wassersteinpar {dad}

2-Wasserstein distance between Gaussian densities given their parameters

Description

The 2-Wasserstein distance between two multivariate (p > 1) or univariate (p = 1) Gaussian densities given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate) (see Details).

Usage

wassersteinpar(mean1, var1, mean2, var2, check = FALSE)

Arguments

mean1

p-length numeric vector: the mean of the first Gaussian density.

var1

p x p symmetric numeric matrix (p > 1) or numeric (p = 1): the covariance matrix (p > 1) or the variance (p = 1) of the first Gaussian density.

mean2

p-length numeric vector: the mean of the second Gaussian density.

var2

p x p symmetric numeric matrix (p > 1) or numeric (p = 1): the covariance matrix (p > 1) or the variance (p = 1) of the second Gaussian density.

check

logical. When TRUE (the default is FALSE) the function checks if the covariance matrices are not degenerate (multivariate case) or if the variances are not zero (univariate case).

Details

The mean vectors (m1 and m2) and variance matrices (v1 and v2) given as arguments (mean1, mean2, var1 and var2) are used to compute the 2-Wasserstein distance between the two Gaussian densities, equal to:

(||m1-m2||_2^2 + trace((v1+v2) - 2*(v2^{1/2} v1 v2^{1/2})^{1/2}))^{1/2}

If p = 1:

((m1-m2)^2 + v1 + v2 - 2*(v1*v2)^{1/2})^{1/2}

Value

The 2-Wasserstein distance between two Gaussian densities.

Be careful! If check = FALSE and one covariance matrix is degenerated (multivariate case) or one variance is zero (univariate case), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Peterson, A., Mueller, H.G (2016). Functional Data Analysis for Density Functions by Transformation to a Hilbert Space. The annals of Statistics, 44 (1), 183-218. DOI: 10.1214/15-AOS1363

Dowson, D.C., Ladau, B.V. (1982). The Fréchet Distance between Multivariate Normal Distributions. Journal of Multivariate Analysis, 12, 450-455.

See Also

wasserstein: 2-Wasserstein distance between Gaussian densities estimated from samples.

Examples

m1 <- c(1,1)
v1 <- matrix(c(4,1,1,9),ncol = 2)
m2 <- c(0,1)
v2 <- matrix(c(1,0,0,1),ncol = 2)
wassersteinpar(m1,v1,m2,v2)