IPPP (version 1.1)

IPPPnthpointdens: Probability Density of Events in an Inhomogeneous Poisson Point Process

Description

Determines values of the probability density function of the n-th point of an inhomogeneous Poisson point process (IPPP), given that a fixed number of events occur. The IPPP is described by a rate function r.

Usage

IPPPnthpointdens(x, n, samplesize, xrate, yrate)

Arguments

x

Vector of real numbers where the value of the probability density is determined

n

Natural number smaller than samplesize. Determines the event for which the probability density is determined. 1=first event from the left, i=i-th event from the left.

samplesize

Natural number. The overall number of events occurring

xrate

Vector of (strictly increasing) real numbers

yrate

Vector of positive real numbers of the same length as xrate. The vectors xrate and yrate form the rate function r in the sense that r=approxfun(xrate,yrate)

Value

A list containing two entries:

x

A duplicate of the input of the same name

densval

A vector consisting of the values of the probability density, evaluated at x

Details

Below min(xrate) and above max(xrate), the rate function r is assumed to have the value zero.

References

Hohmann, Niklas. "Conditional Densities and Simulations of Inhomogeneous Poisson Point Processes: The R package "IPPP"" arXiv 2019. <arXiv:1901.10754>

See Also

IPPPconddens for the probability density of the n-th event above/below the location of some given event.

vignette("IPPP") for an overview of the features of the IPPP package and some background.

Examples

# NOT RUN {
sx=c(1,2,3,4,5)
sy=c(0,1,1,3,0)
sm=c(1,0,1,0,-1)
xrate=seq(1,5,length.out=100)
yrate=splinefunH(sx,sy,sm)(xrate)
#plot rate function
plot(xrate,yrate,type='l',xlim=c(0.5,5.5), main='Rate Function') 

x=seq(0,6,length.out = 1000) #point where the pdf is determined
n=1          #get the pdf of the first point from the left ...
samplesize=5  #... out of a sample of five
ll=IPPPnthpointdens(x,n,samplesize,xrate,yrate)
plot(ll$x,ll$densval,type='l') #plot the resulting pdf
legend('topleft',legend=paste('pdf of point no.',as.character(n),
'out of ',as.character(samplesize)), 'points' ,lty=1)
# }