Description Usage Arguments Details Value References Examples
Compute the maximum likelihood estimates of intensity rates of either exponential polynomial or exponential Fourier series of non-stationary Poisson process models.
1 2 |
data |
point process data. |
mag |
magnitude. |
threshold |
threshold magnitude. |
nparam |
maximum number of parameters. |
nsub |
number of subdivisions in either (0,t) or (0,
|
cycle |
periodicity to be investigated days in a Poisson process model. If zero (default) fit an exponential polynomial model. |
tmpfile |
a character string naming the file to write the process of minimizing by
Davidon-Fletcher-Powell procedure. If "" print the process to the standard
output and if |
nlmax |
the maximum number of steps in the process of minimizing. |
plot |
logical. If |
This function computes the maximum likelihood estimates (MLEs) of the coefficients A_1, A_2,..., A_n is an exponential polynomial
f(t) = exp(A_1 + A_2t + A_3t^2 + ... )
or A_1, A_2, B_2, ..., A_n, B_n in a Poisson process model with an intensity taking the form of an exponential Fourier series
f(t) = exp{ A_1 + A_2cos(2π t/p) + B_2sin(2π t/p) + A_3cos(4π t/p) + B_3sin(4π t/p) +... }
which represents the time varying rate of occurrence (intensity function) of earthquakes in a region.
These two models belong to the family of non-stationary Poisson process. The optimal order n can be determined by minimize the value of the Akaike Information Criterion (AIC).
aic |
AIC. |
param |
parameters. |
aicmin |
minimum AIC. |
maice.order |
number of parameters of minimum AIC. |
time |
time ( |
intensity |
intensity rates. |
Ogata, Y., Katsura, K. and Zhuang, J. (2006) Computer Science Monographs, No.32, TIMSAC84: STATISTICAL ANALYSIS OF SERIES OF EVENTS (TIMSAC84-SASE) VERSION 2. The Institute of Statistical Mathematics.
Ogata, Y. (2006) Computer Science Monographs, No.33, Statistical Analysis of Seismicity - updated version (SASeies2006). The Institute of Statistical Mathematics.
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 | ## The Occurrence Times Data of 627 Blastings
data(Brastings)
# exponential polynomial trend fitting
eptren(Brastings, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(Brastings, nparam = 10, nsub = 1000, cycle = 1)
## Poisson Process data
data(PoissonData)
# exponential polynomial trend fitting
eptren(PoissonData, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(PoissonData, nparam = 10, nsub = 1000, cycle = 1)
## The aftershock data of 26th July 2003 earthquake of M6.2
data(main2003JUL26)
x <- main2003JUL26
# exponential polynomial trend fitting
eptren(x$time, mag = x$magnitude, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(x$time, mag = x$magnitude, nparam = 10, nsub = 1000, cycle = 1)
|
AIC -6820.946962
parameters
-1.987778e+00
AIC -6839.492351
parameters
-2.319355e+00 1.376921e-04
AIC -6839.237636
