Log-likelihood of a spatiotemporal Hawkes process

Evaluates \(\sum_j \log\lambda(t_j,x_j,y_j) - \int \lambda\), with the compensator \(\mu\,T\,A + \sum_i \alpha(1 - e^{-\beta(T - t_i)})\) (the spatial kernel integrates to 1 over the plane; interior-region boundary effects are neglected, standard for events away from the edge).

Usage

morie_hawkes_st_loglik(events, params, end_time = NULL, area = 1)

Arguments

events

Data frame with t, x, y (sorted or not).

params

List list(mu, alpha, beta, sigma).

end_time

Observation horizon \(T\) (default max(t)).

area

Spatial region area \(A\) used for the background term.

Value

Scalar log-likelihood.

Examples

ev <- morie_hawkes_st_simulate(
  list(mu = 0.2, alpha = 0.5, beta = 1, sigma = 0.3),
  end_time = 20, region = c(0, 10, 0, 10), seed = 1)
morie_hawkes_st_loglik(ev, list(mu = 0.2, alpha = 0.5, beta = 1, sigma = 0.3),
                       end_time = 20, area = 100)
#> [1] -1348.227