Fit a Hawkes (self-exciting point process) model by maximum likelihood

Fits a one-dimensional Hawkes process with a constant baseline to a vector of event times. The conditional intensity is $$lambda(t) = nu + eta * sum_{t_j < t} g(t - t_j),$$ where nu = exp(a0) is the baseline rate, eta the branching ratio, and g the chosen triggering kernel.

Usage

morie_hawkes_fit(
  times,
  end_time = NULL,
  kernel = c("exponential", "weibull", "lomax", "gamma")
)

Arguments

times: numeric vector of sorted, non-decreasing event times.
end_time: observation horizon; defaults to the last event time.
kernel: triggering kernel: one of "exponential", "weibull", "lomax", "gamma".

Value

An object of class morie_hawkes_fit: a list with the parameter estimate, loglik, aic, branching_ratio, baseline_rate, n_events, converged and the backend used: from 50 events the fit is rmoriebricklayer::core_hawkes_fit() (bounded, analytic gradient), which also reports at_bound, the parameters on their bound, and a note when the kernel shape is not identified (a Lomax at its exponential limit).

Details

The negative log-likelihood is evaluated by the shared morie C++ core (the same kernels the Python package uses); without a compiled core it falls back to a pure-R $O(n^2)$ likelihood.

Examples

set.seed(1)
ev <- cumsum(rexp(200, rate = 2))
fit <- morie_hawkes_fit(ev, kernel = "exponential")
print(fit)
#> morie Hawkes fit  --  exponential kernel, constant baseline
#>   events: 200   horizon: 100   backend: rmoriebricklayer core
#>   estimate:
#>     a0        0.692284
#>     eta       0.001
#>     beta      0.1
#>   baseline rate nu : 1.99828
#>   branching ratio  : 0.001
#>   logLik: -61.3685   AIC: 128.737   converged: TRUE
#>   vs Poisson: logLik -61.3674  (gain -0.00114)