rassoc {nnspat}

Generation of Points Associated 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=type fashion with the parameter=asc.par which specifies the level of association. 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), say class 1 points).

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 new point from the class 2, say x_{2new}, is generated from a distribution specified by the type argument.

In type I association, i.e., if type="I", first a Uniform(0,1) number, U, is generated. If U \le p, x_{2new} is generated (uniform in the polar coordinates) within a circle with radius equal to the distance to the closest X_1 point, else it is generated uniformly within the smallest bounding box containing X_1 points.

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 \sigma I_2 where I_2 is 2 \times 2 identity matrix.

See Ceyhan (2014) for more detail.

Usage

rassoc(X1, n2, asc.par, type)

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=type sense with these points.

n2

A positive integer representing the number of class 2 points to be generated.

asc.par

A positive real number representing the association parameter. For type="I", it is attraction probability, p, of class 2 points associated with a randomly selected class 1 point; for type="C" or "U", it is the radius of association, r0, of class 2 points associated with a randomly selected class 1 point; for type="G", it is the variance of the Gaussian marginals, where the bivariate normal distribution has covariance \sigma I_2 with I_2 being the 2 \times 2 identity matrix.

type

The type of the association pattern. Takes on values "I", "C", "U" and "G" for types I, C, U and G association patterns (see the description above).

Value

A list with the elements

pat.type

="ref.gen" for the bivariate pattern of association of class 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

The asc.par value specifying the level of association

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

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

Ceyhan E (2014). “Simulation and characterization of multi-class spatial patterns from stochastic point processes of randomness, clustering and regularity.” Stochastic Environmental Research and Risk Assessment (SERRA), 38(5), 1277-1306.

See Also

rassocI, rassocC, rassocU, and rassocG

Examples

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

#with default bounding box (i.e., unit square)
X1<-cbind(runif(n1),runif(n1))

Xdat<-rassoc(X1,n2,asc.par=.05,type="G") #try other types as well
Xdat
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)

#with type U association
Xdat<-rassoc(X1,n2,asc.par=.1,type="U")
Xdat
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)

#with type C association
Xdat<-rassoc(X1,n2,asc.par=.1,type=2) #2 is for "C"
Xdat
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)