eptren: Maximum Likelihood Estimates of Intensity Rates

Description Usage Arguments Details Value References Examples

View source: R/SAPP.R

Description

Compute the maximum likelihood estimates of intensity rates of either exponential polynomial or exponential Fourier series of non-stationary Poisson process models.

Usage

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eptren(data, mag = NULL, threshold = 0.0, nparam, nsub, cycle = 0,
       tmpfile = NULL, nlmax = 1000, plot = TRUE)

Arguments

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), where t is the length of observed time interval of points.

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 NULL (default) no report.

nlmax

the maximum number of steps in the process of minimizing.

plot

logical. If TRUE (default) intensity rates are plotted.

Details

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).

Value

aic

AIC.

param

parameters.

aicmin

minimum AIC.

maice.order

number of parameters of minimum AIC.

time

time ( cycle = 0 ) or superposed occurrence time ( cycle > 0 ).

intensity

intensity rates.

References

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.

Examples

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## 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)   

Example output

 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

SAPP documentation built on July 2, 2020, 2:59 a.m.