linlin: Maximum Likelihood Estimates of Linear Intensity Models

In SAPP: Statistical Analysis of Point Processes

Maximum Likelihood Estimates of Linear Intensity Models

Description

Perform the maximum likelihood estimates of linear intensity models of self-exciting point process with another point process input, cyclic and trend components.

Usage

linlin(external, self.excit, interval, c, d, ax = NULL, ay = NULL, ac = NULL,
       at = NULL, opt = 0, tmpfile = NULL, nlmax = 1000)

Arguments

external	another point process data.
self.excit	self-exciting data.
interval	length of observed time interval of event.
c	exponential coefficient of lgp in self-exciting part.
d	exponential coefficient of lgp in input part.
ax	coefficients of self-exciting response function.
ay	coefficients of input response function.
ac	coefficients of cycle.
at	coefficients of trend.
opt	0 : minimize the likelihood with fixed exponential coefficient c 1 : not fixed d.
tmpfile	a character string naming the file to write the process of minimizing. 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.

Details

The cyclic part is given by the Fourier series, the trend is given by usual polynomial. The response functions of the self-exciting and the input are given by the Laguerre type polynomials (lgp), where the scaling parameters in the exponential function, say c and d, can be different. However, it is advised to estimate c first without the input component, and then to estimate d with the fixed c (this means that the gradient corresponding to the c is set to keep 0), which are good initial estimates for the c and d of the mixed self-exciting and input model.

Note that estimated intensity sometimes happen to be negative on some part of time interval outside the neighborhood of events. this take place more easily the larger the number of parameters. This causes some difficulty in getting the m.l.e., because the negativity of the intensity contributes to the seeming increase of the likelihood.

Note that for the initial estimates of ax(1), ay(1) and at(1), some positive value are necessary. Especially 0.0 is not suitable.

Value

c1	initial estimate of exponential coefficient of lgp in self-exciting part.
d1	initial estimate of exponential coefficient of lgp in input part.
ax1	initial estimates of lgp coefficients in self-exciting part.
ay1	initial estimates of lgp coefficients in the input part.
ac1	initial estimates of coefficients of Fourier series.
at1	initial estimates of coefficients of the polynomial trend.
c2	final estimate of exponential coefficient of lgp in self-exciting part.
d2	final estimate of exponential coefficient of lgp in input part.
ax2	final estimates of lgp coefficients in self-exciting part.
ay2	final estimates of lgp coefficients in the input part.
ac2	final estimates of coefficients of Fourier series.
at2	final estimates of coefficients of the polynomial trend.
aic2	AIC/2.
ngmle	negative max likelihood.
rayleigh.prob	Rayleigh probability.
distance	= \sqrt(rwx^2+rwy^2).
phase	phase.

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. and Akaike, H. (1982) On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes. J. royal statist. soc. b, vol. 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., vol. 34. pp. 373-387.

Examples

data(PProcess)  # point process data 
data(SelfExcit) # self-exciting point process data
linlin(PProcess[1:69], SelfExcit, interval = 20000, c = 0.13, d = 0.026,
       ax = c(0.035, -0.0048), ay = c(0.0, 0.00017), at = c(0.007, -.00000029))

SAPP documentation built on June 7, 2023, 5:45 p.m.