# l2d: L^2 inner product of estimated probability densities

### Description

Computes the L^2 inner product of multivariate probability densities, estimated from samples.

### Usage

 1 l2d(x1, x2, method = "gaussiand", check = FALSE, varw1, varw2) 

### Arguments

 x1 a matrix or data frame of n1 rows (observations) and p columns (variables) or a vector of length n1. x2 matrix or data frame of n2 rows and p columns or vector of length n1. 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 smoothing bandwidth matrices are not degenerate, before computing the inner product. varw1, varw2 a p \times p-symmetric matrix: the smoothing bandwidth for the estimation of the probability densities. If they are omitted, the smoothing bandwidth are computed using the normal reference rule matrix bandwidth (see details).

### Details

• If method = "gaussiand", the mean vectors (m1 and m2) and the variance matrices (v1 and v2) of the two samples are computed, and they are used to compute the inner product equal to:

(2 pi)^{-p/2} |v1+v2|^{-1/2} exp(-(1/2)(m1-m2)'(v1+v2)^{-1}(m1-m2))

• If method = "kern", the densities of both samples are estimated using the Gaussian kernel method. These estimations are then used to compute an estimation of the inner product. if varw1 and varw2 arguments are omitted, the smoothing bandwidths are computed using the normal reference rule matrix bandwidth:

h_1 v_1^{1/2}

where

h_1 = (4/(n_1(p+2)))^{1/(p+4)}

for the first density. Idem for the second density after making the necessary changes.

### Value

Returns the L^2 inner product of the two probability 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

### References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Wand, M., Jones, M. (1995). Kernel smoothing. Chapman and Hall/CRC, London.

Yousfi, S., Boumaza R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density functions. Journal of Statistical Computational and Simulation, 85 (11), 2315-2330.

  1 2 3 4 5 6 7 8 9 10 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) l2d(x1, x2, method = "gaussiand") l2d(x1, x2, method = "kern") l2d(x1, x2, method = "kern", varw1 = v1, varw2 = v2)