mllMRH {MRHawkes}

Minus loglikelihood of an (bivariate) MRHawkes model

Description

Calculates the minus loglikelihood of an (bivariate) MRHawkes model with given immigration hazard functions \mu, offspring density functions h and bracnhing ratios \eta for the event times and types data on the interval [0,cens].

Usage

mllMRH(data, cens, par,
       h1.fn = function(x, p) 1 / p * exp( - x / p),
       h2.fn = function(x, p) 1 / p * exp( - x / p),
       mu1.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))
       },
       mu2.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))
       },
       H1.fn = function(x, p) pexp(x, rate = 1 / p),
       H2.fn = function(x, p) pexp(x, rate = 1 / p),
       Mu1.fn = function(x, p){
         - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, 
                   log.p = TRUE)
       },
       Mu2.fn = function(x, p){
         - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, 
                    log.p = TRUE)
       })

Arguments

data

A two column matrix. The first column contains the event times sorted in ascending order. The second column contains the corresponding event type with the label one or two.

cens

A scalar. The censoring time.

par

A numeric vector. Contains the ten parameters of the model, in order of the immigration parameters \mu(.) for the two renewal distributions, the two offspring parameters h(.) and lastly the four branching ratios \eta.

h1.fn

A (vectorized) function. The offspring density function for type one events.

h2.fn

A (vectorized) function. The offspring density function for type two events.

mu1.fn

A (vectorized) function. The immigration hazard function for events of type one.

mu2.fn

A (vectorized) function. The immigration hazard function for events of type two.

H1.fn

A (vectorized) function. Its value at t gives the integral of the offspring density function from 0 to t for type one events.

H2.fn

A (vectorized) function. Its value at t gives the integral of the offspring density function from 0 to t for type two events.

Mu1.fn

A (vectorized) function. Its value at t gives the integral of the immigrant hazard function from 0 to t for type one events.

Mu2.fn

A (vectorized) function. Its value at t gives the integral of the immigrant hazard function from 0 to t for type two events.

Value

Value of the negative log-liklihood.

Author(s)

Tom Stindl <[e-mail]> Feng Chen <[e-mail]>

Examples

  ## Magnitude 5.5 or greater earthquakes over the 25 year period from 
  ## 01/01/1991 to 31/12/2015.
  data(fivaqks); 
  near.fiji <- grep("Fiji", fivaqks$place)
  near.vanuatu <- grep("Vanuatu", fivaqks$place)
  t.beg <- strptime("1991-01-01 00:00:00", "%Y-%m-%d %H:%M:%S", tz = "UTC")
  t.end <- strptime("2015-12-31 23:59:59", "%Y-%m-%d %H:%M:%S", tz = "UTC")
  t0 <- 0
  t1 <- as.numeric(t.end - t.beg)
  tms <- strptime(fivaqks$time, "%Y-%m-%dT%H:%M:%OSZ", tz = "UTC")
  ts <- as.numeric(tms[-1] - t.beg)
  ts <- c(as.numeric(tms[1] - t.beg)/24, ts)
  ts.fi <- ts[near.fiji]; ts.fi <- ts.fi[ts.fi >= 0 & ts.fi <= t1]
  ts.va <- ts[near.vanuatu]; ts.va <- ts.va[ts.va >=0 & ts.va <= t1]
  ts.c <- c(ts.fi, ts.va)
  z.c <- c(rep(1, times = length(ts.fi)), rep(2, times = length(ts.va)))
  o <- order(ts.c)
  data <- cbind(ts.c[o], z.c[o])
  
  ## calculate the minus loglikelihood of an (bivariate) MRHawkes with some 
  ## parameters the default hazard functions and density functions are Weibull 
  ## and exponential respectivley
  mllMRH(data, cens = t1, par = c(0.488, 20.10, 0.347, 9.53, 461, 720,
                                  0.472, 0.293, 0.399, -0.0774))
                                  
  ## calculate the MLE for the parameter assuming known parametric forms
  ## of the immigrant hazard function and offspring density functions.  
  
    system.time(est <- optim(c(0.488, 20.10, 0.347, 9.53, 461, 720,
                                  0.472, 0.293, 0.399, -0.0774), 
                             mllMRH, data = data, cens = t1,
                             control = list(maxit = 5000, trace = TRUE),
                             hessian = TRUE)
    )
    ## point estimate by MLE
    est$par
    ## standard error estimates:
    diag(solve(est$hessian))^0.5