.573613  75.785109  76.114598  76.408189  76.586568  76.632692
 [19]  76.633796  76.638648  76.709471  76.709733  76.718591  76.874764
 [25]  77.188219  77.249874  77.471797  77.637678  77.886889  78.045568
 [31]  78.076556  78.434520  78.686615  86.715249  90.095710  90.427911
 [37]  90.964471  91.520982  92.098548  96.012734  96.272510  96.590992
 [43]  97.622599 115.366191 115.736537 126.987041 127.166678 130.196480
 [49] 130.744846 131.152401 131.572148 131.594464 132.345522 136.565540
 [55] 138.648479 144.635218 152.930439 153.585790 153.764234 163.108244
 [61] 165.483231 177.072832 180.675613 180.758657 180.803049 180.895075
 [67] 182.664951 188.495101 197.624297 198.592390 209.213220 215.702088
 [73] 218.031418 218.251144 221.885764 223.013282 223.540417 223.575111
 [79] 223.584892 225.263944 225.471317 227.263515 228.051066 228.838638
 [85] 238.115411 239.994557 247.862047 251.455105 252.972228 253.356092
 [91] 262.045763 265.919857 273.403722 274.328668 274.413843 274.587991
 [97] 284.243518 284.382496 285.083803 288.630992 288.880520 289.011315
[103] 289.080715 290.847325 291.802191 294.880243 297.747355 298.951903
[109] 299.368754 302.314798 307.787154 308.194013 308.671945 308.805000
[115] 313.758176 314.311404 315.679332 330.302213 330.305845 331.058093
[121] 331.145052 333.250107 334.201168 334.655222 335.605228 336.118200
[127] 339.576006 340.101627 345.581059 347.093143 355.264601 365.412898
[133] 365.545247 368.029485 373.847905 375.457387 386.877551 387.628010
[139] 393.545095 393.704882 393.796211 397.301128 406.456804 407.964852
[145] 414.878565 420.265028 421.086047 424.146077 424.586278 429.683189
[151] 430.319680 430.830872 435.120156 435.440441 435.713161 436.977991
[157] 437.287701 440.107159 448.154688 448.605610 452.332682 456.394453
[163] 456.454299 460.407261 460.984937 461.530613 462.212987 470.035625
[169] 474.047946 474.244754 494.627975 496.509785 496.586978

$magnitudes
  [1] 3.271330 3.775691 3.483032 3.411822 3.553616 3.107439 3.513167 3.526080
  [9] 3.106770 3.031639 3.267758 3.153741 4.157052 3.272281 3.515521 3.149987
 [17] 3.014855 4.269408 5.288889 3.080203 3.329425 3.000794 3.318630 4.727975
 [25] 3.183519 3.213207 3.133004 3.879214 3.063698 3.008036 3.183639 4.368874
 [33] 3.368419 3.731542 3.515109 3.273371 3.477124 3.094183 3.274306 3.274447
 [41] 3.200721 3.171944 3.081781 3.251054 3.992415 3.246458 3.189450 3.407194
 [49] 3.148268 3.353954 3.383681 3.088985 3.058546 3.390520 3.232699 3.285592
 [57] 3.695502 3.239489 3.367060 3.001803 3.194566 3.150967 3.664976 3.495170
 [65] 3.152175 3.040217 3.906664 3.444618 3.085183 3.314796 3.418288 4.498433
 [73] 3.702639 3.835678 3.134373 3.136512 3.184194 3.994968 3.254063 3.287782
 [81] 3.192002 3.055149 3.048849 3.229577 3.207625 3.033149 3.534319 3.436436
 [89] 3.062425 3.277749 4.164727 3.662748 3.405469 3.693688 3.263176 3.474630
 [97] 3.982337 3.128124 3.573568 3.105369 3.606506 3.376607 3.028696 3.040740
[105] 4.263993 3.939075 3.247976 3.125932 3.262178 3.066905 3.679944 3.077092
[113] 3.049990 3.437656 3.489904 3.236996 3.236955 4.519126 3.038996 3.282670
[121] 3.048136 4.673171 3.282573 3.288135 3.024455 3.756295 3.226708 3.311504
[129] 3.105385 3.263679 3.087584 3.088813 3.383868 3.417251 3.017370 3.088537
[137] 3.461142 3.174261 3.802214 3.130100 3.282180 3.426272 3.802198 3.279816
[145] 3.267836 4.160067 3.208862 3.576921 3.016329 3.074337 3.114391 3.560579
[153] 4.403684 3.644662 3.602848 3.353973 3.061409 3.257035 3.416996 3.836086
[161] 3.256299 3.848662 3.313628 3.066250 3.172714 3.568150 3.795174 3.444665
[169] 3.002827 3.607187 3.231940 3.246752 3.136017

$branching
  [1]   0   0   0   0   0   0   6   0   8   0   0   0   0  13  14  13  16  15
 [19]  18  19  20  19  19  19  14  19  24  24  18  19  30  23  19  32  34  35
 [37]  36   0  34   0   0  41  19   0  44   0  46   0  48  48  50  51  50  44
 [55]   0   0   0  57  58   0   0   0   0   0  64  63   0   0   0   0   0   0
 [73]  72   0  72   0   0   0  78   0  78   0   0  82   0   0   0   0   0  89
 [91]   0   0   0   0  94  94   0  97  97   0   0   0 102 101   0   0   0   0
[109] 107   0   0 111 112 113   0   0   0   0 118 118 120 118   0 122 123   0
[127] 122 127 126   0   0   0   0   0   0   0   0 137   0 139 140   0   0   0
[145]   0   0   0   0 148   0   0 151   0 153 153   0 156   0   0 159   0 160
[163] 162   0   0   0   0   0   0   0   0   0 172