# eptren: Maximum Likelihood Estimates of Intensity Rates In SAPP: Statistical Analysis of Point Processes

## Description

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

## Usage

 ```1 2``` ```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

 ``` 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) ```

### 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.