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

In dad: Three-Way / Multigroup Data Analysis Through Densities

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)

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||:

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.

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.

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)

dad documentation built on Aug. 30, 2023, 5:06 p.m.