distl2dnorm: $L^2$ distance between $L^2$-normed probability densities

Description

$L^2$ distance between two multivariate ($p > 1$) or univariate (dimension: $p = 1$) $L^2$-normed probability densities, estimated from samples, where a $L^2$-normed probability density is the original probability density function divided by its $L^2$-norm.

Usage

distl2dnorm(x1, x2, method = "gaussiand", check = FALSE, varw1 = NULL, varw2 = NULL)

Value

The $L^2$ distance between the two $L^2$-normed densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned can not be considered.

Arguments

x1, x2

the samples from the probability densities (see l2d.

method

string. It can be:

"gaussiand" if the densities are considered to be Gaussian.
"kern" if they are estimated using the Gaussian kernel method.

check

logical. When TRUE (the default is FALSE) the function checks if the covariance matrices (if method = "gaussiand") or smoothing bandwidth matrices (if method = "kern") are not degenerate, before computing the inner product.

Notice that if $p = 1$, it checks if the variances or smoothing parameters are not zero.

varw1, varw2

the bandwidths when the densities are estimated by the kernel method (see l2d.

Author

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

Details

Given densities $f_1$ and $f_2$, the function distl2dnormpar computes the distance between the $L^2$-normed densities $f_1 / ||f_1||$ and $f_2 / ||f_2||$:

$$2 - 2 <f_1, f_2> / (||f_1|| ||f_2||)$$

For some information about the method used to compute the $L^2$ inner product or about the arguments, see l2d.

Examples

Run this code

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
distl2dnorm(x1, x2, method = "gaussiand")
distl2dnorm(x1, x2, method = "kern")
distl2dnorm(x1, x2, method = "kern", varw1 = v1, varw2 = v2)

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distl2dnorm: \(L^2\) distance between \(L^2\)-normed probability densities