# mloglik1d: Minus loglikelihood of an IHSEP model In IHSEP: Inhomogeneous Self-Exciting Process

 mloglik1d R Documentation

## Minus loglikelihood of an IHSEP model

### Description

Calculates the minus loglikelihood of an IHSEP model with given baseline intensity function ν and excitation function g(x)=∑ a_i exp(-b_i x) for event times `jtms` on interval `[0,TT]`.

### Usage

```mloglik1d(jtms, TT, nu, gcoef, Inu)
```

### Arguments

 `jtms` A numeric vector, with values sorted in ascending order. Jump times to fit the inhomogeneous self-exciting point process model on. `TT` A scalar. The censoring time, or the terminal time for observation. Should be (slightly) greater than the maximum of `jtms`. `nu` A (vectorized) function. The baseline intensity function. `gcoef` A numeric vector (of 2k elements), giving the parameters `(a_1,...,a_k,b_1,...,b_k)` of the exponential excitation function g(x)=∑_{i=1}^k a_i*exp(-b_i*x). `Inu` A (vectorized) function. Its value at `t` gives the integral of the baseline intensity function ν from 0 to `t`.

### Details

This function calculates the minus loglikelihood of the inhomegeneous Hawkes model with background intensity function ν(t) and excitation kernel function g(t)=∑_{i=1}^{k} a_i e^{-b_i t} relative to continuous observation of the process from time 0 to time `TT`. Like `mloglik1c`, it takes advantage of the Markovian property of the intensity process and uses external C++ code to speed up the computation.

### Value

The value of the negative log-liklihood.

### Author(s)

Feng Chen <feng.chen@unsw.edu.au>

`mloglik1c`

### Examples

```## simulated data of an IHSEP on [0,1] with baseline intensity function
## nu(t)=200*(2+cos(2*pi*t)) and excitation function
## g(t)=8*exp(-16*t)
data(asep)

## get the birth times of all generations and sort in ascending order
tms <- sort(unlist(asep))
## calculate the minus loglikelihood of an SEPP with the true parameters
mloglik1d(tms,TT=1,nu=function(x)200*(2+cos(2*pi*x)),
gcoef=8*1:2,
Inu=function(y)integrate(function(x)200*(2+cos(2*pi*x)),0,y)\$value)
## calculate the MLE for the parameter assuming known parametric forms
## of the baseline intensity and excitation functions
## Not run:
system.time(est <- optim(c(400,200,2*pi,8,16),
function(p){
mloglik1d(jtms=tms,TT=1,
nu=function(x)p[1]+p[2]*cos(p[3]*x),
gcoef=p[-(1:3)],
Inu=function(y){
integrate(function(x)p[1]+p[2]*cos(p[3]*x),
0,y)\$value
})
},hessian=TRUE,control=list(maxit=5000,trace=TRUE),
method="BFGS")
)
## point estimate by MLE
est\$par
## standard error estimates:
diag(solve(est\$hessian))^0.5

## End(Not run)
```

IHSEP documentation built on Sept. 17, 2022, 1:05 a.m.