MetaModel function

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.

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. arXiv:1905.13695

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

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)}