| mle {hawkesbow} | R Documentation |
Fitting Hawkes processes from continuous data
Description
This function fits a Hawkes process to continuous data by minimizing the likelihood
on the interval [0,\mathrm{end}].
Usage
mle(events, kern, end, init = NULL, opts = NULL, ...)
Arguments
events |
The locations of events (sorted in ascending order) |
kern |
Either a string (partially) matching one of the kernels implemented (see Details), or an object of class Model |
end |
The time until which the process is observed. |
init |
(Optional) Initial values of the optimisation algorithm |
opts |
(Optional) To be passed to |
... |
Additional arguments passed to |
Details
The maximum likelihood estimation procedure has only been implemented for the
exponential and the power law kernels.
For the exponential kernel, the likelihood is computed in O(n) complexity
(as described in details in T. Ozaki and Y. Ogata, “Maximum likelihood
estimation of Hawkes’ self-exciting point processes,” Ann. Inst. Stat. Math.,
vol. 31, no. 1, pp. 145–155, Dec. 1979).
For the power law kernel, the complexity is O(n^2).
Value
Returns a list containing the solution of the optimisation procedure, the object Model
with its parameters updated to the solution, and the output produced by nloptr.
See Also
hawkes() for the simulation of Hawkes processes,
Model for the abstract class, and Exponential for the specific
reproduction kernels.
Examples
# Simulate an exponential Hawkes process with baseline intensity 1,
# reproduction mean 0.5 and exponential fertility function with rate 2.
x = hawkes(100, fun = 1, repr = .5, family = "exp", rate = 1)
# Estimate the parameters from the arrival times of `x` using MLE
opt = mle(x$p, "Exponential", x$end)
opt$par # Estimated parameters
opt$model$ddloglik(x$p, x$end) # Hessian matrix of the log-likelihood