Ogata (1988) ETAS with the modified-Omori (Lomax) triggering kernel
and exponential magnitude productivity: conditional intensity
$$\lambda(t) = \mu + K \sum_{t_i < t} e^{\alpha (m_i - m_0)}
(t - t_i + c)^{-p}.$$
MLE by L-BFGS-B on the exact log-likelihood (direct sum plus the
closed-form kernel integral).
Usage
morie_crim_etas(times, magnitudes = NULL, m0 = NULL, t_max = NULL)
Arguments
- times
Numeric event times (sorted or sortable).
- magnitudes
Numeric marks (same length); constant marks give
a plain Omori-Hawkes process.
- m0
Reference (cutoff) magnitude. Default min(magnitudes).
- t_max
Observation horizon. Default max(times).
Value
List of class "morie_etas": par (mu, K, alpha, c, p),
loglik, branching_ratio, n, converged, call.
References
Ogata (1988) JASA 83(401).
Examples
set.seed(1)
tt <- sort(runif(120, 0, 100))
mm <- rexp(120, 1.5) + 2
morie_crim_etas(tt, mm)
#> ETAS (Ogata 1988), n = 120
#> mu K alpha c p
#> 1.2087 0.0000 0.9177 18.2029 2.0954
#> loglik = -97.24 branching ratio = 0.000
