IarcPEmid.int {pcds}

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

Returns I(p_2 \in N_{PE}(p_1,r,c)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{PE}(p_1,r,c), returns 0 otherwise, where N_{PE}(x,r,c) is the PE proximity region for point x and is constructed with expansion parameter r \ge 1 and centrality parameter c \in (0,1) for the interval (a,b).

PE proximity regions are defined with respect to the middle interval int and vertex regions are based on the center associated with the centrality parameter c \in (0,1). For the interval, int=(a,b), the parameterized center is M_c=a+c(b-a). rv is the index of the vertex region p_1 resides, with default=NULL. If p_1 and p_2 are distinct and either of them are outside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2012, 2016)).

Usage

IarcPEmid.int(p1, x2, int, r, c = 0.5, rv = NULL)

Arguments

p1, x2

1D points; p_1 is the point for which the proximity region, N_{PE}(p_1,r,c) is constructed and p_2 is the point which the function is checking whether its inside N_{PE}(p_1,r,c) or not.

int

A vector of two real numbers representing an interval.

r

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

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).

rv

The index of the vertex region p_1 resides, with default=NULL.

Value

I(p_2 \in N_{PE}(p_1,r,c)) for points p_1 and p_2 that is, returns 1 if p_2 is in N_{PE}(p_1,r,c), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

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

See Also

IarcPEend.int, IarcCSmid.int, and IarcCSend.int

Examples

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

IarcPEmid.int(7,5,int,r,c)
IarcPEmid.int(1,3,int,r,c)