# l2d: L^2 inner product of probability densities In dad: Three-Way Data Analysis Through Densities

## Description

L^2 inner product of two multivariate (p > 1) or univariate (p = 1) probability densities, estimated from samples.

## Usage

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

## 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 n2. `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` p x p symmetric matrices: the smoothing bandwidths for the estimation of the probability densities. If they are omitted, the smoothing bandwidths are computed using the normal reference rule matrix bandwidth (see details).

## Details

• If `method = "gaussiand"`, the mean vectors and the variance matrices (v1 and v2) of the two samples are computed, and they are used to compute the inner product using the `l2dpar` function.

• If `method = "kern"`, the densities of both samples are estimated using the Gaussian kernel method. These estimations are then used to compute 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

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.

l2dpar for Gaussian densities, the parameters being given.

## Examples

 ``` 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) ```

dad documentation built on May 10, 2018, 9:03 a.m.