Metric Unfolding

unfold computes a metric unfolding solution based on a rectangular matrix, that is, reconstructs two sets of points from the distances between points of the first set and the points of the second set.

uapply applies a function the two point sets that are reconstructed by unfold.

Usage

unfold(x,...)

# S3 method for class 'matrix'
unfold(x, ndims=NULL, squared=FALSE, tol=1e-7,
          method=c("Schoenemann", "CG"), ...)

# S3 method for class 'formula'
unfold(x,data=parent.frame(), ...)

# S3 method for class 'unfolding'
biplot(x, dimen=c(1,2), type=attr(x,"biplot_type"),
  xlim, ylim, tpos=c(4,2), tposdim=1,
  asp=1, lty=c(1,2), lwd=c(1,1), pch=c(1,3), cex=c(1,1),
  col=c("black","black"), contour.col="black", contour.lty=1,
  xlab=paste("Dimension ",dimen[1]),
  ylab=paste("Dimension ",dimen[2]),
  ...)

# S3 method for class 'unfolding'
plot(x, y=NULL ,dimen=1, discrete=attr(x,"plot_discrete"),
  use.rownames=discrete, xlab=paste("Dimension ",dimen), ...)

uapply(x,FUN)

Arguments

x

for unfold.matrix: a rectangular matrix that contains distances or squared distances (if argument squared is TRUE). For unfold.formula: a formula which specifies the variables that form the columns of the matrix of distances. For biplot.unfolding and plot.unfolding: an object that contains an unfolding solution.

data

a data frame or an environment that contains variables specified in the formula given as first argument.

ndims

an optional integer value that specifies the dimensionality of the solution. If NULL the dimensionality is selected automatically based on a singular value decomposition of the matrix of squared distances.

squared

a logical value; does the matrix D contain squared distances?

tol

a tolerance value for the convergence of the conjugate gradients method.

method

a method for the iterative computation of the unfolding solution.

y

a dummy argument for compatibility with default methods, ignored.

dimen

for biplot: a two-element integer vector, for plot: a single integer value, that specifies the dimension(s) of the unfolding solution to be plotted.

type

a character vector of length less then or equal to 2. Determines how each of the two point sets of the unfolding solutions are represented in the biplot. Valid choices are

"points"

the respective set of points are plotted as points in the biplot.

"lines"

the points of the respective set are connected by lines.

"both"

the points of the respective set are plotted as points and connected by lines.

"text"

the points of the respective set are represented by the corresponding row names and, if argument tpos is present, by points.

"density"

contour lines are drawn of two-dimensional kernel density estimate for the respective set of points. This biplot type uses the function kde2d of library MASS.

tpos

a two-element integer vector; specifies the position of text labels relative to the points. For the meaning of these integer values see text

tposdim

an integer value; specifies which how elements of tpos are used. Labels of points with negative positions along coordinate axis dimen[tposdim] are positioned according to tpos[1], labels of other points are positioned according to tpos[1].

xlab, ylab, xlim, ylim, asp, lty, lwd, pch, cex, col

arguments passed to base graphics functions

.

contour.col, contour.lty

colour and line type for contour lines, see contour.

discrete

a logical vector of lenght 2; if TRUE, the respective set of points are represented by spikes in theplot, otherwise the set is represented by a graph of a kernel density estimate.

use.rownames

logical; should row names used for annotation?

...

further arguments passed to optim in case of unfold or points in case of the plotting methods.

FUN

a function applied to the two sets of points that result from the unfolding.

Details

unfold first computes an unfolding solution according to Schoenemanns metric unfolding algorithm that uses only linear algebra operations. This preliminary solution is then refined by minimizing the stress using a conjugate-gradients method.

uapply applies a given function to the two sets of points recovered by an unfolding solution. It applies the function to the components A and B of an object of class "unfolding".

Value

unfold returns an object of class "unfolding" with components

A

a numeric matrix representing the first set of points. Each row contains the coordinate of one point of the first set.

B

a numeric matrix representing the second set of points. Each row contains the coordinate of one point of the second set.

fitted

a numeric matrix that contains the fitted squared distances.

stress

A stress value, denotes the "badness of fit".

Examples

r <- seq(from=0,to=2*pi,length=24)
a1 <- cos(r)*4 + 0.00001*rnorm(r)
a2 <- sin(r)*4 + 0.00001*rnorm(r)
b1 <- c(.5,-.5,-.5,.5)*3 + 5
b2 <- c(.5,.5,-.5,-.5)*3 + 1

D1 <- outer(b1,a1,"-")
D2 <- outer(b2,a2,"-")

Dsq <- D1^2+D2^2


Dsq.uf<-unfold(sqrt(Dsq),squared=FALSE)

oldpar <- par(mfrow=c(1,2))
A <- cbind(a1,a2)
B <- cbind(b1,b2)

ltype <- c(rep(1,NROW(A)),rep(2,NROW(A)))

orig <- rbind(A,B)
unfolded <- rbind(Dsq.uf$A,Dsq.uf$B)

xlim <- ylim <- range(orig)#*1.5

plot(A,type="b",pch=1,
    xlim=xlim,ylim=ylim,
    xlab="Dimension 1",ylab="Dimension 2",main=expression("Original data"),asp=1)
lines(B,type="b",pch=3,lty=2)
abline(h=0,v=0,lty=3)

biplot(Dsq.uf,type="b",
    xlim=xlim,ylim=ylim,
    main=expression(paste(italic(unfold)," solution")),asp=1)



par(oldpar)