Overview of the program [eptren]

This program computes the maximum likelihood estimates (MLEs) of the coefficients A1, A2, c, An in the intensity rate with an exponential polynomial

f(t) = exp{A_1 + A_2 t + A_3 t^2 + c }                                           (1)

or A_1, A_2, B_2, c , A_n, B_n in an exponential Fourier series

f(t) = exp{A_1+A_2cos(2t/P)+B_2sin(2t/P)+A_3cos(4t/P)+B_3sin(4t/P)+c }   (2) 
  
which represents the time varying rate of occurrence (intensity function) of earthquakes in a region. A non-stationary Poisson process is assumed for the model of earthquake occurrences. The optimal order n can be determined by minimize the Akaike Information Criterion (AIC) values (cf., Akaike, 1974). 

The maximum number of coefficients is limited to 21 due to the dimension set in the FORTRAN source. The coefficients A_1, A_2, c, A_{21} can be determined for equation (1) and A_1, A_2, B_2, c , A_{10}, B_{10} can be determined for equation (2) at the most.

Since a non-stationary Poisson process is assumed in the data, we need a careful interpretation of the results when the model is applied to data containing earthquake clusters such as aftershocks, foreshocks, and earthquake swarms.

R-modules are also attached in order to display the outputs of numerical data by plotting graphs of the estimated functions.

Structure of the program is the following.

          [eptren]
             |-----[inputs]
             |-----[reduc1]
             |-----[reduc2]
             |-----[davidn]--------------[funct]
             |         |-----[hesian]
             |         |-----[linear]----[funct]
             |
             |-----[fincal]
             |-----[output]---[printr]---[trenfn]
                                   |-----[cyclfn]

1. The data file for this program is always [work.data]. Any other data to be analyzed should have the same format and have to be named as [work.data]. Two test data files [Brastings.data] and [Poisson.data] are attached for this program in the directory of [DataExamples]. When dataset includes remarkable clusters, the interpretation of the trend and cycles has to be careful. 

2. Control file  [eptren.init] is for the selection of either trend or cycle fitting, and the values of the necessary variables, which are also specified and explained.

3. Output files of [eptern] are [out.eptren1] and [out.eptren2]. To make figure R-module [r. eptrend] or [r. epcycle] is used to make R-display and also produces postscript output [eptrend.ps] and [epcycle.ps].

4. Calculated record of each program is stored by the name [*.print] in the directory of [Calculations] for your initial check of the program.

This program was originally designed (January 1985) and revised (December 2005) by Yosihiko Ogata, and programmed and also reprogrammed by Koichi Katsura, Institute of Statistical Mathematics, Tokyo, Japan. The R-module was designed and programmed by Jiancang Zhuang and Ogata (December 2005).

References

Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. Automat. Control, AC-19, 716-723.

Lewis, P.A.W. and G.S. Shedler (1976). Statistical analysis of non-stationary series in a database system, IBM. J. Res. Develop., 20, 465-481.

Maclean, C.J. (1974). Estimation and testing of an exponential polynomial rate function within the non-stationary Poisson process, Biometrika, 61, 81-86.   
           

