em.saltykov: Expectation Maximization (EM) algorithm
In unfoldr: Stereological Unfolding for Spheroidal Particles
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
Estimation of the empirical sphere diameter distribution
Usage
1
em.saltykov(y, bin, maxIt = 32)
Arguments
y
vector of observed absolute frequencies of circle diameters
bin
non-decreasing vector of class limits
maxIt
maximum number of iterations to be used
Details
The function performs the EM algorithm, see reference below.
Value
vector of count data of absolute frequenties of sphere diameters
Author(s)
M. Baaske
References
Ohser, J. and Muecklich, F. Statistical analysis of microstructures in materials science J. Wiley & Sons, 2000
Examples
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## Not run:
## Comment: Simulate a Poisson sphere system,
## intersect, discretize and display results
library(unfoldr)
library(rgl)
library(plotrix)
spheres <- function(spheres, box=NULL, draw.box=FALSE, draw.axis=FALSE, ...) {
xyz <- apply(sapply(spheres, "[[", "center"),1,function(x) x)
sizes <- unlist(lapply(spheres,function(x) x$r))
spheres3d(xyz,radius=sizes,...)
if(draw.axis) {
axes3d(c('x','y','z'), pos=c(1,1,1), tick=FALSE)
#title3d('','','x','y','z')
}
axes3d(edges = "bbox",labels=TRUE,tick=FALSE,box=TRUE,nticks=0,
expand=1.0,xlen=0,xunit=0,ylen=0,yunit=0,zlen=0,zunit=0)
if(draw.box) {
x <- box$xrange[2]
y <- box$yrange[2]
z <- box$zrange[2]
c3d.origin <- rgl::translate3d(rgl::scale3d(rgl::cube3d(col="gray", alpha=0.1),
x/2,y/2,z/2),x/2,y/2,z/2)
rgl::shade3d(c3d.origin)
}
}
col <- c("#0000FF","#00FF00","#FF0000","#FF00FF","#FFFF00","#00FFFF")
#################################################################
## `beta` distribution for radii
#################################################################
lam <- 50
## parameter beta distribution (radii)
theta <- list("size"=list("shape1"=1,"shape2"=10))
# simulation bounding box
box <- list("xrange"=c(-1,4),"yrange"=c(-1.5,3.5),"zrange"=c(0,2))
## simulate and return full spheres system
## intersect with XZ plane and return full list of intersection profiles
## section plane xy
S <- simPoissonSystem(theta,lam,size="rbeta",box=box,type="spheres",
intersect="full",n=c(0,0,1),dz=0,pl=1)
# check resulting distribution
length(S$S)
summary(sapply(S$S,"[[","r"))
theta$size[[1]]/(theta$size[[1]]+theta$size[[2]]) # mean
## interior spheres:
## the ones which intersect one of the lateral planes (without top/bottom planes)
## showing spheres with color intersect
notIn <- sapply(S$S,function(x) !attr(x,"interior"))
spheres(S$S[notIn],box,FALSE,TRUE,color=col)
# not intersecting
In <- sapply(S$S,function(x) attr(x,"interior"))
spheres(S$S[In],box,color="gray")
## full sphere system
# open3d()
# spheres(S$S,box,FALSE,TRUE,color=col)
# planes3d(0,0,1,0,col="darkgray",alpha=1)
## draw intersections
sp <- S$sp
id <- sapply(sp,"[[","id")
open3d()
spheres(S$S[id],box,FALSE,TRUE,color=col)
planes3d(0,0,1,0,col="darkgray",alpha=1)
XYr <- t(sapply(sp,function(s) cbind(s$center[1],s$center[2],s$r)))
# centers
x <- XYr[,1]
y <- XYr[,2]
r <- XYr[,3]
win <- attr(sp,"win")
xlim <- win[[1]]
ylim <- win[[2]]
dev.new()
plot(x,y,type="n",xaxs="i", yaxs="i", xlab="x",ylab="y",xlim=xlim,ylim=ylim)
for(i in 1:nrow(XYr))
draw.circle(x[i],y[i],r[i],nv=100,border=NULL,col="black")
## digitize inersections
## `win` can also be omitted if not different
## from original itersection window (derived from the box)
W <- digitizeProfiles(sp, delta=0.01, win=win)
dim(W)
dev.new()
image(1:nrow(W),1:ncol(W),W,col=gray(1:0))
#################################################################
## Exact simulation of spheres with log normal radii
#################################################################
lam <- 100
## parameter rlnorm distribution (radii)
theta <- list("size"=list("meanlog"=-2.5,"sdlog"=0.5))
# simulation bounding box
box <- list("xrange"=c(0,5),"yrange"=c(0,5),"zrange"=c(0,5))
## simulate only 3D system
S <- simPoissonSystem(theta,lam,size="rlnorm",box=box,type="spheres",
intersect="original", perfect=TRUE, pl=101)
## show
open3d()
spheres(S[1:2000],box,TRUE,TRUE,color=col)
