residuals {hawkesbow}

Residuals of a Hawkes process

Description

Outputs the residuals (values of the compensator at the times of arrival) of a Hawkes process. Useful function for diagnosis through the random time change theorem: the residuals should follow a unit rate Poisson process.

Usage

residuals(x, fun = NULL, repr = NULL, family = NULL, M = NULL, ...)

Arguments

x

A non-negative numeric vector, sorted in ascending order; or an object of class "hawkes" output by function hawkes.

fun

(default = TRUE) A non-negative numeric function or value - intensity (function) of the immigrant process.

repr

(default = NULL) A non-negative numeric value - mean number of offsprings.

family

(default = NULL) A character string "name" naming a distribution with corresponding distribution function dname, or directly the distribution function.

M

(default = NULL) A non-negative numeric value - upper bound on fun(ignored if fun is a numeric value).

...

Additional arguments passed on to the random generation function dname.

Value

The residuals of the process.

Examples

# Simulate an exponential Hawkes process with baseline intensity 1,
# reproduction mean 0.5 and exponential fertility distribution with rate 2.
x <- hawkes(10, fun=1, repr=0.5, family="exp", rate=2)
resid = residuals(x)
resid
plot(resid)
abline(0, 1, col="red", lty="dashed")
# Residuals with a different set of parameters
residuals(x, repr=0.8, rate=3)
# Residuals with a different distribution function
residuals(x, family="chisq", df=2)
# Simulate a Hawkes process with baseline intensity function 1 + sin(x),
# reproduction mean 0.5 and custom [0,1]-triangular fertility function.
x <- hawkes(10, fun=function(y) {1+sin(y)}, M=2, repr=0.5,
            family=function(n) {1 - sqrt(1 - runif(n))})
resid = residuals(x, family=function(y) ifelse(y>0 & y<1, 2-2*y, 0))
plot(resid)
abline(0, 1, col="red", lty="dashed")