preRSum0 | R Documentation |
preRSum0()
function generates a sample of fractal (prefractal) points in R^n with a matrix of random sums of a numerical series.
preRSum0(n=10000, mu=1, eps=1e-9, Z=R2ngon(), P=rep(1/nrow(Z), times=nrow(Z)))
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. |
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)
.
A list with the prefractal ($pre
) and protofractal points ($proto
); distributions of probabilities & coefficients ($distr
).
Pavel V. Moskalev, Alexey G. Bukhovets and Tatyana Ya. Biruchinskay
Bukhovets A.G. and Bukhovets E.A. (2012) Modeling of fractal data structures. Automation and Remote Control, Vol.73, No.2, pp.381-385, doi:10.1134/S0005117912020154.
Bukhovetc A.G. and Biruchinskay T.Y. (2011) Modelling fractal's properties of system objects. Proceedings of Voronezh State University. Series: Systems Analysis and Information Technologies, No.2 (July-December), pp.22-26; in Russian.
R2ngon, preRIFS, plotR2pre
# Example 1a. Sierpinski triangle, 1st order, p=const, mu=const l <- preRSum0() plot(l$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)
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