Procrustes: Procrustean Similarity Transformations

In smacof: Multidimensional Scaling

Procrustean Similarity Transformations

Description

Solves the Procrustean problem of fitting one (MDS) configuration (testee) to another (target) MDS configuration.

Usage

Procrustes(X, Y)

## S3 method for class 'procr'
plot(x, plot.type = "jointplot", plot.dim = c(1,2), main, xlab, ylab, 
                     xlim, ylim, asp = 1, pch = 20, col.X = "cadetblue", 
                     col.Y = "gray", col.Yhat = "coral1", 
                     label.conf = list(label = TRUE, pos = 3, cex = 0.8), 
                     arrows = TRUE, length = 0.10, 
                     legend = list(plot = TRUE, labels = c("Target", "Testee"), 
                                   pos = "bottomright"), ...) 

Arguments

X	Target configuration
Y	Testee configuration
x	Object of class procr
plot.type	Either "jointplot" or "transplot"
plot.dim	Vector with dimensions to be plotted.
main	Plot title.
xlab	Label of x-axis.
ylab	Label of y-axis.
xlim	Scale x-axis.
ylim	Scale y-axis.
pch	Plot symbol.
asp	Aspect ratio.
col.X	Color target configuration.
col.Y	Color testee configuration.
col.Yhat	Color transformed configuration.
label.conf	List with arguments for plotting the labels of the configurations in a configuration plot (logical value whether to plot labels or not, label position, label color).
length	length of the edges of the arrow head (in inches).
arrows	For "transplot" only, whether arrows should be plotted or not.
legend	List with arguments for plotting the legend.
...	Additional plot arguments.

Details

Y is going to be modified by finding an optimal dilation factor, an optimal translation and rotation for Y such that it is a similar as possible to X. X remains untouched.

Value

Returns an object of class procr with:

X	Input target configuration
Y	Input testee configuration
Yhat	Procrustes transformed (fitted) configuration
translation	Translation vector
dilation	Dilation factor
rotation	Rotation-reflection matrix
confdistX	Configuration distances X
confdistY	Configuration distances Y
confdistYhat	Configuration distances of fitted configuration
congcoef	Congruence coefficient
aliencoef	Alienation coefficient
r	Correlation coefficient between X and Y-hat (shrinkage ratio)
pairdist	Pairwise object distances (sorted)

References

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling and Unfolding (2nd ed.). Springer.

Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. doi: 10.18637/jss.v102.i10

Borg, I. and Mair, P. (2022). A note on Procrustean fittings of noisy configurations. Austrian Journal of Statistics, 51, 1-9. doi: 10.17713/ajs.v51i4.1423

Examples

## artificial example:
X <- matrix(c(1, -1, -1, 1, 2, 2, -2, -2), ncol = 2)
Y <- matrix(c(0.07, 0.93, 1.93, 1.07, 2.62, 3.12, 1.38, 0.88), ncol = 2)
op <- par(mfrow = c(1,2))
plot(X[,1], X[,2], xlim = c(-3, 3), ylim = c(-2, 3.5), asp = 1, xlab = "", ylab = "")
rect(-1, -2, 1, 2)
points(Y[,1], Y[,2], xlim = c(-3, 3), col = "gray")
polygon(Y[,1], Y[,2], border = "gray")
fitp <- Procrustes(X, Y)
plot(fitp$Yhat[,1], fitp$Yhat[,2], col = "red", xlim = c(-3, 3), ylim = c(-2, 3.5), 
     asp = 1, xlab = "", ylab = "")
polygon(fitp$Yhat[,1], fitp$Yhat[,2], border = "red")
par(op)

## MDS example:
eastD <- sim2diss(EW_eng$east)
attr(eastD, "Labels") <- abbreviate(attr(eastD, "Labels"))
fit.east <- mds(eastD, type = "ordinal")
westD <- sim2diss(EW_eng$west)
attr(westD, "Labels") <- abbreviate(attr(westD, "Labels"))
fit.west <- mds(westD, type = "ordinal", init = torgerson(eastD))

fit.proc <- Procrustes(fit.east$conf, fit.west$conf)
fit.proc

## Configuration plots; Procrustes plots.
plot(fit.east, main = "MDS East Germany")   ## MDS plot East Germany
plot(fit.west, main = "MDS West Germany")   ## MDS plot West Germany

## Procrustes configurations (X and Yhat)
plot(fit.proc, ylim = c(-1, 1),  col.X = "cadetblue", col.Yhat = "brown", pch = 19, 
    legend = list(pos = "topleft", labels = c("East Germany", "West Germany"))) 
    
## Procrustes transformations (Y and Yhat)
plot(fit.proc, plot.type = "transplot", length = 0.05, ylim = c(-1,1), 
     legend = list(pos = "bottomright", 
     labels = c("West Germany (untransformed)", "West Germany (transformed)")))

smacof documentation built on May 6, 2022, 9:05 a.m.