distl2dnorm function

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

$L^2$ 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.

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

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.

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 $f_2 / ||f_2||$:

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

Returns

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.

Author(s)

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

See Also

distl2d for the distance between two probability densities.

matdistl2dnorm in order to compute pairwise distances between several $L^2$-normed densities.

Examples

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)