Type:	Package
Title:	Renewal Hawkes Process
Version:	1.0
Date:	2022-5-4
Description:	The renewal Hawkes (RHawkes) process (Wheatley, Filimonov, and Sornette, 2016 <doi:10.1016/j.csda.2015.08.007>) is an extension to the classical Hawkes self-exciting point process widely used in the modelling of clustered event sequence data. This package provides functions to simulate the RHawkes process with a given immigrant hazard rate function and offspring birth time density function, to compute the exact likelihood of a RHawkes process using the recursive algorithm proposed by Chen and Stindl (2018) <doi:10.1080/10618600.2017.1341324>, to compute the Rosenblatt residuals for goodness-of-fit assessment, and to predict future event times based on observed event times up to a given time. A function implementing the linear time RHawkes process likelihood approximation algorithm proposed in Stindl and Chen (2021) <doi:10.1007/s11222-021-10002-0> is also included.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Depends:	R (≥ 2.10), IHSEP
NeedsCompilation:	no
Packaged:	2022-05-04 13:17:53 UTC; z3243864
Author:	Feng Chen [aut, cre], Tom Stindl [ctb]
Maintainer:	Feng Chen <feng.chen@unsw.edu.au>
Repository:	CRAN
Date/Publication:	2022-05-05 14:40:09 UTC

Renewal Hawkes Process

Description

The renewal Hawkes (RHawkes) process (Wheatley, Filimonov, and Sornette, 2016 <doi:10.1016/j.csda.2015.08.007>) is an extension to the classical Hawkes self-exciting point process widely used in the modelling of clustered event sequence data. This package provides functions to simulate the RHawkes process with a given immigrant hazard rate function and offspring birth time density function, to compute the exact likelihood of a RHawkes process using the recursive algorithm proposed by Chen and Stindl (2018) <doi:10.1080/10618600.2017.1341324>, to compute the Rosenblatt residuals for goodness-of-fit assessment, and to predict future event times based on observed event times up to a given time. A function implementing the linear time RHawkes process likelihood approximation algorithm proposed in Stindl and Chen (2021) <doi:10.1007/s11222-021-10002-0> is also included.

Details

The DESCRIPTION file:

Index of help topics:

EM1partial              Partial EM algorithm for the RHawkes process,
                        version 1
EM2partial              Partial EM algorithm for the RHawkes process,
                        version 2
RHawkes-package         Renewal Hawkes Process
damllRH                 Dynamically approxomated minus loglikelihood of
                        a RHawkes model
mllRH                   Minus loglikelihood of a RHawkes model
mllRH1                  Minus loglikelihood of a RHawkes model with
                        parent probabilities
mllRH2                  Minus loglikelihood of a RHawkes model with
                        Rosenblatt residuals
pred.den                RHawkes predictive density function
pred.haz                RHawkes predictive hazard function
quake                   An RHawkes earthquake data set
sim.pred                Simulate a fitted RHawkes process model
sim.pred1               Simulate a fitted RHawkes process model for
                        prediction purposes
simRHawkes              Simulate a renewal Hawkes (RHawkes) process
simRHawkes1             Simulate a renewal Hawkes (RHawkes) process
tms                     mid-price change times of the AUD/USD exchange
                        rate

Author(s)

NA

Maintainer: NA

Partial EM algorithm for the RHawkes process, version 1

Description

Calculates the RHawkes model parameters via a partial Expectation-Maximization (EM1) algorithm of Wheatley, Filimonov and Sornette (2016).

Usage

EM1partial(tms, cens, pars, maxiter = 1000, tol = 1e-8,
         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),
         logg.fn = function(x, p){
              dweibull(x, shape = p[1], scale = p[2], log = TRUE) -
              pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE) 
              - (x / p[2])^p[1]},
         Mu.fn = function(x, p){
              - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE)
         })

Arguments

Value

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

## Not run: 
## simulated data
tms <- sort(runif(100,0,100))
## the slower version of the EM algorithms on simulated data with default 
## immigrant hazard function
## and offspring density
system.time( est1 <- EM1partial(tms, 101, c(2,1,0.5,1)) )

## End(Not run)

Partial EM algorithm for the RHawkes process, version 2

Description

Calculates the RHawkes model parameters via a partial Expectation-Maximization (EM2) algorithm of Wheatley, Filimonov and Sornette (2016).