parameters
-2.457719e+00 3.024394e-04 -3.450138e-08
AIC -6842.441583
parameters
-2.177895e+00 -3.857168e-04 3.272058e-07 -5.149384e-11
AIC -6840.460315
parameters
-2.196091e+00 -3.075866e-04 2.525058e-07 -2.661788e-11 -2.679979e-15
AIC -6855.076407
parameters
-2.865567e+00 3.738334e-03 -5.699733e-06 3.347044e-09 -8.152297e-13
6.992849e-17
AIC -6872.954251
parameters
-3.929925e+00 1.152535e-02 -2.142796e-05 1.646287e-08 -6.015396e-12
1.045802e-15 -6.975925e-20
AIC -6874.583056
parameters
-3.380743e+00 6.446511e-03 -7.864977e-06 8.727766e-10 3.018237e-12
-1.722788e-15 3.585472e-19 -2.633018e-23
AIC -6882.490445
parameters
-4.453306e+00 1.879626e-02 -4.993590e-05 6.431146e-08 -4.696754e-11
2.037147e-14 -5.157888e-18 6.999883e-22 -3.918800e-26
AIC -6894.660215
parameters
-6.515836e+00 4.394377e-02 -1.497668e-04 2.480718e-07 -2.304148e-10
1.276257e-13 -4.292136e-17 8.575372e-21 -9.347353e-25 4.277032e-29
minimum AIC -6894.660215
parameters
-6.515836e+00 4.394377e-02 -1.497668e-04 2.480718e-07 -2.304148e-10
1.276257e-13 -4.292136e-17 8.575372e-21 -9.347353e-25 4.277032e-29
AIC 3748.673468
parameters
-1.987778e+00
AIC 2935.917487
parameters
-2.993938e+00 -2.228208e+00 -3.394883e-01
AIC 2795.133600
parameters
-4.591351e+00 -4.349900e+00 -2.211597e+00 -5.115197e-01 -1.646180e+00
AIC 2715.898446
parameters
-3.618100e+00 -2.795589e+00 -7.469769e-01 -9.238863e-02 2.420047e-01
-2.231542e-01 1.119179e+00
AIC 2684.245084
parameters
-3.640266e+00 -2.731892e+00 -9.987121e-01 2.817838e-01 -4.537336e-01
6.111146e-01 5.284477e-01 7.046523e-01 -2.001942e-01
AIC 2678.585427
parameters
-3.746211e+00 -2.760547e+00 -1.283700e+00 4.602742e-01 -5.727876e-01
7.172496e-01 8.983974e-01 4.643990e-01 2.993803e-01 -3.063479e-01
2.610341e-01
AIC 2681.089369
parameters
-3.660020e+00 -2.682327e+00 -1.078178e+00 3.539086e-01 -3.772595e-01
5.807152e-01 8.938055e-01 4.744687e-01 9.816165e-02 -1.346558e-01
8.592809e-02 1.518247e-01 -5.900747e-02
AIC 2679.828724
parameters
-3.727481e+00 -2.854868e+00 -1.092954e+00 1.508242e-01 -4.475855e-01
4.660264e-01 7.296640e-01 5.829483e-01 2.612204e-02 1.766784e-01
2.814307e-01 3.526503e-01 3.465631e-01 2.851646e-02 3.023127e-01
AIC 2683.773579
parameters
-3.739963e+00 -2.870617e+00 -1.121229e+00 1.535853e-01 -4.885361e-01
4.838971e-01 6.927180e-01 6.100064e-01 8.817888e-03 1.914558e-01
2.951647e-01 3.375452e-01 3.858982e-01 -1.152068e-02 3.313167e-01
-3.112560e-02 7.849195e-03
AIC 2686.382934
parameters
-3.870645e+00 -3.042992e+00 -1.332043e+00 1.644134e-01 -7.736482e-01
6.411467e-01 4.630444e-01 8.243812e-01 -1.081159e-01 3.829070e-01
3.273770e-01 3.726283e-01 5.464211e-01 -1.902719e-01 5.277316e-01
-3.001721e-01 7.657180e-02 -1.681358e-01 -2.422645e-02
minimum AIC 2678.585427
parameters
-3.746211e+00 -2.760547e+00 -1.283700e+00 4.602742e-01 -5.727876e-01
7.172496e-01 8.983974e-01 4.643990e-01 2.993803e-01 -3.063479e-01
