sammon Sammon's Non-Linear Mapping

Description

One form of non-metric multidimensional scaling.

Usage

sammon(d, y = cmdscale(d, k), k = 2, niter = 100, trace = TRUE,
       magic = 0.2, tol = 1e-4)

Arguments

d	distance structure of the form returned by dist, or a full, symmetric matrix. Data are assumed to be dissimilarities or relative distances, but must be positive except for self-distance. This can contain missing values.
y	An initial configuration. If none is supplied, cmdscale is used to provide the classical solution. (If there are missing values in d, an initial configuration must be provided.) This must not have duplicates.
k	The dimension of the configuration.
niter	The maximum number of iterations.
trace	Logical for tracing optimization. Default TRUE.
magic	initial value of the step size constant in diagonal Newton method.
tol	Tolerance for stopping, in units of stress.

Details

This chooses a two-dimensional configuration to minimize the stress, the sum of squared differences between the input distances and those of the configuration, weighted by the distances, the whole sum being divided by the sum of input distances to make the stress scale-free.

An iterative algorithm is used, which will usually converge in around 50 iterations. As this is necessarily an O(n^2) calculation, it is slow for large datasets. Further, since the configuration is only determined up to rotations and reflections (by convention the centroid is at the origin), the result can vary considerably from machine to machine. In this release the algorithm has been modified by adding a step-length search (magic) to ensure that it always goes downhill.

Value

Two components:

points	A two-column vector of the fitted configuration.
stress	The final stress achieved.

Side Effects

If trace is true, the initial stress and the current stress are printed out every 10 iterations.

References

Sammon, J. W. (1969) A non-linear mapping for data structure analysis. IEEE Trans. Comput., C-18 401–409.

Ripley, B. D. (1996) Pattern Recognition and Neural Networks. Cambridge University Press.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

See Also

cmdscale, isoMDS

Examples

swiss.x <- as.matrix(swiss[, -1])
swiss.sam <- sammon(dist(swiss.x))
plot(swiss.sam$points, type = "n")
text(swiss.sam$points, labels = as.character(1:nrow(swiss.x)))