Usage

    
EM2partial(tms, cens, pars, maxiter = 1000, tol = 1e-8,
         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),
         logg.fn = function(x, p){
              dweibull(x, shape = p[1], scale = p[2], log = TRUE) -
              pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE) 
              - (x / p[2])^p[1]},
         Mu.fn = function(x, p){
              - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE)
         })

Arguments

Value

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

## Not run: 
## simulated data
tms <- sort(runif(100,0,100))
## the quicker version on simulated data with default immigrant hazard function
## and offspring density
system.time( est2 <- EM2partial(tms, 101, c(2,1,0.5,1)) )

## End(Not run)

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

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 <feng.chen@unsw.edu.au>

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)

Minus loglikelihood of a RHawkes model

Description

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

Usage

mllRH(tms, cens, par, 
      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)
      })

Arguments

Value

The value of the negative log-likelihood.

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

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))
## 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,
                        control = list(maxit = 5000, trace = TRUE),
                        hessian = TRUE)
            )
## point estimate by MLE
est$par
## standard error estimates:
diag(solve(est$hessian))^0.5

## End(Not run)

Minus loglikelihood of a RHawkes model with parent probabilities

Description

Calculates the minus loglikelihood of a RHawkes model with given immigration hazard function \mu, offspring density function h and branching ratio \eta for event times tms on interval [0,cens]. The same as mllRH although this version also returns the parent probabilities.

Usage

mllRH1(tms, cens, par, 
      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)
        })

Arguments

Value

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

See Also

mllRH

Examples

tmp <- mllRH1(sort(runif(1000,0,1000)), 1001, c(2,1,0.5,1))
for(i in 1:tmp$n)
  cat(exp(tmp$log.p[i*(i - 1)/2 + 1:i]), "\n")    

Minus loglikelihood of a RHawkes model with Rosenblatt residuals

Description

Calculates the minus loglikelihood of a RHawkes model with given immigration hazard function \mu, offspring density function h and branching ratio \eta for event times tms on interval [0,cens]. The same as mllRH although this version also returns the Rosenblatt residuals.

Usage

mllRH2(tms, cens, par, 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)
       })

Arguments

Details

Calculate the RHawkes point process Rosenblatt residuals

Value

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

See Also

mllRH

Examples

## Not run: 
tmp <- mllRH2(sort(runif(1000,0,1000)),1001,c(2,1,0.5,1))
par(mfrow=c(1,2))
qqunif<-function(dat,...){
  dat<-sort(as.numeric(dat));
  n<-length(dat);
  pvec<-ppoints(n);
  plot(pvec,dat,xlab="Theoretical Quantiles",
       ylab="Sample Quantiles",main="Uniform Q-Q Plot",...)
}
qqunif(tmp$U)
acf(tmp$U)
ks.test(tmp$U,"punif")

## End(Not run)

RHawkes predictive density function

Description

Calculates the predictive density of the next observed event time after the censoring time cens based on observations over the interval [0,cens].

Usage

pred.den(x, tms, cens, par, 
        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)
          })

Arguments

Value

The predictive density of the next event evaluated at x.

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

data(quake);
tms <- sort(quake$time);
# add some random noise to the identical event times
tms[213:214] <- tms[213:214] + 
                    sort(c(runif(1, -1, 1)/(24*60), runif(1, -1, 1)/(24*60)))
curve(pred.den(x, tms = tms, cens = 35064, par= c(0.314, 22.2, 1266, 0.512)) 
        ,0 ,2000, col = 2, lty = 2)

RHawkes predictive hazard function

Description

Calculates the predictive hazard function of the next observed event time after the censoring time cens based on observations over the interval [0,cens].

Usage

pred.haz(x, tms, cens, par, 
        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)
        })

Arguments

Value

The predictive hazard rate of the next event evaluated at x.

