# Function to produce a sequence of meta models that are the solutions of the RKHS Ridge Group Sparse or RKHS Group Lasso optimization problems. Calculates the Gram matrices $K_v$ for a chosen reproducing kernel and fits the solution of an RKHS ridge group sparse or an RKHS group lasso problem for each pair of penalty parameters $(\mu,\gamma)$, for the Gaussian regression model. ```r RKHSMetMod(Y, X, kernel, Dmax, gamma, frc, verbose) ``` ## Arguments - `Y`: Vector of response observations of size $n$. - `X`: Matrix of observations with $n$ rows and $d$ columns. - `kernel`: Character, indicates the type of the reproducing kernel: matern $($matern kernel$)$, brownian $($brownian kernel$)$, gaussian $($gaussian kernel$)$, linear $($linear kernel$)$, quad $($quadratic kernel$)$. See the function `calc_Kv` - `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 $2,\ldots$. See the function `calc_Kv` - `gamma`: Vector of non negative scalars, values of the penalty parameter $\gamma$ in decreasing order. If $\gamma=0$ the function solves an RKHS Group Lasso problem and for $\gamma>0$ it solves an RKHS Ridge Group Sparse problem. - `frc`: Vector of positive scalars. Each element of the vector sets a value to the penalty parameter $\mu$, $\mu=\mu_{max}/(\sqrt{n}\times frc)$. The value $\mu_{max}$ is calculated by the program. See the function `mu_max`. - `verbose`: Logical, if TRUE, prints: the group $v$ for which the correction of Gram matrix $K_v$ is done, and for each pair of the penalty parameters $(\mu,\gamma)$: the number of current iteration, active groups and convergence criterias. Set as FALSE by default. ## Details Details. ## Returns List of l components, with l equals to the number of pairs of the penalty parameters $(\mu,\gamma)$. Each component of the list is a list of $3$ components "mu", "gamma" and "Meta-Model": - **mu**: Positive scalar, penalty parameter $\mu$ associated with the estimated Meta-Model. - **gamma**: Positive scalar, an element of the input vector gamma associated with the estimated Meta-Model. - **Meta-Model**: An RKHS Ridge Group Sparse or RKHS Group Lasso object associated with the penalty parameters mu and gamma: - **intercept**: Scalar, estimated value of intercept. - **teta**: Matrix with vMax rows and $n$ columns. Each row of the matrix is the estimated vector $\theta_{v}$ for $v=1,...,$vMax. - **fit.v**: Matrix with $n$ rows and vMax columns. Each row of the matrix is the estimated value of $f_{v}=K_{v}\theta_{v}$. - **fitted**: Vector of size $n$, indicates the estimator of $m$. - **Norm.n**: Vector of size vMax, estimated values for the Ridge penalty norm. - **Norm.H**: Vector of size vMax, estimated values of the Sparse Group penalty norm. - **supp**: Vector of active groups. - **Nsupp**: Vector of the names of the active groups. - **SCR**: Scalar, equals to $\Vert Y-f_{0}-\sum_{v}K_{v}\theta_{v}\Vert ^{2}$. - **crit**: Scalar, indicates the value of the penalized criteria. - **gamma.v**: Vector of size vMax, coefficients of the Ridge penalty norm, $\sqrt{n}\gamma\times$gama_v. - **mu.v**: Vector of size vMax, coefficients of the Group Sparse penalty norm, $n\mu\times$mu_v. - **iter**: List of two components: maxIter, and the number of iterations until the convergence is achieved. - **convergence**: TRUE or FALSE. Indicates whether the algorithm has converged or not. - **RelDiffCrit**: Scalar, value of the first convergence criteria at the last iteration, $\Vert\frac{\theta_{lastIter}-\theta_{lastIter-1}}{\theta_{lastIter-1}}\Vert ^{2}$. - **RelDiffPar**: Scalar, value of the second convergence criteria at the last iteration, $\frac{crit_{lastIter}-crit_{lastIter-1}}{crit_{lastIter-1}}$. ## 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 For the case $\gamma=0$ the outputs "mu"$=\mu_g$ and "Meta-Model" is the same as the one returned by the function `RKHSgrplasso`. ## See Also `calc_Kv`, `mu_max`, `RKHSgrplasso`, `pen_MetMod` ## 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" frc <- c(10,100) gamma <- c(.5,.01,.001,0) result <- RKHSMetMod(Y,X,kernel,Dmax,gamma,frc,FALSE) l <- length(result) for(i in 1:l){print(result[[i]]$mu)} for(i in 1:l){print(result[[i]]$gamma)} for(i in 1:l){print(result[[i]]$`Meta-Model`$Nsupp)} ```

MetaModel function