fct_kv_EvaluesQ function

Function to calculate the Gram matrices and their eigenvalues and eigenvectors for a chosen reproducing kernel.

Calculates the Gram matrices $K_v$ for $v=1,\ldots,$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.

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

Arguments

• 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 $2,\ldots$.
• 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 $\lambda_{min} < \lambda_{max}\times$tol, then the correction to $K_v$ is done (see details). Set as $1e^{-8}$ by default.

Details

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

Returns

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 $K_v$.

References

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

Author(s)

Halaleh Kamari

Note

Note.

See Also

RKHSMetaMod

Examples

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