fdc3s {SECP}

Mass fractal dimension of a 3D cluster

Description

fdc3s() function uses a linear regression model for statistical estimation of the mass fractal dimension of a cluster on 3D square lattice with iso- & isotropic sets cover.

Usage

fdc3s(acc=ssi30(x=95), bnd=isc3s(k=12, x=dim(acc)))

Arguments

acc

an accessibility matrix for 3D 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 3D square lattice is formed from scalable cubes with variable sizes 2r+1 and a fixed point in the lattice center.

The anisotropic set cover on 3D square lattice is formed from scalable cuboids with variable sizes r+1 and a fixed face along the lattice boundary.

The percolation is simulated on 3D 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 3D square lattice with iso- & anisotropic sets cover.

Author(s)

Pavel V. Moskalev

See Also

fdc2s, fds2s, fds3s

Examples

# # # # # # # # # # # # # # # # #
# Example 1: Isotropic set cover
# # # # # # # # # # # # # # # # #
pc <- .311608
p1 <- pc - .02
p2 <- pc + .02
lx <- 33; ss <- (lx+1)/2
set.seed(20120627); ac1 <- ssi30(x=lx, p=p1)
set.seed(20120627); ac2 <- ssi30(x=lx, p=p2)
bnd <- isc3s(k=9, x=dim(ac1))
fd1 <- fdc3s(acc=ac1, bnd=bnd)
fd2 <- fdc3s(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 <- z <- seq(lx)
y1 <- ac1[,ss,]; y2 <- ac2[,ss,]
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, z, y1, cex.main=1,
      main=paste("Isotropic set cover and\n",
                 "a 3D cluster in the y=",ss," slice with\n",
                 "(1,0)-neighborhood and p=",
                 round(p1, digits=3), sep=""))
rect(bnd["x1",], bnd["z1",], bnd["x2",], bnd["z2",])
abline(h=ss, lty=2); abline(v=ss, lty=2)
image(x, z, y2, cex.main=1,
      main=paste("Isotropic set cover and\n",
                 "a 3D cluster in the y=",ss," slice with\n",
                 "(1,0)-neighborhood and p=",
                 round(p2, digits=3), sep=""))
rect(bnd["x1",], bnd["z1",], bnd["x2",], bnd["z2",])
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 1: Anisotropic set cover, dir=3
# # # # # # # # # # # # # # # # #
pc <- .311608
p1 <- pc - .02
p2 <- pc + .02
lx <- 33; ss <- (lx+1)/2
ssz <- seq(lx^2+lx+2, 2*lx^2-lx-1)
set.seed(20120627); ac1 <- ssi30(x=lx, p=p1, set=ssz, all=FALSE)
set.seed(20120627); ac2 <- ssi30(x=lx, p=p2, set=ssz, all=FALSE)
bnd <- asc3s(k=9, x=dim(ac1), dir=3)
fd1 <- fdc3s(acc=ac1, bnd=bnd)
fd2 <- fdc3s(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 <- z <- seq(lx)
y1 <- ac1[,ss,]; y2 <- ac2[,ss,]
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, z, y1, cex.main=1,
      main=paste("Anisotropic set cover and\n",
                 "a 3D cluster in the y=",ss," slice with\n",
                 "(1,0)-neighborhood and p=",
                 round(p1, digits=3), sep=""))
rect(bnd["x1",], bnd["z1",], bnd["x2",], bnd["z2",])
abline(v=ss, lty=2)
image(x, z, y2, cex.main=1,
      main=paste("Anisotropic set cover and\n",
                 "a 3D cluster in the y=",ss," slice with\n",
                 "(1,0)-neighborhood and p=",
                 round(p2, digits=3), sep=""))
rect(bnd["x1",], bnd["z1",], bnd["x2",], bnd["z2",])
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)