num.arcsCSint {pcds}

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the interval - one interval case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the data points in Xp in the one middle interval case. It also provides number of vertices (i.e., number of data points inside the intervals) and indices of the data points that reside in the intervals.

The data points could be inside or outside the interval is int=(a,b).

CS proximity region is constructed with an expansion parameter t>0 and a centrality parameter c \in (0,1). CS proximity region is constructed for both points inside and outside the interval, hence the arcs may exist for all points inside or outside the interval.

See also (Ceyhan (2016)).

Usage

num.arcsCSint(Xp, int, t, c = 0.5)

Arguments

Xp

A set of 1D points which constitute the vertices of CS-PCD.

int

A vector of two real numbers representing an interval.

t

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

c

A positive real number in (0,1) parameterizing the center inside int=(a,b) with the default c=.5. For the interval, int=(a,b), the parameterized center is M_c=a+c(b-a).

Value

A list with the elements

desc

A short description of the output: number of arcs and quantities related to the interval

num.arcs

Total number of arcs in all intervals (including the end-intervals), i.e., the number of arcs for the entire CS-PCD

num.in.range

Number of Xp points in the interval int

num.in.ints

The vector of number of Xp points in the partition intervals (including the end-intervals)

int.num.arcs

The vector of the number of arcs of the components of the CS-PCD in the partition intervals (including the end-intervals)

data.int.ind

A vector of indices of partition intervals in which data points reside. Partition intervals are numbered from left to right with 1 being the left end-interval.

ind.left.end, ind.mid, ind.right.end

Indices of data points in the left end-interval, middle interval, and right end-interval (respectively)

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the end points of the support interval int.

vertices

Vertices of the digraph, Xp.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

num.arcsCSmid.int, num.arcsCSend.int, and num.arcsPEint

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
set.seed(1)
Xp<-runif(n,a,b)
Narcs = num.arcsCSint(Xp,int,t,c)
Narcs
summary(Narcs)
plot(Narcs)