Fit a spatiotemporal Hawkes process by maximum likelihood

Maximises morie_hawkes_st_loglik over $(\mu, \alpha, \beta, \sigma)$ on the log scale (positivity) via stats::optim (L-BFGS-B). $\alpha$ is unconstrained above here; check fit$params$alpha < 1 for a stable fit.

Usage

morie_hawkes_st_fit(events, end_time = NULL, area = 1, start = NULL)

Arguments

events: Data frame with t, x, y.
end_time: Horizon $T$ (default max(t)).
area: Spatial region area $A$.
start: Optional named starting list list(mu, alpha, beta, sigma).

Value

A list of class morie_hawkes_st_fit: params (named list), loglik, n, convergence (0 = success).

Details

Identifiability. The background/branching split ($\mu$ vs $\alpha$) is weakly identified at small samples, and the likelihood neglects spatial edge effects (offspring that drift outside the region are attributed to background). Interpret $\mu$ vs $\alpha$ cautiously and prefer large regions relative to $\sigma$ and long records.

Examples

ev <- morie_hawkes_st_simulate(
  list(mu = 0.2, alpha = 0.5, beta = 1, sigma = 0.4),
  end_time = 40, region = c(0, 10, 0, 10), seed = 7)
morie_hawkes_st_fit(ev, end_time = 40, area = 100)$params
#> $mu
#> [1] 0.2916207
#> 
#> $alpha
#> [1] 0.2937365
#> 
#> $beta
#> [1] 1.73148
#> 
#> $sigma
#> [1] 0.3187456
#>