# 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. ```r 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. ## Author(s) Halaleh Kamari ## Note Note. ## See Also `RKHSMetaMod` ## Examples ```r 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 ```

fct_kv_EvaluesQ function