2.610341e-01
AIC -34948.694674
parameters
-6.719430e-01
AIC -34947.288978
parameters
-6.984868e-01 1.057334e-05
AIC -34946.023292
parameters
-7.371036e-01 5.642344e-05 -9.139568e-09
AIC -34944.023680
parameters
-7.381641e-01 5.893394e-05 -1.038873e-08 1.662081e-13
AIC -34942.263000
parameters
-7.682036e-01 1.772408e-04 -1.162210e-07 3.299465e-11 -3.277889e-15
AIC -34940.312156
parameters
-7.530933e-01 8.801552e-05 7.857379e-09 -3.295030e-11 1.153115e-14
-1.183480e-18
AIC -34938.397580
parameters
-7.747926e-01 2.675485e-04 -3.487105e-07 2.512146e-10 -9.479155e-14
1.750437e-17 -1.244916e-21
AIC -34937.166599
parameters
-8.458696e-01 1.044638e-03 -2.427516e-06 2.549222e-09 -1.354722e-12
3.796251e-16 -5.348098e-20 2.982297e-24
AIC -34938.443948
parameters
-1.008756e+00 3.280636e-03 -1.013818e-05 1.377131e-08 -9.731835e-12
3.853791e-15 -8.624252e-19 1.018940e-22 -4.940700e-27
AIC -34936.634772
parameters
-1.051899e+00 4.005242e-03 -1.326126e-05 1.954106e-08 -1.532236e-11
6.971689e-15 -1.898807e-18 3.045398e-22 -2.644183e-26 9.546751e-31
minimum AIC -34948.694674
parameters
-6.719430e-01
AIC 8538.940185
parameters
-6.719429e-01
AIC 8542.937999
parameters
-6.719433e-01 7.087410e-05 1.306741e-03
AIC 8546.632569
parameters
-6.720033e-01 7.429736e-05 1.297407e-03 -1.468064e-02 -4.874285e-03
AIC 8548.118363
parameters
-6.724951e-01 -1.781599e-04 1.064885e-03 -1.466143e-02 -4.890223e-03
-2.144415e-02 -3.886646e-02
AIC 8551.499430
parameters
-6.726158e-01 2.681432e-04 8.632811e-04 -1.450098e-02 -4.832583e-03
-2.145191e-02 -3.886004e-02 1.773956e-02 1.305705e-02
AIC 8553.599192
parameters
-6.729949e-01 5.600329e-04 1.159915e-03 -1.529546e-02 -5.141845e-03
-2.159176e-02 -3.913959e-02 1.776369e-02 1.306389e-02 -5.192381e-03
-3.825034e-02
AIC 8556.872293
parameters
-6.731421e-01 1.813550e-04 8.917706e-04 -1.526638e-02 -4.886691e-03
-2.188613e-02 -3.958847e-02 1.779290e-02 1.287170e-02 -5.183334e-03
-3.824396e-02 1.095387e-02 -2.121617e-02
AIC 8560.549810
parameters
-6.732044e-01 -1.260564e-05 8.367736e-04 -1.545774e-02 -5.135068e-03
-2.191323e-02 -3.941933e-02 1.768909e-02 1.252489e-02 -5.126004e-03
-3.836174e-02 1.095900e-02 -2.121185e-02 1.110390e-02 -1.138614e-02
AIC 8560.375110
parameters
-6.739932e-01 4.348250e-04 9.479805e-04 -1.485469e-02 -4.789173e-03
-2.144968e-02 -3.843227e-02 1.793473e-02 1.197191e-02 -5.045277e-03
-3.710031e-02 1.067451e-02 -2.086400e-02 1.109814e-02 -1.141520e-02
-4.803230e-02 3.113475e-02
AIC 8562.565622
parameters
-6.743398e-01 -4.114549e-04 3.159351e-04 -1.468348e-02 -4.570659e-03
-2.126116e-02 -3.802860e-02 1.778069e-02 1.266165e-02 -4.696554e-03
-3.732511e-02 1.022304e-02 -2.018761e-02 1.081018e-02 -1.133624e-02
-4.804033e-02 3.112800e-02 -3.759249e-02 -2.759948e-03
minimum AIC 8538.940185
parameters
-6.719429e-01
AIC -31082.473755
parameters
4.815524e+00
AIC -31721.801816
parameters
5.622249e+00 -1.021647e-01
