fdc2s {SECP}

Mass fractal dimension of a 2D cluster

Description

fdc2s() function uses a linear regression model for statistical estimation of the mass fractal dimension of a cluster on 2D square lattice with iso- & anisotropic sets cover.

Usage

fdc2s(acc=ssi20(x=95), bnd=isc2s(k=12, x=dim(acc)))

Arguments

acc

an accessibility matrix for 2D square percolation lattice.

bnd

bounds for the iso- or anisotropic set cover.

Details

The mass fractal dimension for a cluster is equal to the coefficient of linear regression between log(n) and log(r), where n is an absolute frequency of the total cluster sites which are bounded elements of iso- & anisotropic sets cover.

The isotropic set cover on 2D square lattice is formed from scalable squares with variable sizes 2r+1 and a fixed point in the lattice center.

The anisotropic set cover on 2D square lattice is formed from scalable rectangles with variable sizes r+1 and a fixed edge along the lattice boundary.

The percolation is simulated on 2D square lattice with uniformly weighted sites and the constant parameter p.

The isotropic cluster is formed from the accessible sites connected with initial sites subset.

If acc[e]<p then e is accessible site; if acc[e]==1 then e is non-accessible site; if acc[e]==2 then e belong to a sites cluster.

Value

A linear regression model for statistical estimation of the mass fractal dimension of a cluster on 2D square lattice with iso- & anisotropic sets cover.

Author(s)

Pavel V. Moskalev

References

Moskalev P.V., Grebennikov K.V. and Shitov V.V. (2011) Statistical estimation of percolation cluster parameters. Proceedings of Voronezh State University. Series: Systems Analysis and Information Technologies, No.1 (January-June), pp.29-35, arXiv:1105.2334v1; in Russian.

See Also

fdc3s, fds2s, fds3s

Examples

# # # # # # # # # # # # # # # # #
# Example 1: Isotropic set cover
# # # # # # # # # # # # # # # # #
pc <- .592746
p1 <- pc - .03
p2 <- pc + .03
lx <- 33; ss <- (lx+1)/2
set.seed(20120627); ac1 <- ssi20(x=lx, p=p1)
set.seed(20120627); ac2 <- ssi20(x=lx, p=p2)
bnd <- isc2s(k=9, x=dim(ac1))
fd1 <- fdc2s(acc=ac1, bnd=bnd)
fd2 <- fdc2s(acc=ac2, bnd=bnd)
n1 <- fd1$model[,"n"]; n2 <- fd2$model[,"n"]
r1 <- fd1$model[,"r"]; r2 <- fd2$model[,"r"]
rr <- seq(min(r1)-.2, max(r1)+.2, length=100)
nn1 <- predict(fd1, newdata=list(r=rr), interval="conf")
nn2 <- predict(fd2, newdata=list(r=rr), interval="conf")
s1 <- paste(round(confint(fd1)[2,], digits=3), collapse=", ")
s2 <- paste(round(confint(fd2)[2,], digits=3), collapse=", ")
x <- y <- seq(lx)
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, y, ac1, cex.main=1,
      main=paste("Isotropic set cover and a 2D cluster of\n",
                 "sites with (1,0)-neighborhood and p=",
                 round(p1, digits=3), sep=""))
rect(bnd["x1",], bnd["y1",], bnd["x2",], bnd["y2",])
abline(h=ss, lty=2); abline(v=ss, lty=2)
image(x, y, ac2, cex.main=1,
      main=paste("Isotropic set cover and a 2D cluster of\n",
                 "sites with (1,0)-neighborhood and p=",
                 round(p2, digits=3), sep=""))
rect(bnd["x1",], bnd["y1",], bnd["x2",], bnd["y2",])
abline(h=ss, lty=2); abline(v=ss, lty=2)
plot(r1, n1, pch=3, ylim=range(c(n1,n2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s1,")", sep=""))
matlines(rr, nn1, lty=c(1,2,2), col=c("black","red","red"))
plot(r2, n2, pch=3, ylim=range(c(n1,n2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s2,")", sep=""))
matlines(rr, nn2, lty=c(1,2,2), col=c("black","red","red"))

## Not run: 
# # # # # # # # # # # # # # # # #
# Example 2: Anisotropic set cover, dir=2
# # # # # # # # # # # # # # # # #
pc <- .592746
p1 <- pc - .03
p2 <- pc + .03
lx <- 33; ss <- (lx+1)/2; ssy <- seq(lx+2, 2*lx-1)
set.seed(20120627); ac1 <- ssi20(x=lx, p=p1, set=ssy, all=FALSE)
set.seed(20120627); ac2 <- ssi20(x=lx, p=p2, set=ssy, all=FALSE)
bnd <- asc2s(k=9, x=dim(ac1), dir=2)
fd1 <- fdc2s(acc=ac1, bnd=bnd)
fd2 <- fdc2s(acc=ac2, bnd=bnd)
n1 <- fd1$model[,"n"]; n2 <- fd2$model[,"n"]
r1 <- fd1$model[,"r"]; r2 <- fd2$model[,"r"]
rr <- seq(min(r1)-.2, max(r1)+.2, length=100)
nn1 <- predict(fd1, newdata=list(r=rr), interval="conf")
nn2 <- predict(fd2, newdata=list(r=rr), interval="conf")
s1 <- paste(round(confint(fd1)[2,], digits=3), collapse=", ")
s2 <- paste(round(confint(fd2)[2,], digits=3), collapse=", ")
x <- y <- seq(lx)
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, y, ac1, cex.main=1,
      main=paste("Anisotropic set cover and a 2D cluster of\n",
                 "sites with (1,0)-neighborhood and p=",
                 round(p1, digits=3), sep=""))
rect(bnd["x1",], bnd["y1",], bnd["x2",], bnd["y2",])
abline(v=ss, lty=2)
image(x, y, ac2, cex.main=1,
      main=paste("Anisotropic set cover and a 2D cluster of\n",
                 "sites with (1,0)-neighborhood and p=",
                 round(p2, digits=3), sep=""))
rect(bnd["x1",], bnd["y1",], bnd["x2",], bnd["y2",])
abline(v=ss, lty=2)
plot(r1, n1, pch=3, ylim=range(c(n1,n2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s1,")", sep=""))
matlines(rr, nn1, lty=c(1,2,2), col=c("black","red","red"))
plot(r2, n2, pch=3, ylim=range(c(n1,n2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s2,")", sep=""))
matlines(rr, nn2, lty=c(1,2,2), col=c("black","red","red"))

## End(Not run)