## check!
mean(log(sapply(S,"[[","r")))
sd(log(sapply(S,"[[","r")))
#################################################################
## Planar section
#################################################################
# planar section of exact simulated `rlnorm` sphere system:
# returns diameters for those section profiles
# which have their centers inside the intersection window
sp <- planarSection(S,d=2.5,intern=TRUE,pl=1)
hist(sp)
summary(sp)
mean(log(sp/2))
sd(log(sp/2))
#################################################################
## General intersection, all objects (inter=FALSE)
#################################################################
SP <- intersectSystem(S, 2.5, n=c(0,0,1), intern=FALSE, pl=1)
## show in 3D
id <- sapply(SP,"[[","id")
open3d()
spheres(S[id],box,TRUE,color=col)
planes3d(0,0,-1,2.5,col="black",alpha=1)
## 2D sections
XYr <- t(sapply(SP,function(s) cbind(s$center[1],s$center[2],s$r)))
# centers
x <- XYr[,1]
y <- XYr[,2]
r <- XYr[,3]
xlim <- box$xrange
ylim <- box$yrange
dev.new()
plot(x,y,type="n",xaxs="i", yaxs="i", xlab="x",ylab="y",xlim=xlim,ylim=ylim)
for(i in 1:nrow(XYr))
draw.circle(x[i],y[i],r[i],nv=100,border=NULL,col="black")
# digitize
W <- digitizeProfiles(SP, delta=0.01)
dev.new()
image(1:nrow(W),1:ncol(W),W,col=gray(1:0))
#################################################################
## Unfolding
#################################################################
ret <- unfold(sp,nclass=25)
## Point process intensity
cat("Intensities: ", sum(ret$N_V)/25, "vs.",lam,"\n")
## original diameters
r3d <- unlist(lapply(S,function(x) 2*x$r))
rest3d <- unlist(lapply(2:(length(ret$breaks)),
function(i) rep(ret$breaks[i],sum(ret$N_V[i-1]))))
op <- par(mfrow = c(1, 2))
hist(r3d[r3d<=max(ret$breaks)], breaks=ret$breaks, main="Radius 3D",
freq=FALSE, col="gray",xlab="r")
hist(rest3d, breaks=ret$breaks,main="Radius estimated",
freq=FALSE, col="gray", xlab="r")
par(op)
#################################################################
## Update intersection: find objects which intersect bounding box
#################################################################
idx <- updateIntersections(S)
sum(!idx) # objects intersecting
id <- which( idx != 1)
open3d()
spheres(S[id],box,TRUE,TRUE,color=col)
#################################################################
## user-defined simulation function
#################################################################
lam <- 50
theta <- list("p1"=-2.5,"p2"=0.5)
myfun <- function(p1,p2) { c("r"=rlnorm(1,p1,p2)) }
box <- list("xrange"=c(0,5),"yrange"=c(0,5),"zrange"=c(0,5))
S <- simPoissonSystem(theta,lam,rjoint=myfun,box=box,type="spheres",pl=1)
r <- sapply(S,"[[","r")
mean(log(r))
sd(log(r))
open3d()
spheres(S,box,TRUE,TRUE,color=col)
## End(Not run)
Example output
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl.init' failed, running with 'rgl.useNULL = TRUE'.
