rassocI: Generation of Points Associated in the Type I Sense with a...

In nnspat: Nearest Neighbor Methods for Spatial Patterns

Generation of Points Associated in the Type I Sense with a Given Set of Points

Description

An object of class "SpatPatterns".

Generates n_2 2D points associated with the given set of points (i.e. reference points) X_1 in the type I fashion with circular (or radial) between class attraction parameter p, which is a probability value between 0 and 1. The generated points are intended to be from a different class, say class 2 (or X_2 points) than the reference (i.e. X_1 points, say class 1 points, denoted as X1 as an argument of the function). To generate n_2 (denoted as n2 as an argument of the function) X_2 points, n_2 of X_1 points are randomly selected (possibly with replacement) and for a selected X1 point, say x_{1ref}, a Uniform(0,1) number, U, is generated. If U ≤ p, a new point from the class 2, say x_{2new}, is generated within a circle with radius equal to the distance to the closest X_1 point (uniform in the polar coordinates), else the new point is generated uniformly within the smallest bounding box containing X_1 points. That is, if U ≤ p, x_{2new} = x_{1ref}+r_u c(\cos(t_u),\sin(t_u)) where r_u \sim U(0,rad) and t_u \sim U(0, 2π) with rad=\min(d(x_{1ref},X_1\setminus \{x_{1ref}\})), else x_{2new} \sim rect(X_1) where rect(X_1) is the smallest bounding box containing X_1 points. Note that, the level of association increases as p increases, and the association vanishes when p approaches to 0.

Type I association is closely related to Type C association in \insertCiteceyhan:serra-2014;textualnnspat, see the function rassocC and also other association types. In the type C association pattern the new point from the class 2, x_{2new}, is generated (uniform in the polar coordinates) within a circle centered at x_{1ref} with radius equal to r_0, in type U association pattern x_{2new} is generated similarly except it is uniform in the circle. In type G association, x_{2new} is generated from the bivariate normal distribution centered at x_{1ref} with covariance σ I_2 where I_2 is 2 \times 2 identity matrix.

Usage

rassocI(X1, n2, p)

Arguments

X1	A set of 2D points representing the reference points, also referred as class 1 points. The generated points are associated in a type I sense (in a circular/radial fashion) with these points.
n2	A positive integer representing the number of class 2 (i.e. X_2) points to be generated.
p	A real number between 0 and 1 representing the attraction probability of class 2 points associated with a randomly selected class 1 point (see the description below).

Value

A list with the elements

pat.type	equals "ref.gen" for the bivariate pattern of association of class 2 (i.e. X_2) points with the reference points (i.e. X_1), indicates reference points are required to be entered as an argument in the function
type	The type of the point pattern
parameters	Radial (i.e. circular) between class attraction parameter controlling the level of association
gen.points	The output set of generated points (i.e. class 2 points) associated with reference (i.e. X_1 points)
ref.points	The input set of reference points X_1, i.e., points with which generated class 2 points are associated.
desc.pat	Description of the point pattern
lab	The class labels of the generated points, it is NULL for this function, since only class 2 points are generated in this pattern
mtitle	The "main" title for the plot of the point pattern
num.points	The vector of two numbers, which are the number of generated class 2 points and the number of reference (i.e. X_1) points.
xlimit,ylimit	The possible ranges of the x- and y-coordinates of the generated and the reference points

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

rassocC, rassocG, rassocU, and rassoc

Examples

n1<-20; n2<-1000;  #try also n1<-10; n2<-1000;

p<- .75 #try also .25, .5, .9, runif(1)
#with default bounding box (i.e., unit square)
X1<-cbind(runif(n1),runif(n1))  #try also X1<-1+cbind(runif(n1),runif(n1))

Xdat<-rassocI(X1,n2,p)
Xdat
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)

nnspat documentation built on Aug. 30, 2022, 9:06 a.m.