Transform samples to uniform scale

Since Kullback-Leibler divergence is scale-invariant, its sample-based approximations can be computed on a conveniently chosen scale. This helper functions transforms each variable in a way that all marginal distributions of the joint dataset \((X,Y)\) are uniform. In this way, the scales of different variables are rendered comparable, with the idea of a better performance of neighbour-based methods in this situation.

Usage

to_uniform_scale(X, Y)

Arguments

X, Y

n-by-d and m-by-d matrices, representing n samples from the true distribution \(P\) and m samples from the approximate distribution \(Q\), both in d dimensions. Vector input is treated as a column matrix. Y can be left blank if q is specified (see below).

Value

A list with fields X and Y, containing the transformed samples.

Examples

# 2D example
n <- 10L
X <- cbind(rnorm(n, mean = 0, sd = 3),
           rnorm(n, mean = 1, sd = 2))
Y <- cbind(rnorm(n, mean = 1, sd = 2),
           rnorm(n, mean = 0, sd = 2))
to_uniform_scale(X, Y)
#> $X
#>       [,1] [,2]
#>  [1,] 0.25 0.70
#>  [2,] 0.10 0.60
#>  [3,] 0.70 0.85
#>  [4,] 0.60 0.20
#>  [5,] 0.90 0.50
#>  [6,] 0.05 0.75
#>  [7,] 0.30 0.95
#>  [8,] 1.00 0.80
#>  [9,] 0.40 0.65
#> [10,] 0.85 0.90
#> 
#> $Y
#>       [,1] [,2]
#>  [1,] 0.55 0.30
#>  [2,] 0.20 0.35
#>  [3,] 0.50 0.10
#>  [4,] 0.45 0.40
#>  [5,] 0.65 0.55
#>  [6,] 0.35 0.15
#>  [7,] 0.80 1.00
#>  [8,] 0.95 0.25
#>  [9,] 0.15 0.05
#> [10,] 0.75 0.45
#>