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

data(quake);
tms <- sort(quake$time);
# add some random noise to the identical event times
tms[213:214] <- tms[213:214] + 
                    sort(c(runif(1, -1, 1)/(24*60), runif(1, -1, 1)/(24*60)))
curve(pred.haz(x, tms = tms, cens = 35064, par= c(0.314, 22.2, 1266, 0.512)) 
        ,0 ,2000, col = 2, lty = 2)

An RHawkes earthquake data set

Description

An earthquake data set containing the earthquake occurrence times near the Japan region previously examined by Ogata (1998).

Usage

data(quake)

Format

The format is a vector of the arrival/birth times of earthquakes.

Details

Times of arrivals of earthquake occurrences in a vector in ascending order.

Source

Simulated by a call to the function simHawkes1.

Examples

## Not run: 
data(quake)
## number of earthquake occurrences
nrow(quake)

## End(Not run)

Simulate a fitted RHawkes process model

Description

Simulate a fitted RHawkes process model after the censoring time cens to a future time point cens.tilde.

Usage

sim.pred(tms, re.dist = rweibull, par, 
    par.redist = list(shape = par[1], scale = par[2]), 
    h.fn = function(x, p) dexp(x, rate = 1 / p), p.ofd = par[3], 
    branching.ratio = par[4], cens, cens.tilde = cens * 1.5, 
    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))
    })

Arguments

Value

A numeric vector that contains the simulated event times from censoring time cens up until cens.tilde

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

N <- 5; i <- 0;
data(quake); tms <- sort(quake$time);
# add some random noise the simultaneous occurring event times
tms[213:214] <- tms[213:214] + 
                    sort(c(runif(1, -1, 1)/(24*60), runif(1, -1, 1)/(24*60)))
# simulate future event time based on MLE fitted RHawkes model
times <- replicate(N,
                   {cat(i<<-i+1,'\n');
                   sim.pred(tms = tms, par = c(0.314, 22.2, 1266, 0.512),
                            cens=35063)
                   })
plot(NA,NA,xlim=c(0,35063*1.5),ylim=c(0,max(lengths(times))+nrow(quake)),
     xlab="time",ylab="Sample path")
lines(c(0,quake$time),0:nrow(quake),type="s")
for(i in 1:N)
    lines(c(tail(quake$time,1),times[[i]]),nrow(quake)+0:length(times[[i]]),
          type="s",lty=2)

Simulate a fitted RHawkes process model for prediction purposes

Description

Simulate a fitted RHawkes process model from the censoring time cens to a future time point cens.tilde, conditional on the observed event times until the censoring time.

Usage

sim.pred1(tms, par, re.dist = rweibull, 
    par.redist = list(shape = par[1], scale = par[2]), 
    of.dis="exp", par.ofdis = list(rate=par[3]), 
    branching.ratio = par[4], cens=tail(tms,1)+mean(diff(tms))/2, 
    cens.tilde = cens * 1.5, 
    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))
    })

Arguments

Value

A numeric vector that contains the simulated event times from censoring time cens up until cens.tilde

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

See Also

sim.pred.

Examples

N <- 5; i <- 0;
data(quake); tms <- sort(quake$time);
# add some random noise the simultaneous occurring event times
tms[213:214] <- tms[213:214] + 
                    sort(c(runif(1, -1, 1)/(24*60), runif(1, -1, 1)/(24*60)))
# simulate future event time based on MLE fitted RHawkes model
times <- replicate(N,
                   {cat(i<<-i+1,'\n');
                   sim.pred1(tms = tms, par = c(0.314, 22.2, 1266, 0.512),
                             cens=35063)
                   })
plot(NA,NA,xlim=c(0,35063*1.5),ylim=c(0,max(lengths(times))+nrow(quake)),
     xlab="time",ylab="Sample path")
lines(c(0,quake$time),0:nrow(quake),type="s")
for(i in 1:N)
    lines(c(tail(quake$time,1),times[[i]]),nrow(quake)+0:length(times[[i]]),
          type="s",lty=2)

Simulate a renewal Hawkes (RHawkes) process

Description

Simulate a renewal Hawkes (RHawkes) process with given renewal immigration distribution function, offspring density function and branching ratio.