Attaching package: ‘plotrix’
The following object is masked from ‘package:rgl’:
mtext3d
Size parameters: mx=1.000000, sdx=10.000000
Spheres simulation with `rbeta` (perfect=0):
Mean number: 50.000000 (Box volume 50.000000)
Set label 'N'.
Done. Simulated 2534 spheres.
Converting spheres ...
Converting 108 discs.
Getting plane indices: [0 1 ]
[1] 2534
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000087 0.0277244 0.0677158 0.0914636 0.1312227 0.5309968
[1] 0.09090909
null
2
[1] 500 500
dev.new(): using pdf(file="Rplots1.pdf")
Size parameters: mx=-2.500000, sdx=0.500000
Cumulative sum of probabilities: 0.896168, 0.996196, 0.999949, 1.000000 ( mu=139.482809 )
Spheres simulation with `rlnorm` (perfect=1):
Mean number: 100.000000 (exact simulation: 139.482809)
Set label 'N'.
perfect
Done. Simulated 13978 spheres.
Converting spheres ...
null
3
[1] -2.473883
[1] 0.5009234
Converting 486 discs.
Getting plane indices: [0 1 ]
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.005738 0.102507 0.160404 0.179879 0.234117 0.658794
[1] -2.580125
[1] 0.6209039
Converting 542 discs.
Getting plane indices: [0 1 ]
null
4
dev.new(): using pdf(file="Rplots2.pdf")
dev.new(): using pdf(file="Rplots3.pdf")
Intensities: 105.895 vs. 100
Warning: stack imbalance in '<-', 2 then 1
[1] 2740
null
5
Spheres simulation by `myfun`
Box volume: 125.000000, lam: 50.000000, number of spheres: 6100
Done. Simulated 6100 spheres.
Converting spheres ...
[1] -2.497105
[1] 0.4907834
null
6
unfoldr documentation built on Sept. 25, 2021, 1:07 a.m.
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Description Usage Arguments Details Value Author(s) References Examples
Description
Estimation of the empirical sphere diameter distribution
Usage
1 | em.saltykov(y, bin, maxIt = 32)
|
Arguments
y |
vector of observed absolute frequencies of circle diameters |
bin |
non-decreasing vector of class limits |
maxIt |
maximum number of iterations to be used |
Details
The function performs the EM algorithm, see reference below.
Value
vector of count data of absolute frequenties of sphere diameters
Author(s)
M. Baaske
References
Ohser, J. and Muecklich, F. Statistical analysis of microstructures in materials science J. Wiley & Sons, 2000
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 | ## Not run:
## Comment: Simulate a Poisson sphere system,
## intersect, discretize and display results
library(unfoldr)
library(rgl)
library(plotrix)
spheres <- function(spheres, box=NULL, draw.box=FALSE, draw.axis=FALSE, ...) {
xyz <- apply(sapply(spheres, "[[", "center"),1,function(x) x)
sizes <- unlist(lapply(spheres,function(x) x$r))
spheres3d(xyz,radius=sizes,...)
