HawkesProcesses.jl

fit(events::Array{<:Number, 1}, maxT::Number, its::Int64)

Arguments

• events The array of event times to fit the Hawkes process too.
• maxT The boundary time over which the events were observed.
• its Number of iterations to sample for.

Notes

• All events must be unique

hierarchical_fit(eventsTimesList, maxT, its)

Arguments

• eventTimesList An array of arrays of events.
• maxT The boundary time over which the events were observed.
• its Number for of iterations to sample for.

intensity(ts, events, background, kappa, kernel)

Evaluate the intensity over a timegrid for some observed events with given Hawkes parameters.

Arguments

• ts::Array{<:Number,1}: time grid to evalue the intensity over
• events::Array{<:Number,1}: Events of the process
• background: Background rate
• kappa::Float64: Kappa value
• kernel::Function: Kernel function

Notes

• kappa must be between 0 and 1

compensator(ts::Number, events::Array{<:Number}, bg::Number, kappa::Number, kern::Distributions.Distribution)

The compensator or integrated intensity function.

Arguments

• ts::Number time to which evaluate the integrated intensity function.
• events::Array{<:Number} events from the intensity function.
• bg::Number Background rate.
• kappa::Number Kappa parameter
• kern::Distributions.Distribution Distribution of the kernel.

Examples

simulate(bg::Number, kappa::Float64, kern::Function, maxT::Number)

Simulate a Hawkes process between 0 and maxT with parameters bg, kappa, kern.

Arguments

• bg: The background rate of the Hawkes process. Constant or positive function.
• kappa: The kappa parameter of the Hawkes process.
• kern : The kernel function of the Hawkes process.
• maxT : Maximum time that the Hawkes process will be simulated for.

Notes

• kappa must be between 0 and 1 for a stable Hawkes process.

Examples

``julia kern_f(x) = pdf.(Distributions.Exponential(1/0.5), x)

simevents = simulate(0.5, 0.5, kern_f, 100) ``

likelihood(events::Array{<:Number, 1}, background::Float64, kappa::Float64, kernel::Distributions.Distribution, maxT::Number)

Calculate the log likelihood of a Hawkes process for given parameters.

Arguments

• events Vector of events to calculate the likelihood for.
• background Background rate.
• kappa Kappa parameter.
• kernel Function or distribution of the kernel.
• maxT Maximum time of the process.

Notes

• The kernel function must be a proper probability distribution.

Examples

time_change_null(events, maxT)

Calculate the residuals for some events from a constant Poisson process.

Arguments

• events::Array{<:Number} the events from the process.
• maxT::Number the maximum window time.

time_change_hawkes(events, bg, kappa, kern)

Calculate the residuals using the time change theorem for a Hawkes process.

Arguments

• events::Array{<:Number} the events from the process.
• bg the background rate of the Hawkes process.
• kappa the kappa parameter of the Hawkes process.
• kern the kernel distriubtion of the Hawkes process.