pivot_mds {bigmds}

Pivot MDS

Description

Pivot MDS, introduced in the literature of graph layout algorithms, is similar to Landmark MDS (landmark_mds()) but it uses the distance information between landmark and non-landmark points to improve the initial low dimensional configuration, as more relations than just those between landmark points are taken into account.

Usage

pivot_mds(x, num_pivots, r)

Arguments

x

A matrix with n individuals (rows) and k variables (columns).

num_pivots

Number of pivot points to obtain an initial MDS configuration. It is equivalent to l parameter used in interpolation_mds(), divide_conquer_mds() and fast_mds(). Therefore, it is the size for which classical MDS can be computed efficiently (using cmdscale function). It means that if \bar{l} is the limit size for which classical MDS is applicable, then l\leq \bar{l}.

r

Number of principal coordinates to be extracted.

Value

Returns a list containing the following elements:

points

A matrix that consists of n individuals (rows) and r variables (columns) corresponding to the principal coordinates. Since we are performing a dimensionality reduction, r<<k

eigen

The first r largest eigenvalues: \lambda_i, i \in \{1, \dots, r\} , where each \lambda_i is obtained from applying classical MDS to the first data subset.

References

Delicado P. and C. Pachón-García (2021). Multidimensional Scaling for Big Data. https://arxiv.org/abs/2007.11919.

Brandes U. and C. Pich (2007). Eigensolver Methods for Progressive Multidimensional Scaling of Large Data. Graph Drawing.

Borg, I. and P. Groenen (2005). Modern Multidimensional Scaling: Theory and Applications. Springer.

Gower JC. (1968). Adding a point to vector diagrams in multivariate analysis. Biometrika.

Examples

set.seed(42)
x <- matrix(data = rnorm(4 * 10000), nrow = 10000) %*% diag(c(9, 4, 1, 1))
mds <- pivot_mds(x = x, num_pivots = 200, r = 2)
head(mds$points)
mds$eigen