What is MDS

In general, multidimensional scaling deals with constructing a low dimensional map of n samples such that the Euclidean distances between the samples match a given set of dissimilarities Δ : n × n as closely as possible. Here the focus is on Principal Coordinate Analysis (PCO), an approach based on the singular value decomposition of a matrix. However, the regression biplot provides a general structure for fitting any 2D map of samples with biplot axes.

Regression biplot

The function regress accepts as arguments an object of class biplot. The call to the function biplot should contain the data set that will be used to construct the biplot axes. The second argument Z contains the coordinates of the samples in the MDS map. By default linear regression axes will be fitted to the plot. If X denote the data matrix, the directions of the biplot axes are found by solving the regression equation

X = ZH′ + E. The matrix H′ = (Z′Z)⁻¹Z′X and calibration of the axes proceed analogous to equation (2) in the biplotEZ vignette. The coordinates of the marker /mu on biplot axis j is given as follows

$$ p_{\mu} = \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}\mathbf{h}_{(j)}. $$

biplot(rock) |> 
  regress(Z = MASS::sammon(dist(scale(rock), method="manhattan"))$points) |> 
  plot()
#> Initial stress        : 0.02554
#> stress after  10 iters: 0.01094, magic = 0.500
#> stress after  20 iters: 0.01080, magic = 0.500
#> stress after  30 iters: 0.01078, magic = 0.500

More flexibility is provided by approximating the biplot axes using B-splines. A calibrated trajectory represented in a matrix H : m × 2 snakes through the samples points Z such that the marker on the trajectory which is closest to a particular sample gives the predicted value of that sample for the particular variable. To retain smoothness of the trajectory H, B-splines are used.

biplot(rock) |> 
  regress(Z = MASS::sammon(dist(scale(rock), method="manhattan"))$points,
          axes = "splines") |> 
  plot()
#> Initial stress        : 0.02554
#> stress after  10 iters: 0.01094, magic = 0.500
#> stress after  20 iters: 0.01080, magic = 0.500
#> stress after  30 iters: 0.01078, magic = 0.500

#> Calculating spline axis for variable 1 
#> Calculating spline axis for variable 2 
#> Calculating spline axis for variable 3 
#> Calculating spline axis for variable 4

Principal Coordinate Analysis (PCO) biplots

Let Δ̃ be the n × n matrix containing the values $-\frac{1}{2}\delta_{ij}^2$ where δ_ij represent the dissimilarity between objects i and j. If it is possible to find a set of coordinates Y : n × m where typically m = n − 1 such that the Euclidean distances between the rows of Y exactly match the dissimilarities δ_ij, the dissimilarity is known as a Euclidean embeddable distance metric. J. C. Gower (1982) shows that if the distances δ_ij are Euclidean embeddable, then

(I − 1s′)Δ̃(I − s1′)

is positive semi-definite.

The Euclidean representation of the samples is obtained as $\mathbf{Y=V\Lambda}^{\frac{1}{2}}$ where (I − 1s′)Δ̃(I − s1′) = VΛV′. Since the coordinates in Y is already referred to their principal axes with Y′Y = Λ, the representation of the samples in a 2D biplot is obtained from the first two columns of Y.

In addition to the exact biplot axis representations discussed in J. C. Gower, Lubbe, and Roux (2011) approximate axes can be obtained. Linear axes are fitted with the regression method. The variables in the data matrix X : n × p can be represented as biplot axes in the PCO biplot with the sample points $\mathbf{Z=V\Lambda}^{\frac{1}{2}}\mathbf{J}_2$ according to the regression method discussed in section 2 above.

biplot(rock, scale = TRUE) |> PCO() |> plot()

Using B-splines instead of linear regression provides the user with more flexibility. This is achieved by setting the argument axes = "splines".

biplot(rock, scale = TRUE) |> PCO(axes = "splines") |> plot()

#> Calculating spline axis for variable 1 
#> Calculating spline axis for variable 2 
#> Calculating spline axis for variable 3 
#> Calculating spline axis for variable 4

By default the distance metric used for the analysis is Euclidean distance. The user can also specify the distance matrix Dmat as an n × n matrix of a dist object. As illustration of a metric that is Euclidean embeddable, the Clark’s distance

$$ \delta_{ij}^2 = \sum_{k=1}^{p}\left(\frac{x_{ik}-x_{jk}}{x_{ik}+x_{jk}}\right)^2 $$

as defined by John C. Gower and Ngouenet (2005) is calculated below and used for constructing a PCO biplot. Note that the metric is only defined for strictly positive values, so that the data is scaled to values between 1 and 2.

Clark.dist <- function(X)
{
  n <- nrow(X)
  p <- ncol(X)
  Dmat <- matrix (0, nrow=n, ncol=n)
  for (i in 1:(n-1))
    for (j in (i+1):n)
      Dmat[i,j] <- sum(((X[i,] - X[j,])/(X[i,] + X[j,]))^2)
  sqrt(Dmat + t(Dmat))
}
my.data <- scale(rock, center=apply(rock,2,min), scale=diff(apply(rock,2,range)))+1
biplot(rock) |> PCO(Dmat = Clark.dist (rock), axes = "splines") |> plot()

#> Calculating spline axis for variable 1 
#> Calculating spline axis for variable 2 
#> Calculating spline axis for variable 3 
#> Calculating spline axis for variable 4

Alternatively, the user can specify a function which computes a distance matrix or dist object. The Manhattan distance is not Euclidean embeddable, but the square root of the distance is. The function sqrtManhattan is included as an example of a function computing a Euclidean embeddable dist object.

sqrtManhattan
#> function (X) 
#> {
#>     sqrt(stats::dist(X, method = "manhattan"))
#> }
#> <bytecode: 0x5598dc91e638>
#> <environment: namespace:biplotEZ>
biplot(rock, scaled = TRUE) |> PCO(dist.func = sqrtManhattan) |> plot()