# Function to fit a solution with q active groups of an RKHS Group Lasso problem. Fits a solution of the group lasso problem based on RKHS, with $q$ active groups in the obtained solution for the Gaussian regression model. It determines $\mu_{g}(q)$, for which the number of active groups in the solution of the RKHS group lasso problem is equal to $q$, and returns the RKHS meta model associated with $\mu_{g}(q)$. ```r grplasso_q(Y, Kv, q, rat, Num) ``` ## Arguments - `Y`: Vector of response observations of size $n$. - `Kv`: List of eigenvalues and eigenvectors of positive definite Gram matrices $K_v$ and their associated group names. It should have the same format as the output of the function `calc_Kv` (see details). - `q`: Integer, the number of active groups in the obtained solution. - `rat`: Positive scalar, used to restrict the minimum value of $\mu_g$, to be evaluted in the RKHS Group Lasso algorithm, $\mu_{min}=\mu_{max}/rat$. The value $\mu_{max}$ is calculated inside the program, see function `mu_max`. - `Num`: Integer, used to restrict the number of different values of the penalty parameter $\mu_g$ to be evaluated in the RKHS Group Lasso algorithm, until it achieves $\mu_g(q)$: for Num $= 1$ the program is done for $3$ values of $\mu_g$, $\mu_{1}=(\mu_{min}+\mu_{max})/2$, $\mu_{2}=(\mu_{min}+\mu_{1})/2$ or $\mu_{2}=(\mu_{1}+\mu_{max})/2$ depending on the value of $q$ associated with $\mu_{1}$, $\mu_{3}=\mu_{min}$. ## Details Input Kv should contain the eigenvalues and eigenvectors of positive definite Gram matrices $K_v$. It is necessary to set input "correction" in the function `calc_Kv` equal to "TRUE". ## Returns List of $4$ components: "mus", "qs", "mu", "res": - **mus**: Vector, values of the evaluated penalty parameters $\mu_g$ in the RKHS group lasso algorithm until it achieves $\mu_{g}(q)$. - **qs**: Vector, number of active groups associated with each value of $\mu_g$ in mus. - **mu**: Scalar, value of $\mu_{g}(q)$. - **res**: An RKHS Group Lasso object: - **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.H**: Vector of size vMax, estimated values of the 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. - **MaxIter**: Integer, number of iterations until convergence is reached. - **convergence**: TRUE or FALSE. Indicates whether the algorithm has converged or not. - **RelDiffCrit**: Scalar, value of the first convergence criteria at the last iteration, $\frac{crit_{lastIter}-crit_{lastIter-1}}{crit_{lastIter-1}}$. - **RelDiffPar**: Scalar, value of the second convergence criteria at the last iteration, $\Vert\frac{\theta_{lastIter}-\theta_{lastIter-1}}{\theta_{lastIter-1}}\Vert ^{2}$. ## 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 `calc_Kv`, `mu_max` ## 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, TRUE, TRUE) result <- grplasso_q(Y,Kv,5,100 ,Num=10) result$mu result$res$Nsupp ```

calc_q_function function