fds3s {SECP}

Mass fractal dimension of sampling 3D clusters

Description

fds3s() function uses a linear regression model for statistical estimation of the mass fractal dimension of sampling clusters on 3D square lattice with iso- & anisotropic sets cover.

Usage

fds3s(rfq=fssi30(x=95), bnd=isc3s(k=12, x=dim(rfq)))

Arguments

rfq

relative sampling frequencies for sites of the percolation lattice.

bnd

bounds for the iso- or anisotropic set cover.

Details

The mass fractal dimension for sampling clusters is equal to the coefficient of linear regression between log(w) and log(r), where w is a relative sampling frequency of the total 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.

Each element of the matrix rfq is equal to the relative frequency with which the 3D square lattice site belongs to a cluster sample.

Value

A linear regression model for statistical estimation of the mass fractal dimension of sampling clusters on 3D square lattice with iso- & anisotropic sets cover.

Author(s)

Pavel V. Moskalev

See Also

fds2s, fdc2s, fdc3s

Examples

# # # # # # # # # # # # # # # # #
# Example 1: Isotropic set cover
# # # # # # # # # # # # # # # # #
pc <- .311608
p1 <- pc - .01
p2 <- pc + .01
lx <- 33; ss <- (lx+1)/2
rf1 <- fssi30(n=100, x=lx, p=p1)
rf2 <- fssi30(n=100, x=lx, p=p2)
bnd <- isc3s(k=9, x=dim(rf1))
fd1 <- fds3s(rfq=rf1, bnd=bnd)
fd2 <- fds3s(rfq=rf2, bnd=bnd)
w1 <- fd1$model[,"w"]; w2 <- fd2$model[,"w"]
r1 <- fd1$model[,"r"]; r2 <- fd2$model[,"r"]
rr <- seq(min(r1)-.2, max(r1)+.2, length=100)
ww1 <- predict(fd1, newdata=list(r=rr), interval="conf")
ww2 <- 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 <- rf1[,ss,]; y2 <- rf2[,ss,]
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, z, y1, zlim=c(0, 3*mean(y1)), cex.main=1,
      main=paste("Isotropic set cover and a 3D clusters\n",
                 "frequency 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, zlim=c(0, 3*mean(y2)), cex.main=1,
      main=paste("Isotropic set cover and a 3D clusters\n",
                 "frequency 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, w1, pch=3, ylim=range(c(w1,w2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s1,")", sep=""))
matlines(rr, ww1, lty=c(1,2,2), col=c("black","red","red"))
plot(r2, w2, pch=3, ylim=range(c(w1,w2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s2,")", sep=""))
matlines(rr, ww2, lty=c(1,2,2), col=c("black","red","red"))

## Not run: 
# # # # # # # # # # # # # # # # #
# Example 2: Anisotropic set cover, dir=3
# # # # # # # # # # # # # # # # #
pc <- .311608
p1 <- pc - .01
p2 <- pc + .01
lx <- 33; ss <- (lx+1)/2
ssz <- seq(lx^2+lx+2, 2*lx^2-lx-1)
rf1 <- fssi30(n=100, x=lx, p=p1, set=ssz, all=FALSE)
rf2 <- fssi30(n=100, x=lx, p=p2, set=ssz, all=FALSE)
bnd <- asc3s(k=9, x=dim(rf1), dir=3)
fd1 <- fds3s(rfq=rf1, bnd=bnd)
fd2 <- fds3s(rfq=rf2, bnd=bnd)
w1 <- fd1$model[,"w"]; w2 <- fd2$model[,"w"]
r1 <- fd1$model[,"r"]; r2 <- fd2$model[,"r"]
rr <- seq(min(r1)-.2, max(r1)+.2, length=100)
ww1 <- predict(fd1, newdata=list(r=rr), interval="conf")
ww2 <- 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 <- rf1[,ss,]; y2 <- rf2[,ss,]
par(mfrow=c(2,2), mar=c(3,3,3,1), mgp=c(2,1,0))
image(x, z, y1, zlim=c(0, .3), cex.main=1,
      main=paste("Anisotropic set cover and a 3D clusters\n",
                 "frequency 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, zlim=c(0, .3), cex.main=1,
      main=paste("Anisotropic set cover and a 3D clusters\n",
                 "frequency 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, w1, pch=3, ylim=range(c(w1,w2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s1,")", sep=""))
matlines(rr, ww1, lty=c(1,2,2), col=c("black","red","red"))
plot(r2, w2, pch=3, ylim=range(c(w1,w2)), cex.main=1,
     main=paste("0.95 confidence interval for the mass\n",
                "fractal dimension is (",s2,")", sep=""))
matlines(rr, ww2, lty=c(1,2,2), col=c("black","red","red"))

## End(Not run)