mle: 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, ...)

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.

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 nloptr
...: Additional arguments passed to nloptr

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 \(O(n^2)\).

Examples

Run this code

# 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