funsMuVarUndCS2D {pcds.ugraph}

Returns the mean and (asymptotic) variance of edge density of underlying or reflexivity graphs of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D uniform data in one triangle

Description

The mean and (asymptotic) variance functions for the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs): muOrCS2D and asy.varOrCS2D for the underlying graph and muAndCS2D and asy.varAndCS2D for the reflexivity graph.

muOrCS2D and muAndCS2D return the mean of the (edge) density of the underlying or reflexivity graphs of CS-PCDs, respectively, for 2D uniform data in a triangle. Similarly, asy.varOrCS2D and asy.varAndCS2D return the asymptotic variance of the edge density of the underlying or reflexivity graphs of CS-PCDs, respectively, for 2D uniform data in a triangle.

CS proximity regions are defined with expansion parameter t > 0 with respect to the triangle in which the points reside and edge regions are based on center of mass, CM of the triangle.

See also (Ceyhan (2016)).

Usage

muOrCS2D(t)

muAndCS2D(t)

mu.undCS2D(t, ugraph = c("underlying", "reflexivity"))

asy.varOrCS2D(t)

asy.varAndCS2D(t)

asy.var.undCS2D(t, ugraph = c("underlying", "reflexivity"))

Arguments

t

A positive real number which serves as the expansion parameter in CS proximity region.

ugraph

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

Value

mu.undCS2D returns the mean and asy.varUndOrCS2D returns the (asymptotic) variance of the edge density of the underlying graph of the CS-PCD for uniform data in any triangle if ugraph="underlying", and those of the reflexivity graph if ugraph="reflexivity". The functions muOrCS2D, muAndCS2D, asy.varOrCS2D, and asy.varAndCS2D are the corresponding mean and asymptotic variance functions for the edge density of the reflexivity graph of the CS-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 muCS2D, and asy.varCS2D,

Examples

#\donttest{
mu.undCS2D(1.2)
mu.undCS2D(1.2,ugraph="r")

tseq<-seq(0.01,10,by=.05)
ltseq<-length(tseq)

muOR = muAND <- vector()
for (i in 1:ltseq)
{
  muOR<-c(muOR,mu.undCS2D(tseq[i]))
  muAND<-c(muAND,mu.undCS2D(tseq[i],ugraph="r"))
}

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

#\donttest{
asy.var.undCS2D(1.2)
asy.var.undCS2D(1.2,ugraph="r")

asy.varOrCS2D(.2)

tseq<-seq(.05,25,by=.05)
ltseq<-length(tseq)

avarOR<-avarAND<-vector()
for (i in 1:ltseq)
{
  avarOR<-c(avarOR,asy.var.undCS2D(tseq[i]))
  avarAND<-c(avarAND,asy.var.undCS2D(tseq[i],ugraph="r"))
}

oldpar <- par(mar=c(5,5,4,2))
plot(tseq, 4*avarAND,type="l",lty=2,col=2,xlab="t",
     ylab=expression(paste(sigma^2,"(t)")),xlim=range(tseq))
lines(tseq,4*avarOR,type="l")
legend(18,.1,
       legend=c(expression(paste(sigma["underlying"]^"2","(t)")),
                 expression(paste(sigma["reflexivity"]^"2","(t)")) ),
       lty=1:2,col=1:2)

par(oldpar)
#}