funsMuVarUndPE2D {pcds.ugraph}

Returns the mean and (asymptotic) variance of edge density of underlying or reflexivity graph of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D uniform data in one triangle

Description

The mean and (asymptotic) variance functions for the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs): muOrPE2D and asy.varOrPE2D for the underlying graph and muAndPE2D and asy.varAndPE2D for the reflexivity graph.

muOrPE2D and muAndPE2D return the mean of the (edge) density of the underlying or reflexivity graph of PE-PCDs, respectively, for 2D uniform data in a triangle. Similarly, asy.varOrPE2D and asy.varAndPE2D return the asymptotic variance of the edge density of the underlying or reflexivity graph of PE-PCDs, respectively, for 2D uniform data in a triangle.

PE proximity regions are defined with expansion parameter r \ge 1 with respect to the triangle in which the points reside and vertex regions are based on center of mass, CM of the triangle.

See also (Ceyhan (2016)).

Usage

muOrPE2D(r)

muAndPE2D(r)

mu.undPE2D(r, ugraph = c("underlying", "reflexivity"))

asy.varOrPE2D(r)

asy.varAndPE2D(r)

asy.var.undPE2D(r, ugraph = c("underlying", "reflexivity"))

Arguments

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

ugraph

The type of the graph based on PE-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Value

mu.undPE2D returns the mean and asy.varUndOrPE2D returns the (asymptotic) variance of the edge density of the underlying graph of the PE-PCD for uniform data in any triangle if ugraph="underlying", and those of the reflexivity graph if ugraph="reflexivity". The functions muOrPE2D, muAndPE2D, asy.varOrPE2D, and asy.varAndPE2D are the corresponding mean and asymptotic variance functions for the edge density of the reflexivity graph of the PE-PCD, respectively, for uniform data in any triangle.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

mu.undCS2D, asy.var.undCS2D, muPE2D, asy.varPE2D, muAndCS2D, and asy.varAndCS2D

Examples

#\donttest{
mu.undPE2D(1.2)
mu.undPE2D(1.2,ugraph="r")

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

muOR = muAND <- vector()
for (i in 1:lrseq)
{
  muOR<-c(muOR,mu.undPE2D(rseq[i]))
  muAND<-c(muAND,mu.undPE2D(rseq[i],ugraph="r"))
}

plot(rseq, muOR,type="l",xlab="r",ylab=expression(mu(r)),lty=1,
     xlim=range(rseq),ylim=c(0,1))
lines(rseq,muAND,type="l",lty=2,col=2)
legend("bottomright", inset=.02,
       legend=c(expression(mu[or](r)),expression(mu[and](r))),
       lty=1:2,col=1:2)
#}

#\donttest{
asy.var.undPE2D(1.2)
asy.var.undPE2D(1.2,ugraph="r")

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

avarOR<-avarAND<-vector()
for (i in 1:lrseq)
{
  avarOR<-c(avarOR,asy.var.undPE2D(rseq[i]))
  avarAND<-c(avarAND,asy.var.undPE2D(rseq[i],ugraph="r"))
}

oldpar <- par(mar=c(5,5,4,2))
plot(rseq, avarAND,type="l",lty=2,col=2,xlab="r",
     ylab=expression(paste(sigma^2,"(r)")),xlim=range(rseq))
lines(rseq,avarOR,type="l")
legend(3.75,.02,
       legend=c(expression(paste(sigma["underlying"]^"2","(r)")),
                 expression(paste(sigma["reflexivity"]^"2","(r)")) ),
       lty=1:2,col=1:2)

par(oldpar)
#}