Ground cost that measures distance in units of the data's own spread

Euclidean cost silently declares the coordinates equally important and uncorrelated; on any real feature set that is false, and the transport plan inherits the lie. Whitening by the covariance removes both the scale and the correlation, so the cost is invariant to any full-rank affine recoding of the data.

Usage

Otmd(X, Y, Sigma)

Arguments

X

Source points, n by d.

Y

Target points, m by d.

Sigma

Covariance, symmetric positive definite.

Value

List with C, cost, n, m, d.

Details

Formula: \(_ij = (x_i - y_j)' Sigma^{-1} (x_i - y_j)\).

References

De Maesschalck, R., Jouan-Rimbaud, D. and Massart, D. L. (2000). Chemometrics and Intelligent Laboratory Systems 50(1):1-18. doi:10.1016/S0169-7439(99)00047-7 .

Examples

Otmd(X = c(1, 2, 3, 4, 5, 6, 7, 8), Y = c(1, 2, 3, 4, 5, 6, 7, 8), Sigma = 0.5)
#> $C
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,]    0    2    8   18   32   50   72   98
#> [2,]    2    0    2    8   18   32   50   72
#> [3,]    8    2    0    2    8   18   32   50
#> [4,]   18    8    2    0    2    8   18   32
#> [5,]   32   18    8    2    0    2    8   18
#> [6,]   50   32   18    8    2    0    2    8
#> [7,]   72   50   32   18    8    2    0    2
#> [8,]   98   72   50   32   18    8    2    0
#> 
#> $cost
#> [1] 0
#> 
#> $n
#> [1] 8
#> 
#> $m
#> [1] 8
#> 
#> $d
#> [1] 1
#> 
#> $method
#> [1] "Mahalanobis ground cost"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"