damllRH {RHawkes}

Dynamically approxomated minus loglikelihood of a RHawkes model

Description

Calculates an apprximation to the minus loglikelihood of a RHawkes model with given immigration hazard function \mu, offspring birth time density function h and branching ratio \eta relative to event times tms on interval [0,cens].

Usage

damllRH(tms, cens, par, q=0.999, qe=0.999,
      h.fn = function(x, p) dexp(x, rate = 1 / p), 
      mu.fn = function(x, p) {
        exp(dweibull(x, shape = p[1], scale = p[2], log = TRUE) - 
        pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE))
      }, 
      H.fn = function(x, p) pexp(x, rate = 1 / p), 
      Mu.fn = function(x, p) {
        -pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE)
      },
      keepB=FALSE,
      H.inv=function(x,p)qexp(x,rate=1/p) )

Arguments

tms

A numeric vector, with values sorted in ascending order. Event times to fit the RHawkes point process model.

cens

A numericl scalar. The censoring time.

par

A numeric vector containing the parameters of the model, in order of the immigration parameters, in \mu(.), offspring distribution parameters, in h(.), and lastly the branching ratio \eta(.).

q

A numeric scalar in (0,1] and close to 1, which controls how far we look back when truncating the distribution of the most recent immigrant.

qe

A numeric scalar in (0,1] and close to 1, which controls how to truncation is used in the offspring birth time distribution.

h.fn

A (vectorized) function. The offspring birth time density function.

mu.fn

A (vectorized) function. The immigrant waiting time hazard function.

H.fn

A (vectorized) function. Its value at t gives the integral of the offspring birth time density function from 0 to t.

Mu.fn

A (vectorized) function. Its value at t gives the integral of the immigrant waiting time hazard function from 0 to t.

keepB

A boolean scalar, indicating whether the looking back values B_i should be part of the output or not.

H.inv

A (vectorized) function, giving the inverse function of the integral of the excitation.

Value

A scalar giving the value of the (approximate) negative log-likelihood, when keepB is FALSE (the default); A list with components mll, whhich contains the value of the negative log-likelihood, Bs, which gives the look-back order of the truncation of the distribution of the last immigrant, and Bes, which gives the look-forward order in determining how far into the future the excitation effect is allowed to last.

Author(s)

Feng Chen <[e-mail]>

Examples

## Not run: 
## earthquake times over 96 years
data(quake);
tms <- sort(quake$time);
# add some random noise to the simultaneous occurring event times
tms[213:214] <- tms[213:214] + 
                    sort(c(runif(1, -1, 1)/(24*60), runif(1, -1, 1)/(24*60)))


## calculate the minus loglikelihood of an RHawkes with some parameters 
## the default hazard function and density functions are Weibull and 
## exponential respectively
mllRH(tms, cens = 96*365.25 , par = c(0.5, 20, 1000, 0.5))
damllRH(tms, cens = 96*365.25 , par = c(0.5, 20, 1000, 0.5),q=1,qe=1)
## calculate the MLE for the parameter assuming known parametric forms
## of the immigrant hazard function and offspring density functions.  
system.time(est <- optim(c(0.5, 20, 1000, 0.5), 
                        mllRH, tms = tms, cens = 96*365.25,
                        mu.fn=function(x,p)p[1]/p[2]*(x/p[2])^(p[1]-1),
                        Mu.fn=function(x,p)(x/p[2])^p[1],
                        control = list(maxit = 5000, trace = TRUE),
                        hessian = TRUE)
            )
system.time(est1 <- optim(c(0.5, 20, 1000, 0.5), 
                        function(p){
                            if(any(p<0)||p[4]<0||p[4]>=1)
                                return(Inf);
                            damllRH(tms = tms, cens = 96*365.25,
                                    mu.fn=function(x,p)p[1]/p[2]*(x/p[2])^(p[1]-1),
                                    Mu.fn=function(x,p)(x/p[2])^p[1],
                                    par=p,q=0.999999,qe=0.999999)
                        },
                        control = list(maxit = 5000, trace = TRUE),
                        hessian = TRUE)
            )
## point estimate by MLE
est$par
est1$par
## standard error estimates:
diag(solve(est$hessian))^0.5
diag(solve(est1$hessian))^0.5

## End(Not run)