AIC -31735.389914
parameters
5.752456e+00 -1.569067e-01 3.338481e-03
AIC -31746.893407
parameters
5.886683e+00 -2.676148e-01 2.024223e-02 -6.557509e-04
AIC -31856.510480
parameters
6.274930e+00 -8.183626e-01 1.716911e-01 -1.439579e-02 3.903766e-04
AIC -31905.374605
parameters
6.530752e+00 -1.398155e+00 4.213518e-01 -5.333309e-02 2.888323e-03
-5.607666e-05
AIC -31924.658721
parameters
6.693869e+00 -1.934425e+00 7.540239e-01 -1.314154e-01 1.126285e-02
-4.711299e-04 7.709399e-06
AIC -31926.004508
parameters
6.759561e+00 -2.224341e+00 9.986040e-01 -2.113321e-01 2.384907e-02
-1.493119e-03 4.889665e-05 -6.516814e-07
AIC -31931.718100
parameters
6.860822e+00 -2.798592e+00 1.631037e+00 -4.849194e-01 8.269357e-02
-8.403248e-03 4.993332e-04 -1.594508e-05 2.108210e-07
AIC -31938.886232
parameters
6.971855e+00 -3.592573e+00 2.737260e+00 -1.095986e+00 2.538601e-01
-3.549544e-02 3.025126e-03 -1.533196e-04 4.240019e-06 -4.921460e-08
minimum AIC -31938.886232
parameters
6.971855e+00 -3.592573e+00 2.737260e+00 -1.095986e+00 2.538601e-01
-3.549544e-02 3.025126e-03 -1.533196e-04 4.240019e-06 -4.921460e-08
AIC -17587.777268
parameters
4.815570e+00
AIC -17607.560231
parameters
4.811750e+00 1.421437e-01 -2.147652e-02
AIC -17611.230549
parameters
4.809553e+00 1.391681e-01 -2.552836e-02 4.842838e-02 6.578027e-02
AIC -17608.656410
parameters
4.809328e+00 1.376618e-01 -2.635339e-02 4.812782e-02 6.366655e-02
3.565598e-03 3.511790e-02
AIC -17605.871086
parameters
4.809067e+00 1.371395e-01 -2.613830e-02 4.663273e-02 6.349841e-02
2.711055e-03 3.326766e-02 1.394328e-02 2.942081e-02
AIC -17602.227518
parameters
4.808953e+00 1.368650e-01 -2.608699e-02 4.639229e-02 6.358158e-02
1.872969e-03 3.315150e-02 1.350501e-02 2.845042e-02 7.287139e-03
1.606164e-02
AIC -17599.832226
parameters
4.808611e+00 1.365087e-01 -2.602693e-02 4.579720e-02 6.382870e-02
1.465420e-03 3.354824e-02 1.181306e-02 2.872050e-02 5.719056e-03
1.435043e-02 2.530204e-02 2.760591e-02
AIC -17605.521787
parameters
4.806557e+00 1.349376e-01 -2.632390e-02 4.497708e-02 6.422307e-02
1.100673e-04 3.405327e-02 1.064849e-02 2.956136e-02 1.481374e-03
1.473401e-02 2.209395e-02 2.307004e-02 5.445846e-02 7.421021e-02
AIC -17610.252942
parameters
4.804728e+00 1.313130e-01 -2.507258e-02 4.365301e-02 6.437303e-02
-6.680954e-04 3.457566e-02 9.491727e-03 3.052205e-02 7.546003e-04
1.580201e-02 1.825956e-02 2.436158e-02 5.055810e-02 7.093846e-02
6.668741e-02 5.664139e-02
AIC -17607.671319
parameters
4.804524e+00 1.299362e-01 -2.544712e-02 4.210557e-02 6.443794e-02
-1.198090e-03 3.456019e-02 9.196009e-03 3.066868e-02 2.230035e-04
1.605830e-02 1.788318e-02 2.475936e-02 4.899468e-02 7.104570e-02
6.552747e-02 5.494682e-02 1.985910e-02 2.915904e-02
minimum AIC -17611.230549
parameters
4.809553e+00 1.391681e-01 -2.552836e-02 4.842838e-02 6.578027e-02
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