preRSum0 {RIFS}

Prefractal points in R^n generated with a matrix of random sums

Description

preRSum0() function generates a sample of fractal (prefractal) points in R^n with a matrix of random sums of a numerical series.

Usage

preRSum0(n=10000, mu=1, eps=1e-9, Z=R2ngon(),
         P=rep(1/nrow(Z), times=nrow(Z)))

Arguments

n

a number of prefractal points.

mu

a partition coefficient for iterative segments.

eps

an error of a random sum of a numerical series.

Z

a set of protofractal points.

P

a probability distribution of protofractal points.

Details

A protofractal set Z is a discrete or continuous set, which in the iterative process generates a prefractal set X.

A prefractal set S%*%Z is a sample of a fractal set generates with a matrix of random sums S of a numerical series: S[i,j] <- sum(X[l==j]),
where i in seq(n); j in seq(k); k <- nrow(Z); X <- mu/(mu+1)^seq(m); m <- 1-log(eps*mu)/log(1+mu); l <- sample.int(k, size=m, prob=P, replace=TRUE).

Value

A list with the prefractal ($pre) and protofractal points ($proto); distributions of probabilities & coefficients ($distr).

Author(s)

Pavel V. Moskalev, Alexey G. Bukhovets and Tatyana Ya. Biruchinskay

References

Bukhovets A.G. and Bukhovets E.A. (2012) Modeling of fractal data structures. Automation and Remote Control, Vol.73, No.2, pp.381-385, doi:10.1134/S0005117912020154.
Bukhovetc A.G. and Biruchinskay T.Y. (2011) Modelling fractal's properties of system objects. Proceedings of Voronezh State University. Series: Systems Analysis and Information Technologies, No.2 (July-December), pp.22-26; in Russian.

See Also

R2ngon, preRIFS, plotR2pre

Examples

# Example 1a. Sierpinski triangle, 1st order, p=const, mu=const
l <- preRSum0()
plot(l$proto, asp=1, col="red",
     main="Prefractal points for 3-gon: k=3; p=1/3; mu=1")
points(l$pre, pch=46, col="red")

## Not run: 
# Example 1b. Sierpinski triangle, 1st order, p=var, mu=const
l <- preRSum0(P=c(2,2,5)/9)
plot(l$proto, asp=1, col="red",
     main="Prefractal points for 3-gon: k=3; p=(2,2,5)/9; mu=1")
points(l$pre, pch=46, col="red")

# Example 2a. Sierpinski square, 2nd order, p=const, mu=const
l <- preRSum0(Z=R2ngon(4,2), mu=2)
plot(l$proto, asp=1, col="red",
     main="Prefractal points for 4-gon: k=8, p=1/8, mu=2")
points(l$pre, pch=46, col="red")

# Example 2b. Sierpinski square, 2nd order, p=var, mu=const
l <- preRSum0(Z=R2ngon(4,2), P=2^abs(seq(-3,4))/45, mu=2)
plot(l$proto, asp=1, col="red",
     main="Prefractal points for 4-gon: k=8, p=2^|-3:4|/45, mu=2")
points(l$pre, pch=46, col="red")

## End(Not run)