if(draw.axis) {
axes3d(c('x','y','z'), pos=c(1,1,1), tick=FALSE)
#title3d('','','x','y','z')
}
axes3d(edges = "bbox",labels=TRUE,tick=FALSE,box=TRUE,nticks=0,
expand=1.0,xlen=0,xunit=0,ylen=0,yunit=0,zlen=0,zunit=0)
if(draw.box) {
x <- box$xrange[2]
y <- box$yrange[2]
z <- box$zrange[2]
c3d.origin <- rgl::translate3d(rgl::scale3d(rgl::cube3d(col="gray", alpha=0.1),
x/2,y/2,z/2),x/2,y/2,z/2)
rgl::shade3d(c3d.origin)
}
}
col <- c("#0000FF","#00FF00","#FF0000","#FF00FF","#FFFF00","#00FFFF")
#################################################################
## `beta` distribution for radii
#################################################################
lam <- 50
## parameter beta distribution (radii)
theta <- list("size"=list("shape1"=1,"shape2"=10))
# simulation bounding box
box <- list("xrange"=c(-1,4),"yrange"=c(-1.5,3.5),"zrange"=c(0,2))
## simulate and return full spheres system
## intersect with XZ plane and return full list of intersection profiles
## section plane xy
S <- simPoissonSystem(theta,lam,size="rbeta",box=box,type="spheres",
intersect="full",n=c(0,0,1),dz=0,pl=1)
# check resulting distribution
length(S$S)
summary(sapply(S$S,"[[","r"))
theta$size[[1]]/(theta$size[[1]]+theta$size[[2]]) # mean
## interior spheres:
## the ones which intersect one of the lateral planes (without top/bottom planes)
## showing spheres with color intersect
notIn <- sapply(S$S,function(x) !attr(x,"interior"))
spheres(S$S[notIn],box,FALSE,TRUE,color=col)
# not intersecting
In <- sapply(S$S,function(x) attr(x,"interior"))
spheres(S$S[In],box,color="gray")
## full sphere system
# open3d()
# spheres(S$S,box,FALSE,TRUE,color=col)
# planes3d(0,0,1,0,col="darkgray",alpha=1)
## draw intersections
sp <- S$sp
id <- sapply(sp,"[[","id")
open3d()
spheres(S$S[id],box,FALSE,TRUE,color=col)
planes3d(0,0,1,0,col="darkgray",alpha=1)
XYr <- t(sapply(sp,function(s) cbind(s$center[1],s$center[2],s$r)))
# centers
x <- XYr[,1]
y <- XYr[,2]
r <- XYr[,3]
win <- attr(sp,"win")
xlim <- win[[1]]
ylim <- win[[2]]
dev.new()
plot(x,y,type="n",xaxs="i", yaxs="i", xlab="x",ylab="y",xlim=xlim,ylim=ylim)
for(i in 1:nrow(XYr))
draw.circle(x[i],y[i],r[i],nv=100,border=NULL,col="black")
## digitize inersections
## `win` can also be omitted if not different
## from original itersection window (derived from the box)
W <- digitizeProfiles(sp, delta=0.01, win=win)
dim(W)
dev.new()
image(1:nrow(W),1:ncol(W),W,col=gray(1:0))
#################################################################
## Exact simulation of spheres with log normal radii
#################################################################
lam <- 100
## parameter rlnorm distribution (radii)
theta <- list("size"=list("meanlog"=-2.5,"sdlog"=0.5))
# simulation bounding box
box <- list("xrange"=c(0,5),"yrange"=c(0,5),"zrange"=c(0,5))
## simulate only 3D system
S <- simPoissonSystem(theta,lam,size="rlnorm",box=box,type="spheres",
intersect="original", perfect=TRUE, pl=101)
## show
open3d()
spheres(S[1:2000],box,TRUE,TRUE,color=col)
