group_lasso_function function

Function to fit a solution of an RKHS Group Lasso problem.

Fits the solution of an RKHS group lasso problem for the Gaussian regression model.

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

Arguments

• 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 $\mu_g$ 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.

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

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 $\theta_{v}$ 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 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. 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.

Author(s)

Halaleh Kamari

Note

Note.

See Also

calc_Kv

Examples

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