PtProcess (version 3.3-16)

gif: General Notes on Ground Intensity Functions

Description

This page contains general notes about the required structure of ground intensity functions (including those that are not conditional on their history) to be used with this package.

Value

The returned value is either \(\lambda_g(t_i|{\cal H}_t)\), where the \(t_i\) are specified within evalpts; or $$\int \lambda_g(t|{\cal H}_t) dt$$ where the limits of the integral are specified by the function argument TT.

Forms of Usage

The usage of a ground intensity function takes two forms, one to evaluate the gif at specified evalpts, or to evaluate the integral of the gif on the interval TT, each shown below, respectively.

gif(data, evalpts, params, tplus=FALSE)

gif(data, NULL, params, TT)

Arguments

All ground intensity functions should be defined to contain the following arguments, in the order below, even though they may not be required (see Details below).

data

a data frame containing the history of the process, denoted below as \( {\cal H}_t\). It should contain all variables that are required to evaluate the gif function, though can contain others too. No history is represented as NULL.

evalpts

a object containing the values at which the gif function is to be evaluated, consistent with what is required by the gif function.

params

vector containing values of the parameters required by the gif function.

TT

vector of length 2, being the time interval over which the integral of the ground intensity function is to be evaluated.

tplus

logical, \(\lambda_g(t|{\cal H}_t)\) is evaluated as \(\lambda_g(t^+|{\cal H}_t)\) if TRUE, else \(\lambda_g(t^-|{\cal H}_t)\). It is important if a “jump” occurs at \(t\).

Function Attributes

Each function should have some of the following attributes if it is to be used in conjunction with residuals.mpp or simulate.mpp:

rate

must be specified if the default method for simulate.mpp is to be used. Takes the values "bounded", "decreasing" or "increasing"; see Details.

regions

an expression giving the number of regions; required with linksrm_gif.

Details

Note that the gif functions not only evaluate values of \(\lambda_g(t_i|{\cal H}_t)\), but also the integral. The value of the ground intensity function is returned at each time point specified in evalpts when TT==NA. If TT is not missing, the integral between TT[1] and TT[2] of the ground intensity function is calculated. In this last situation, anything assigned to the argument evalpts will have no effect.

At the moment, we have the following types of processes: those jump processes that are conditional on their history (etas_gif, srm_gif, linksrm_gif), and non-homogeneous Poisson processes that are not conditional on their history (simple_gif). Another case is where we have a collection of point like “regions” (or lattice nodes), each with their own ground intensity function, but where each is also dependent on what is happening in the other regions (linksrm_gif).

Functions have been given an attribute “rate”, taking the values of "bounded", "decreasing" or "increasing". This is used within the simulation function simulate.mpp which uses the thinning method. This method requires a knowledge of the maximum of \(\lambda_g(t|{\cal H}_t)\) in a given interval. The argument tplus is also used by the simulation routine, where it is necessary to determine the value of the intensity immediately after a simulated event.

See Also

etas_gif, expfourier_gif, exppoly_gif, fourier_gif, linksrm_gif, poly_gif, simple_gif, srm_gif

Examples

# NOT RUN {
#  Ogata's Data: ground intensity function
#  evaluate lambda_g(t) at certain times

data(Ogata)

p <- c(0.02, 70.77, 0.47, 0.002, 1.25)
times <- sort(c(seq(0, 800, 0.5), Ogata$time))
TT <- c(0, 800)

plot(times, log(etas_gif(Ogata, times, params=p)), type="l",
     ylab=expression(paste(log, " ", lambda[g](t))),
     xlab=expression(t), xlim=TT)

#  Evaluate the integral
#   The first form below is where the arguments are in their
#   default positions, the 2nd is where they are not, hence
#   their names must be specified

print(etas_gif(Ogata, NULL, p, TT))
#  or
print(etas_gif(Ogata, params=p, TT=TT))
# }