LOE: LOE for a unweighted Graph.
In loe: Local Ordinal Embedding

Description Usage Arguments Value Author(s) Examples

Performs LOE for a given unweighted grah.

1 2	LOE(ADM, p=2, c=0.1,eps= 1e-5,maxit =1000,method=c("BFGS","SD","MM"), iniX = "auto",report=100,DEL=1,H=0.5)

`ADM`	The adjacency matrix.
`p`	Number of dimensions.
`c`	Scale parameter which only takes strictly positive value.
`eps`	Convergence criterion for the majorization algorithm or the steepest descent algorithm.
`maxit`	Maximum number of iteretions.
`method`	If "BFGS", then the BFGS method is used for optimizing the stress function. If "SD", then the steepest descent method is used. If "MM", then the majorization minimization algortihm is used.
`iniX`	Matrix with starting values for embedding (optional). If "auto", then Laplacian eigenmaps is used as a starting value.
`report`	The frequency of reports. Defaults to every 100 iterations.
`DEL`	The initial step size in the steepest descent algorithm. Defaults to 1.
`H`	The rate parameter of the backtracking line search in the steepest descent algorithm. This parameter only takes value in (0,1). Defaults to 0.5.

LOE returns a list with components:

`X`	The best corrdinate matrix with p columns whose rows give the coordinates of the vertexes.
`str`	If `method` is "BFGS", then the value of the stress function of LOE corresponding to `X` is returned. If "SD" or "MM", then the vector of values on each iteration is returned.

Yoshikazu Terada

#################
#Realizable case#
#################
library(igraph)
###############################
#Create a toy data
x <- seq(-5,5,by=1)
y <- seq(1,6,by=1)
hx1 <- seq(-3.5,-1.5,by=0.5)
hx2 <- seq(1.5,3.5,by=0.5)
hy <- seq(2.5,4.5,by=0.5)
D1 <- matrix(0,66,2)
for(i in 1:11){
	for(j in 1:6){
		D1[i+11*(j-1),] <- c(x[i],y[j])
	}
}
D2n <- matrix(0,25,2)
D2p <- matrix(0,25,2)
for(i in 1:5){
	for(j in 1:5){
		D2n[i+5*(j-1),] <- c(hx1[i],hy[j])
		D2p[i+5*(j-1),] <- c(hx2[i],hy[j])
	}
}
D2n <- D2n[-c(7,9,17,19),]
D2p <- D2p[-c(7,9,17,19),]
Data <- rbind(D1,D2n,D2p)
Data <- scale(Data[order(Data[,1]),], scale=FALSE)
N <- nrow(Data)
#Visualization of the original data
plot(Data,pch=20,xlab="",ylab="",cex=1,col=rainbow(N,start=.7,end=.9),
xlim=c(-7,7),ylim=c(-7,7),main="Original data")

#Creating a k-NN graph based on Data
DM <- as.matrix(dist(Data))
ADM <- make.kNNG(DM,k=25)

#plot of the adjacency matrix
AADM <- ADM
diag(AADM) <- NA
image(AADM[N:1,],col=topo.colors(3),ann=FALSE,axes=FALSE)

#Apply some graph embedding methods
LE <-spec.emb(A=ADM,2,norm=FALSE)
result.LOE <- LOE(ADM=ADM,p=2,c=0.1,method="BFGS",report=1000,iniX=LE)

#Procrustes transform
library(vegan)
LOEX <- procrustes(X=Data,Y=result.LOE$X)$Yrot
plot(LOEX,pch=20,xlab="",ylab="",cex=1,col=rainbow(N,start=.7,end=.9),
xlim=c(-7,7),ylim=c(-7,7),main="LOE")
###############################

#############
#This function provide appropriate vectors of xlim and ylim
#for given embedding matrix X.
#############
make.lim <- function(X){
			mima <- matrix(0, 2,2)
			mima[,1] <- apply(X, 2, min)
			mima[,2] <- apply(X, 2, max)
			han <- mima[,2] - mima[,1]
			cent <-  (mima[,2] + mima[,1])/2
			tmpr <- max(han)+max(han)*0.05
			for(s in 1:2){
				mima[s,] <- c(cent[s]-tmpr/2,cent[s]+tmpr/2)
			}
			return(mima)
		}
##############################
#Standered graph-drawing task#
##############################
###############################
ADM <- as.matrix( get.adjacency(graph.famous("Thomassen")) )
TG <- graph.adjacency(ADM)

#Apply some graph embedding methods
LE <-spec.emb(A=ADM,2,norm=FALSE)
KK <- layout.kamada.kawai(TG,maxiter=1000,start=LE)
FR <- layout.fruchterman.reingold(TG,maxiter=1000,start=LE)
result.LOE <- LOE(ADM=ADM,p=2,c=0.1,method="MM",report=1000,maxit=2000)


#Visualization of embeddings
par(mfrow=c(2,3),oma = c(0, 0, 4, 0))
#plot of the adjacency matrix
	AADM <- ADM
	N <- nrow(AADM)
	diag(AADM) <- NA
	image(AADM[N:1,],col=topo.colors(3),ann=FALSE,axes=FALSE)
#plot of Laplacian eigenmaps
	tmplim <- make.lim(LE)
	vsize <- (tmplim[1,2] -tmplim[1,1])*4
	plot(TG, layout=LE, main="Laplacian eigenmaps",
		vertex.size=vsize,vertex.color="blue",vertex.label.color="white",vertex.label=NA,
		edge.arrow.size=0.1,xlim=tmplim[1,],ylim=tmplim[2,],axes=TRUE,rescale=FALSE)
#plot of Kamada and Kawai
	tmplim <- make.lim(KK)
	vsize <- (tmplim[1,2] -tmplim[1,1])*4
	plot(TG, layout=KK, main="Kamada and Kawai",
		vertex.size=vsize,vertex.color="blue",vertex.label.color="white",vertex.label=NA,
		edge.arrow.size=0.1,xlim=tmplim[1,],ylim=tmplim[2,],axes=TRUE,rescale=FALSE)
#plot of Fruchterman Reingold
	tmplim <- make.lim(FR)
	vsize <- (tmplim[1,2] -tmplim[1,1])*4
	plot(TG, layout=FR, main="Fruchterman Reingold",
		vertex.size=vsize,vertex.color="blue",vertex.label.color="white",vertex.label=NA,
		edge.arrow.size=0.1,xlim=tmplim[1,],ylim=tmplim[2,],axes=TRUE,rescale=FALSE)
#plot of LOE
	tmplim <- make.lim(result.LOE$X)
	vsize <- (tmplim[1,2] -tmplim[1,1])*4
	plot(TG, layout=result.LOE$X, main="LOE",
		vertex.size=vsize,vertex.color="blue",vertex.label.color="white",vertex.label=NA,
		edge.arrow.size=0.1,xlim=tmplim[1,],ylim=tmplim[2,],axes=TRUE,rescale=FALSE)
	plot(result.LOE$str,type="l",xlab="Number of iter.", ylab="Stress")
#Make title
	mtext(side = 3, line=1, outer=TRUE, text = "Thomassen", cex=2)
###############################

Loading required package: MASS

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union

initial  value 590.780031 
final  value 0.000000 
converged
Loading required package: permute

Attaching package: 'permute'

The following object is masked from 'package:igraph':

    permute

Loading required package: lattice
This is vegan 2.4-4

Attaching package: 'vegan'

The following object is masked from 'package:igraph':

    diversity

initial stress= 7.531723 
iter 1000 stress = 3.078888 
iter (final) 1494 stress = 3.068628 
converged