# Metric Unfolding¶

## Description¶

`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,...)
## S4 method for signature 'matrix'
unfold(x, ndims=NULL, squared=FALSE, tol=1e-7,
method=c("Schoenemann", "CG"), ...)
## S4 method for signature 'formula'
unfold(x,data=parent.frame(), ...)
## S4 method for signature '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]),
...)
## S4 method for signature '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.

## 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”.

## 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"`

.

## 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)
```