# linlin {SAPP}

Maximum Likelihood Estimates of Linear Intensity Models
Package:
SAPP
Version:
1.0.3

### 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
write the process of minimizing to tmpfile.
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.

### Values

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

Y.Ogata, K.Katsura and J.Zhuang (2006) Computer Science Monographs, No.32, TIMSAC84: STATISTICAL ANALYSIS OF SERIES OF EVENTS (TIMSAC84-SASE) VERSION 2. The Institute of Statistical Mathematics.

Y.Ogata and H.Akaike (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.

Y.Ogata, H.Akaike and K.Katsura (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, 20000, 0.13, 0.026,
c(0.035,-0.0048), c(0.0,0.00017),, c(0.007,-.00000029) )```

Documentation reproduced from package SAPP, version 1.0.3. License: GPL (>= 2)