hspec-class {emhawkes}

An S4 class to represent an exponential marked Hawkes model

Description

This class represents a specification of a marked Hawkes model with exponential kernel. The intensity of the ground process is defined by:

\lambda(t) = \mu + \int_{(-\infty,t)\times E} ( \alpha + g(u, z) ) e^{-\beta (t-u)} M(du \times dz).

For more details, please see the vignettes.

Details

\mu is base intensity. This is generally a constant vector but can be extended to stochastic processes. Currently, piecewise constant mu is also possible. mu is left continuous.

\alpha is a constant matrix which represents impacts on intensities after events. It is represented by slot alpha.

g is for non-constant parts of the impact. It may depend on any information generated by N, \lambda, z and so on. It is represented by slot impact.

\beta is a constant matrix for exponential decay rates. It is represented by slot beta.

z is mark and represented by slot rmark.

mu, alpha and beta are required slots for every exponential Hawkes model. rmark and impact are additional slots.

Slots

mu

Numeric value or matrix or function, if numeric, automatically converted to matrix.

alpha

Numeric value or matrix or function, if numeric, automatically converted to matrix, exciting term.

beta

Numeric value or matrix or function, if numeric, automatically converted to matrix, exponential decay.

eta

Numeric value or matrix or function, if numeric, automatically converted to matrix, impact by additional mark.

dimens

Dimension of the model.

rmark

A function that generates mark for counting process, for simulation.

dmark

A density function for mark, for estimation.

impact

A function that describe the after impact of mark to lambda whose first argument is always param.

type_col_map

Mapping between type and column number of kernel used for multi-kernel model.

Examples

MU <- matrix(c(0.2), nrow = 2)
ALPHA <- matrix(c(0.75, 0.92, 0.92, 0.75), nrow = 2, byrow=TRUE)
BETA <- matrix(c(2.25, 2.25, 2.25, 2.25), nrow = 2, byrow=TRUE)
mhspec2 <- new("hspec", mu=MU, alpha=ALPHA, beta=BETA)
mhspec2