# hfit: Perform Maximum Likelihood Estimation In ksublee/emhawkes: Exponential Multivariate Hawkes Model

 hfit R Documentation

## Perform Maximum Likelihood Estimation

### Description

Generic function hfit. A method for estimating the parameters of the exponential Hawkes model. The reason for being constructed as the S4 method is as follows. First, to represent the structure of the model as an hspec object. There are numerous variations on the mutlivariate marked Hawkes model. Second, to convey the starting point of numerical optimization. The parameter values assigned to the hspec slots become initial values. This function uses `maxLik` for the optimizer.

### Usage

```hfit(
object,
inter_arrival = NULL,
type = NULL,
mark = NULL,
N = NULL,
Nc = NULL,
lambda0 = NULL,
N0 = NULL,
mylogLik = NULL,
reduced = TRUE,
hess = NULL,
constraint = NULL,
method = "BFGS",
verbose = FALSE,
...
)

## S4 method for signature 'hspec'
hfit(
object,
inter_arrival = NULL,
type = NULL,
mark = NULL,
N = NULL,
Nc = NULL,
lambda0 = NULL,
N0 = NULL,
mylogLik = NULL,
reduced = TRUE,
hess = NULL,
constraint = NULL,
method = "BFGS",
verbose = FALSE,
...
)
```

### Arguments

 `object` `hspec-class`. This object includes the parameter values `inter_arrival` inter-arrival times of events. Includes inter-arrival for events that occur in all dimensions. Start with zero. `type` a vector of dimensions. Distinguished by numbers, 1, 2, 3, and so on. Start with zero. `mark` a vector of mark (jump) sizes. Start with zero. `N` a matrix of counting processes `Nc` a matrix of cumulated counting processes `lambda0` the inital values of lambda component. Must have the same dimensional matrix (n by n) with hspec. `N0` the initial values of N. `mylogLik` user defined log likelihood function. mylogLik function should have 'object' argument, cosistent with `hspec`. `reduced` When TRUE, reduced estimation performed. `grad` gradient matrix for the likelihood function. For more information, see `maxLik`. `hess` Hessian matrix for the likelihood function. For more information, see `maxLik`. `constraint` constraint matrix. For more information, see `maxLik`. `method` method for optimization. For more information, see `maxLik`. `verbose` If TRUE, print the progress of the estimation. `...` other parameters for optimization. For more information, see `maxLik`.

`hspec-class`, `hsim,hspec-method`

### Examples

```
# example 1
mu <- c(0.1, 0.1)
alpha <- matrix(c(0.2, 0.1, 0.1, 0.2), nrow=2, byrow=TRUE)
beta <- matrix(c(0.9, 0.9, 0.9, 0.9), nrow=2, byrow=TRUE)
h <- new("hspec", mu=mu, alpha=alpha, beta=beta)
res <- hsim(h, size=100)
summary(hfit(h, res\$inter_arrival, res\$type))

# example 2

mu <- matrix(c(0.08, 0.08, 0.05, 0.05), nrow = 4)
alpha <- function(param = c(alpha11 = 0, alpha12 = 0.4, alpha33 = 0.5, alpha34 = 0.3)){
matrix(c(param["alpha11"], param["alpha12"], 0, 0,
param["alpha12"], param["alpha11"], 0, 0,
0, 0, param["alpha33"], param["alpha34"],
0, 0, param["alpha34"], param["alpha33"]), nrow = 4, byrow = TRUE)
}
beta <- matrix(c(rep(0.6, 8), rep(1.2, 8)), nrow = 4, byrow = TRUE)

impact <- function(param = c(alpha1n=0, alpha1w=0.2, alpha2n=0.001, alpha2w=0.1),
n=n, N=N, ...){

Psi <- matrix(c(0, 0, param['alpha1w'], param['alpha1n'],
0, 0, param['alpha1n'], param['alpha1w'],
param['alpha2w'], param['alpha2n'], 0, 0,
param['alpha2n'], param['alpha2w'], 0, 0), nrow=4, byrow=TRUE)

ind <- N[,"N1"][n] - N[,"N2"][n] > N[,"N3"][n] - N[,"N4"][n] + 0.5

km <- matrix(c(!ind, !ind, !ind, !ind,
ind, ind, ind, ind,
ind, ind, ind, ind,
!ind, !ind, !ind, !ind), nrow = 4, byrow = TRUE)

km * Psi
}
h <- new("hspec",
mu = mu, alpha = alpha, beta = beta, impact = impact)
hr <- hsim(h, size=100)
plot(hr\$arrival, hr\$N[,'N1'] - hr\$N[,'N2'], type='s')
lines(hr\$N[,'N3'] - hr\$N[,'N4'], type='s', col='red')
fit <- hfit(h, hr\$inter_arrival, hr\$type)
summary(fit)

# example 3

mu <- c(0.15, 0.15)
alpha <- matrix(c(0.75, 0.6, 0.6, 0.75), nrow=2, byrow=TRUE)
beta <- matrix(c(2.6, 2.6, 2.6, 2.6), nrow=2, byrow=TRUE)
rmark <- function(param = c(p=0.65), ...){
rgeom(1, p=param[1]) + 1
}
impact <- function(param = c(eta1=0.2), alpha, n, mark, ...){
ma <- matrix(rep(mark[n]-1, 4), nrow = 2)
alpha * ma * matrix( rep(param["eta1"], 4), nrow=2)
}
h1 <- new("hspec", mu=mu, alpha=alpha, beta=beta,
rmark = rmark,
impact=impact)
res <- hsim(h1, size=100, lambda0 = matrix(rep(0.1,4), nrow=2))

fit <- hfit(h1,
inter_arrival = res\$inter_arrival,
type = res\$type,
mark = res\$mark,
lambda0 = matrix(rep(0.1,4), nrow=2))
summary(fit)