RKHSMetMod {RKHSMetaMod}

Function to produce a sequence of meta models that are the solutions of the RKHS Ridge Group Sparse or RKHS Group Lasso optimization problems.

Description

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.

Usage

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.

Value

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\timesgama_v.

mu.v

Vector of size vMax, coefficients of the Group Sparse penalty norm, n\mu\timesmu_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}}.

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.

Author(s)

Halaleh Kamari

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>

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)}