fct_kv_EvaluesQ: Function to calculate the Gram matrices and their eigenvalues...
In RKHSMetaMod: Ridge Group Sparse Optimization Problem for Estimation of a Meta Model Based on Reproducing Kernel Hilbert Spaces

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Calculates the Gram matrices K_v for v=1,…,vMax, and returns their associated eigenvalues and eigenvectors. The calculated Gram matrices may be not positive definite. The option "correction" of this function allows to replace the matrices K_v that are not positive definite by their "nearest positive definite" matrices.

1	calc_Kv(X, kernel, Dmax, correction, verbose, tol)

`X`	Matrix of observations with n rows and d columns.
`kernel`	Character, the type of the reproducing kernel: matern (matern kernel), brownian (brownian kernel), gaussian (gaussian kernel), linear (linear kernel), quad (quadratic kernel).
`Dmax`	Integer, between 1 and d, indicates the order of interactions considered in the meta model: Dmax=1 is used to consider only the main effects, Dmax=2 to include the main effects and the interactions of order 2,….
`correction`	Logical, if TRUE, the program makes the correction to the matrices K_v that are not positive definite (see details). Set as TRUE by default.
`verbose`	Logical, if TRUE, the group v for which the correction is done is printed. Set as TRUE by default.
`tol`	Scalar, used if correction is TRUE. For each matrix K_v if λ_{min} < λ_{max}\timestol, then the correction to K_v is done (see details). Set as 1e^{-8} by default.

Let λ_{v,i},i=1,...,n be the eigenvalues associated with matrix K_v. Set λ_{max}={max}_{i}λ_{v,i} and λ_{min}={min}_{i}λ_{v,i}. The eigenvalues of K_v that is not positive definite are replaced by λ_{v,i}+epsilon, with espilon=λ_{max}\timestol. The value of tol depends on the type of the kernel and it is chosen small.

List of two components "names.Grp" and "kv":

`names.Grp`	Vector of size vMax, indicates the name of groups included in the meta model.
`kv`	List of vMax components with the same names as the vector names.Grp. Each element of the list is a list of two components "Evalues" and "Q":
`Evalues`	Vector of size n, eigenvalues of each Gram matrix K_v.
`Q`	Matrix with n rows and n columns, eigenvectors of each Gram matrix K_v.

Note.

Halaleh Kamari

Kamari, H., Huet, S. and Taupin, M.-L. (2019) RKHSMetaMod : An R package to estimate the Hoeffding decomposition of an unknown function by solving RKHS Ridge Group Sparse optimization problem. <arXiv:1905.13695>

RKHSMetaMod

d <- 3
n <- 50
library(lhs)
X <- maximinLHS(n, d)
c <- c(0.2,0.6,0.8)
F <- 1;for (a in 1:d) F <- F*(abs(4*X[,a]-2)+c[a])/(1+c[a])
epsilon <- rnorm(n,0,1);sigma <- 0.2
Y <- F + sigma*epsilon
Dmax <- 3
kernel <- "matern"
Kv <- calc_Kv(X, kernel, Dmax)
names <- Kv$names.Grp
Eigen.val1 <- Kv$kv$v1.$Evalues
Eigen.vec1 <- Kv$kv$v1.$Q