일 변량 지수 혹 프로세스의 MLE 찾기

단 변량 지수 혹 프로세스는 다음과 같은 이벤트 도달률을 갖는 자체 자극 포인트 프로세스입니다.

$\lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$

여기서, 은 이벤트 도착 시간입니다.$t_1,..t_n$

로그 우도 함수는

$- t_n \mu + \frac{\alpha}{\beta} \sum{( e^{-\beta(t_n-t_i)}-1 )} + \sum\limits_{i<j}{\ln(\mu+\alpha e^{-\beta(t_j-t_i)})}$

재귀 적으로 계산할 수 있습니다.

$- t_n \mu + \frac{\alpha}{\beta} \sum{( e^{-\beta(t_n-t_i)}-1 )} + \sum{\ln(\mu+\alpha R(i))}$

$R(i) = e^{-\beta(t_i-t_{i-1})} (1+R(i-1))$

$R(1) = 0$

What numerical methods can I use to find the MLE? What is the simplest practical method to implement?

The Nelder-Mead simplex algorithm seems to work well.. It is implemented in Java by the Apache Commons Math library at https://commons.apache.org/math/ . I've also written a paper about the Hawkes processes at Point Process Models for Multivariate High-Frequency Irregularly Spaced Data .

felix, using exp/log transforms seems to ensure positivity of the parameters. As for the small alpha thing, search the arxiv.org for a paper called "limit theorems for nearly unstable hawkes processes"

I solved this problem using the nlopt library. I found a number of the methods converged quite quickly.

You could also do a simple maximization. In R:

neg.loglik <- function(params, data, opt=TRUE) {
  mu <- params[1]
  alpha <- params[2]
  beta <- params[3]
  t <- sort(data)
  r <- rep(0,length(t))
  for(i in 2:length(t)) {
    r[i] <- exp(-beta*(t[i]-t[i-1]))*(1+r[i-1])
  }
  loglik <- -tail(t,1)*mu
  loglik <- loglik+alpha/beta*sum(exp(-beta*(tail(t,1)-t))-1)
  loglik <- loglik+sum(log(mu+alpha*r))
  if(!opt) {
    return(list(negloglik=-loglik, mu=mu, alpha=alpha, beta=beta, t=t,
                r=r))
  }
  else {
    return(-loglik)
  }
}

# insert your values for (mu, alpha, beta) in par
# insert your times for data
opt <- optim(par=c(1,2,3), fn=neg.loglik, data=data)