Kruskal's Non-metric Multidimensional Scaling
One form of non-metric multidimensional scaling
isoMDS(d, y = cmdscale(d, k), k = 2, maxit = 50, trace = TRUE, tol = 1e-3, p = 2)
Shepard(d, x, p = 2)
- 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. Both missing and infinite values are allowed.
- An initial configuration. If none is supplied,
cmdscaleis used to provide the classical solution, unless there are missing or infinite dissimilarities.
- The desired dimension for the solution, passed to
- The maximum number of iterations.
- Logical for tracing optimization. Default
- convergence tolerance.
- Power for Minkowski distance in the configuration space.
- A final configuration.
This chooses a k-dimensional (default k = 2) configuration to minimize the stress, the square root of the ratio of the sum of squared differences between the input distances and those of the configuration to the sum of configuration distances squared. However, the input distances are allowed a monotonic transformation.
An iterative algorithm is used, which will usually converge in around 10 iterations. As this is necessarily an $O(n^2)$ calculation, it is slow for large datasets. Further, since for the default $p = 2$ 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.
- Two components:
points A k-column vector of the fitted configuration. stress The final stress achieved (in percent).
trace is true, the initial stress and the current stress
are printed out every 5 iterations.
T. F. Cox and M. A. A. Cox (1994, 2001) Multidimensional Scaling. Chapman & Hall.
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.
swiss.x <- as.matrix(swiss[, -1]) swiss.dist <- dist(swiss.x) swiss.mds <- isoMDS(swiss.dist) plot(swiss.mds$points, type = "n") text(swiss.mds$points, labels = as.character(1:nrow(swiss.x))) swiss.sh <- Shepard(swiss.dist, swiss.mds$points) plot(swiss.sh, pch = ".") lines(swiss.sh$x, swiss.sh$yf, type = "S")