# mds: Multidimensional Scaling In pvermees/IsoplotR: Statistical Toolbox for Radiometric Geochronology

## Description

Performs classical or nonmetric Multidimensional Scaling analysis

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```mds(x, ...) ## Default S3 method: mds(x, classical = FALSE, plot = TRUE, shepard = FALSE, nnlines = FALSE, pos = NULL, col = "black", bg = "white", xlab = "", ylab = "", ...) ## S3 method for class 'detritals' mds(x, classical = FALSE, plot = TRUE, shepard = FALSE, nnlines = FALSE, pos = NULL, col = "black", bg = "white", xlab = "", ylab = "", hide = NULL, ...) ```

## Arguments

 `x` a dissimilarity matrix OR an object of class `detrital` `...` optional arguments to the generic `plot` function `classical` logical flag indicating whether classical (`TRUE`) or nonmetric (`FALSE`) MDS should be used `plot` show the MDS configuration (if `shepard=FALSE`) or Shepard plot (if `shepard=TRUE`) on a graphical device `shepard` logical flag indicating whether the graphical output should show the MDS configuration (`shepard=FALSE`) or a Shepard plot with the 'stress' value. This argument is only used if `plot=TRUE`. `nnlines` if `TRUE`, draws nearest neighbour lines `pos` a position specifier for the labels (if `par('pch')!=NA`). Values of 1, 2, 3 and 4 indicate positions below, to the left of, above and to the right of the MDS coordinates, respectively. `col` plot colour (may be a vector) `bg` background colour (may be a vector) `xlab` a string with the label of the x axis `ylab` a string with the label of the y axis `hide` vector with indices of aliquots that should be removed from the plot.

## Details

Multidimensional Scaling (MDS) is a dimension-reducting technique that takes a matrix of pairwise ‘dissimilarities’ between objects (e.g., age distributions) as input and produces a configuration of two (or higher-) dimensional coordinates as output, so that the Euclidean distances between these coordinates approximate the dissimilarities of the input matrix. Thus, an MDS-configuration serves as a ‘map’ in which similar samples cluster closely together and dissimilar samples plot far apart. In the context of detrital geochronology, the dissimilarity between samples is given by the statistical distance between age distributions. There are many ways to define this statistical distance. `IsoplotR` uses the Kolmogorov-Smirnov (KS) statistic due to its simplicity and the fact that it behaves like a true distance in the mathematical sense of the word (Vermeesch, 2013). The KS-distance is given by the maximum vertical distance between two `cad` step functions. Thus, the KS-distance takes on values between zero (perfect match between two age distributions) and one (no overlap between two distributions). Calculating the KS-distance between samples two at a time populates a symmetric dissimilarity matrix with positive values and a zero diagonal. `IsoplotR` implements two algorithms to convert this matrix into a configuration. The first (‘classical’) approach uses a sequence of basic matrix manipulations developed by Young and Householder (1938) and Torgerson (1952) to achieve a linear fit between the KS-distances and the fitted distances on the MDS configuration. The second, more sophisticated (‘nonmetric’) approach subjects the input distances to a transformation f prior to fitting a configuration:

δ_{i,j} = f(KS_{i,j})

where KS_{i,j} is the KS-distance between samples i and j (for 1 ≤q i \neq j ≤q n) and δ_{i,j} is the ‘disparity’ (Kruskal, 1964). Fitting an MDS configuration then involves finding the disparity transformation that maximises the goodness of fit (or minimises the ‘stress’) between the disparities and the fitted distances. The latter two quantities can also be plotted against each other as a 'Shepard plot'.

## Value

Returns an object of class `MDS`, i.e. a list containing the following items:

points

a two-column vector of the fitted configuration

classical

a logical flag indicating whether the MDS configuration was obtained by classical (`TRUE`) or nonmetric (`FALSE`) MDS

diss

the dissimilarity matrix used for the MDS analysis

stress

(only if `classical=TRUE`) the final stress achieved (in percent)

## References

Kruskal, J., 1964. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29 (1), 1-27.

Torgerson, W. S. Multidimensional scaling: I. Theory and method. Psychometrika, 17(4): 401-419, 1952.

Vermeesch, P., 2013. Multi-sample comparison of detrital age distributions. Chemical Geology, 341, pp.140-146.

Young, G. and Householder, A. S. Discussion of a set of points in terms of their mutual distances. Psychometrika, 3(1):19-22, 1938.

`cad`, `kde`
 ```1 2 3 4``` ```data(examples) mds(examples\$DZ,nnlines=TRUE,pch=21,cex=5) dev.new() mds(examples\$DZ,shepard=TRUE) ```