Usage

simRHawkes(re.dist = rweibull, par.redist = list(shape = 1, scale = 1), 
           ofspr.den = function(x, p.ofd) 1 / p.ofd * exp(-x / p.ofd), 
           p.ofd = 1, branching.ratio = 0.5, cens = 1, B = 10, B0 = 50, 
           max.ofspr.den = max(optimize(ofspr.den, c(0, cens), maximum = TRUE, 
           p = p.ofd)$obj, ofspr.den(0, p.ofd), ofspr.den(cens, p.ofd)) * 1.1)

Arguments

Details

The function works by simulating the arrival times of immigrants according to the renewal immigration distribution. The birth times of offspring from each immigrant are then simulated according to an inhomogeneous Poisson processes with appropriate intensity functions.

Value

A numeric vector of all pooled events (immigration/birth) times of all generations 0, 1, ...

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

B <- 10; i <- 0;
tms <- replicate(B,
                  {cat(i<<-i+1,'\n');
                  simRHawkes(par.redist = list(shape = 3, scale = 1),
                      p.ofd = 0.5, branching.ratio = 0.5,
                      cens = 100)
                  })

Simulate a renewal Hawkes (RHawkes) process

Description

Simulate a renewal Hawkes (RHawkes) process with given renewal immigration distribution function, offspring density function and branching ratio.

Usage

simRHawkes1(re.dist = rweibull, par.redist = list(shape = 1, scale = 1), 
            of.dis = "exp", par.ofdis = list(rate=1),
            branching.ratio = 0.5, cens = 1, B = 10, B0 = 50, 
            flatten=TRUE)

Arguments

Details

The function works by simulating the arrival times of immigrants according to the renewal immigration distribution. The birth times of offspring from each immigrant are then simulated according to an inhomogeneous Poisson processes with appropriate intensity functions.

Value

A numeric vector of pooled event (immigration/offspring birth) times of all generations 0, 1, ..., if flatten=TRUE; A list with two components: immitimes for immigrant arrival times, and offspringtimes for birth times of offspring due to different immigrants.

Author(s)

Feng Chen <feng.chen@unsw.edu.au> Tom Stindl <t.stindl@unsw.edu.au>

Examples

tms <- simRHawkes1(par.redist = list(shape = 3, scale = 1),
                   par.ofdis = list(rate=0.5), branching.ratio = 0.5,
                   cens = 50)
plot(stepfun(tms,0:length(tms)),do.points=FALSE,vertical=FALSE,xlim=c(0,50))

tms.clust <- simRHawkes1(par.redist = list(shape = 3, scale = 1),
                         par.ofdis = list(rate=0.5), branching.ratio = 0.5,
                         cens = 50,flatten=FALSE)
plot(c(0,50),c(0, 1+(nt <-length(it <- tms.clust$immitimes))),
     type="n",xlab="time",ylab="cluster")
segments(x0 = it,y0=-0.2,y1=0.2)
for(i in 1:nt)
    segments(x0 = c(it[i],it[i]+tms.clust$offspringtimes[[i]]),
             y0=i-0.2,y1=i+0.2)
abline(h=0:(nt+1),col="light gray",lty=2)
segments(x0=unlist(lapply(1:nt,function(i)c(it[i],it[i]+tms.clust$offspringtimes[[i]]))),
         y0=nt+1-0.2,y1=nt+1+0.2)

mid-price change times of the AUD/USD exchange rate

Description

A financial data set containing the mid-price changes of the AUD/USD foreign exchange rate during the trading week from 20:00:00 GMT on Sunday 19/07/2015 to 21:00:00 GMT Friday 24/07/2015.

Usage

data(tms)

Format

The format is a list of the arrival times of mid price changes that occur every hour in 121 non-overlapping windows.

Details

Times of arrivals of mid-price changes is listed together in ascending order.

Source

Simulated by a call to the function simHawkes1.

Examples

data(tms)
## number of non over-lapping hourly windows
length(tms)