calc_q_function: Function to fit a solution with q active groups of an RKHS...
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

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 μ_{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 μ_{g}(q).

1	grplasso_q(Y, Kv, q, rat, Num)

`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 μ_g, to be evaluted in the RKHS Group Lasso algorithm, μ_{min}=μ_{max}/rat. The value μ_{max} is calculated inside the program, see function `mu_max`.
`Num`	Integer, used to restrict the number of different values of the penalty parameter μ_g to be evaluated in the RKHS Group Lasso algorithm, until it achieves μ_g(q): for Num = 1 the program is done for 3 values of μ_g, μ_{1}=(μ_{min}+μ_{max})/2, μ_{2}=(μ_{min}+μ_{1})/2 or μ_{2}=(μ_{1}+μ_{max})/2 depending on the value of q associated with μ_{1}, μ_{3}=μ_{min}.

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".

List of 4 components: "mus", "qs", "mu", "res":

`mus`	Vector, values of the evaluated penalty parameters μ_g in the RKHS group lasso algorithm until it achieves μ_{g}(q).
`qs`	Vector, number of active groups associated with each value of μ_g in mus.
`mu`	Scalar, value of μ_{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 θ_{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}θ_{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}-∑_{v}K_{v}θ_{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{θ_{lastIter}-θ_{lastIter-1}}{θ_{lastIter-1}}\Vert ^{2}.

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>

calc_Kv, mu_max

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