Overview of the program [linlin]

This program performs the maximum likelihood estimates of linear intensity models of self-exciting point process with another point process input, cyclic and trend components. The cyclic part is given by the Fourier series, the trend is given by usual polynomials. The response functions to the self-exciting and the external inputs are given by the Laguerre-type polynomials, where the scaling parameters in the exponential function, say c and d, can be different. However it is advised to fit a model without the input component firstly to estimate an initial c, and then to fit a model with both the self-exciting and the external input component to estimate an initial d with c being fixed (this means that the gradient corresponding to c is fixed at 0), which are good initial estimates for the c and d of the mixed self-exciting and input model. It is recommended to use Akaike Information Criterion (AIC; Akaike, 1974) for selecting optimal orders of respective components. Finally, it should be noted that estimated intensity sometimes happens to be negative on some parts of the time interval outside the neighbourhood of events. This take place more easily the larger the number of parameters. This causes some difficulty in getting the MLE, because the negativity of the intensity contributes to the seeming increase of the likelihood. See the references below.

     Structure of the program

          [linlin]
             |---[input]
             |---[comfac]
             |---[cycle]
             |---[dav]-------[davidn]------------[funct1 or 2]
                                  |---[hesian]
                                  |---[linear]---[funct1 or 2]

Remarks:

1. The input data file for this program is always named [linlin.data]. Before analyzing any other data, record them in the same format and name the file [linlin.data]. 

2. The input control file is [linlin.init], in which you can select a model with a possible combination of the trend, the self-exciting or/and the external response components, and in which you can specify necessary variables and initial parameter values.

3. Calculation results of [linlin] are stored in the file [linlin.print]. In the directory of [Calculations], there is an example file [linlin.print] 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.

References

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

Ogata. Y. and Akaike, H. (1982). On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes, J. Roy. Statist. Soc. Ser. B, 44, pp. 102-107.

Ogata, Y., Akaike, H. and Katsura, K. (1982). The application of linear intensity models to the investigation of causal relations between a point process and another stochastic process, Ann. Inst. Statist. Math., 34. pp. 373-387.

Ogata, Y. and Katsura, K. (1986). Point process model with linearly parametrized intensity for the application to earthquake data, Essays in Time Series and Allied Processes (Festscrift for Prof.     E.J. Hannan), J. Gani and M.B. Priestley eds., J. Appl. Probab. 23A, pp.291-310.



