group_lasso_function: Function to fit a solution of an RKHS Group Lasso problem.
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 the solution of an RKHS group lasso problem for the Gaussian regression model.

1	RKHSgrplasso(Y, Kv, mu, maxIter, verbose)

`Y`	Vector of response observations of size n.
`Kv`	List, includes the eigenvalues and eigenvectors of the positive definite Gram matrices K_v, v=1,...,vMax and their associated group names. It should have the same format as the output of the function `calc_Kv` (see details).
`mu`	Positive scalar, value of the penalty parameter μ_g in the RKHS Group Lasso problem.
`maxIter`	Integer, shows the maximum number of loops through all groups. Set as 1000 by default.
`verbose`	Logical, if TRUE, prints: the number of current iteration, active groups and convergence criterias. Set as FALSE by default.

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".

Estimated RKHS meta model, list with 13 components:

`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 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>

Meier, L. Van de Geer, S. and Buhlmann, P. (2008) The group LASSO for logistic regression. Journal of the Royal Statistical Society Series B. 70. 53-71. 10.1111/j.1467-9868.2007.00627.x.

calc_Kv

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)
matZ <- Kv$kv
mumax <- mu_max(Y, matZ)
mug <- mumax/10
gr <- RKHSgrplasso(Y,Kv, mug , 1000, FALSE)
gr$Nsupp