## check!
mean(log(sapply(S,"[[","r")))
sd(log(sapply(S,"[[","r")))
#################################################################
## Planar section
#################################################################
# planar section of exact simulated `rlnorm` sphere system:
# returns diameters for those section profiles
# which have their centers inside the intersection window
sp <- planarSection(S,d=2.5,intern=TRUE,pl=1)
hist(sp)
summary(sp)
mean(log(sp/2))
sd(log(sp/2))
#################################################################
## General intersection, all objects (inter=FALSE)
#################################################################
SP <- intersectSystem(S, 2.5, n=c(0,0,1), intern=FALSE, pl=1)
## show in 3D
id <- sapply(SP,"[[","id")
open3d()
spheres(S[id],box,TRUE,color=col)
planes3d(0,0,-1,2.5,col="black",alpha=1)
## 2D sections
XYr <- t(sapply(SP,function(s) cbind(s$center[1],s$center[2],s$r)))
# centers
x <- XYr[,1]
y <- XYr[,2]
r <- XYr[,3]
xlim <- box$xrange
ylim <- box$yrange
dev.new()
plot(x,y,type="n",xaxs="i", yaxs="i", xlab="x",ylab="y",xlim=xlim,ylim=ylim)
for(i in 1:nrow(XYr))
draw.circle(x[i],y[i],r[i],nv=100,border=NULL,col="black")
# digitize
W <- digitizeProfiles(SP, delta=0.01)
dev.new()
image(1:nrow(W),1:ncol(W),W,col=gray(1:0))
#################################################################
## Unfolding
#################################################################
ret <- unfold(sp,nclass=25)
## Point process intensity
cat("Intensities: ", sum(ret$N_V)/25, "vs.",lam,"\n")
## original diameters
r3d <- unlist(lapply(S,function(x) 2*x$r))
rest3d <- unlist(lapply(2:(length(ret$breaks)),
function(i) rep(ret$breaks[i],sum(ret$N_V[i-1]))))
op <- par(mfrow = c(1, 2))
hist(r3d[r3d<=max(ret$breaks)], breaks=ret$breaks, main="Radius 3D",
freq=FALSE, col="gray",xlab="r")
hist(rest3d, breaks=ret$breaks,main="Radius estimated",
freq=FALSE, col="gray", xlab="r")
par(op)
#################################################################
## Update intersection: find objects which intersect bounding box
#################################################################
idx <- updateIntersections(S)
sum(!idx) # objects intersecting
id <- which( idx != 1)
open3d()
spheres(S[id],box,TRUE,TRUE,color=col)
#################################################################
## user-defined simulation function
#################################################################
lam <- 50
theta <- list("p1"=-2.5,"p2"=0.5)
myfun <- function(p1,p2) { c("r"=rlnorm(1,p1,p2)) }
box <- list("xrange"=c(0,5),"yrange"=c(0,5),"zrange"=c(0,5))
S <- simPoissonSystem(theta,lam,rjoint=myfun,box=box,type="spheres",pl=1)
r <- sapply(S,"[[","r")
mean(log(r))
sd(log(r))
open3d()
spheres(S,box,TRUE,TRUE,color=col)
## End(Not run)
|
Example output
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl.init' failed, running with 'rgl.useNULL = TRUE'.
Attaching package: ‘plotrix’
The following object is masked from ‘package:rgl’:
mtext3d
Size parameters: mx=1.000000, sdx=10.000000
Spheres simulation with `rbeta` (perfect=0):
Mean number: 50.000000 (Box volume 50.000000)
Set label 'N'.
Done. Simulated 2534 spheres.
Converting spheres ...
Converting 108 discs.
Getting plane indices: [0 1 ]
[1] 2534
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000087 0.0277244 0.0677158 0.0914636 0.1312227 0.5309968
[1] 0.09090909
null
2
[1] 500 500
dev.new(): using pdf(file="Rplots1.pdf")
Size parameters: mx=-2.500000, sdx=0.500000
Cumulative sum of probabilities: 0.896168, 0.996196, 0.999949, 1.000000 ( mu=139.482809 )
Spheres simulation with `rlnorm` (perfect=1):
Mean number: 100.000000 (exact simulation: 139.482809)
Set label 'N'.
perfect
Done. Simulated 13978 spheres.
Converting spheres ...
null
3
[1] -2.473883
[1] 0.5009234
Converting 486 discs.
Getting plane indices: [0 1 ]
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.005738 0.102507 0.160404 0.179879 0.234117 0.658794
[1] -2.580125
[1] 0.6209039
Converting 542 discs.
Getting plane indices: [0 1 ]
null
4
dev.new(): using pdf(file="Rplots2.pdf")
dev.new(): using pdf(file="Rplots3.pdf")
Intensities: 105.895 vs. 100
Warning: stack imbalance in '<-', 2 then 1
[1] 2740
null
5
Spheres simulation by `myfun`
Box volume: 125.000000, lam: 50.000000, number of spheres: 6100
Done. Simulated 6100 spheres.
Converting spheres ...
[1] -2.497105
[1] 0.4907834
null
6
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