An S4 Class Representing an Exponential Marked Hawkes Model

This class defines a marked Hawkes model with an exponential kernel. The intensity of the ground process is expressed as: $$\lambda(t) = \mu + \int_{(-\infty,t)\times E} ( \alpha + g(u, z) ) e^{-\beta (t-u)} M(du \times dz).$$ For more details, refer to the vignettes.

Details

$\mu$ is base intensity, typically a constant vector or a function.

$\alpha$ is a constant matrix representing the impact on intensities after events, stored in the alpha slot.

$\beta$ is a constant matrix for exponential decay rates, stored in the beta slot.

$z$ represents the mark and can be generated by rmark slot.

$g$ is represented by eta when it is linear function of $z$, and by impact when it is a genenral function.

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

Slots

mu: A numeric value, matrix, or function. If numeric, it is automatically converted to a matrix.
alpha: A numeric value, matrix, or function. If numeric, it is automatically converted to a matrix, representing the exciting term.
beta: A numeric value, matrix, or function. If numeric, it is automatically converted to a matrix, representing the exponential decay.
eta: A numeric value, matrix, or function. If numeric, it is automatically converted to a matrix, representing the impact of an additional mark.
impact: A function describing the after-effects of the mark on $\lambda$, with the first argument always being param.
dimens: The dimension of the model.
rmark: A function that generates marks for the counting process, used in simulations.
dmark: A density function for the mark, used in estimation.
type_col_map: A mapping between type and column number of the kernel used in multi-kernel models.
rresidual: A function for generating residuals, analogous to the R random number generator function, specifically for the discrete Hawkes model.
dresidual: A density function for the residual.
presidual: A distribution function for the residual.
qresidual: A quantile function for the residual.

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
#> An object of class "hspec" of 2-dimensional Hawkes process
#> 
#> Slot mu: 
#>      [,1]
#> [1,]  0.2
#> [2,]  0.2
#> 
#> Slot alpha: 
#>      [,1] [,2]
#> [1,] 0.75 0.92
#> [2,] 0.92 0.75
#> 
#> Slot beta: 
#>      [,1] [,2]
#> [1,] 2.25 2.25
#> [2,] 2.25 2.25
#>