Create a Point Configuration from a Distance Matrix

Source: R/getConfig.R

getConfig - given an nxn Euclidean distance matrix, produces a d-dimensional point configuration of size n via eigendecomposition

getConfig(D, d)

Arguments

D

an nxn Euclidean distance matrix
d

the dimension for the configuration

Value

Y

an nxd matrix containing the d-dimensional point configuration

Accuracy

the ratio of the sum of retained eigenvalues to the sum of all n eigenvalues obtained during decomposition

Details

Given a distance matrix D, transform to a semi-definite matrix S using the linear transformation \(\tau(D)\). Using S, compute the eigen-decomposition \(S = ULV'\), where L is a diagonal matrix containing the singular-values of S, and the columns of U contain the eigen-vectors. A point configuration X is then computed as:

$$X = US^{.5}$$

To compute a configuration in d dimensions, the first d eigenvalues of S are used.

Examples

set.seed(1337)
D <- matrix(c(0,3,4,3,4,3,
             3,0,1,NA,5,NA,
             4,1,0,5,NA,5,
             3,NA,5,0,1,NA,
             4,5,NA,1,0,5,
             3,NA,5,NA,5,0),byrow=TRUE, nrow=6)
d <- 3
DStar <- dpf(D,d)$D

getConfig(DStar,3)

#> $X
#>               [,1]       [,2]        [,3]
#> [1,] -5.135640e-07  0.9009879  1.46957978
#> [2,]  2.122227e+00 -0.6863731  0.06370942
#> [3,]  2.829508e+00 -1.2090178 -0.41237683
#> [4,] -2.120510e+00 -0.6847331  0.05928362
#> [5,] -2.827222e+00 -1.2157403 -0.40825964
#> [6,] -4.002138e-03  2.8948764 -0.77193636
#> 
#> $Accuracy
#> [1] 1
#>