Package 'SEPaLS'

Bootstrap function for SEPaLS estimator.

Description

Bootstrap function for SEPaLS estimator.

Usage

bootstrap.SEPaLS(
  X,
  Y,
  yn,
  type = c("vMF", "Laplace"),
  mu0 = NULL,
  kappa0 = NULL,
  lambda = NULL,
  B = 20
)

Arguments

X	$(n\times p)$-dimensional matrix of the covariates.
Y	$(n)$-dimensional vector of the response.
yn	$y_n$ the quantile corresponding to lowest values of $Y$s to put in the tail.
type	character, whether vMF for von Mises-Fisher prior or Laplace for Laplace prior. See details.
mu0	$\mu_0$, unitary $(p)$-dimensional vector. The direction parameter for the vMF prior.
kappa0	$\kappa_0$, positive. The concentration parameter for the vMF prior.
lambda	$\lambda$, positive. The concentration parameter for the Laplace prior.
B	positive integer. The number of bootstrap samples on which estimate the SEPaLS directions. Default to 20.

Value

A list with two elements:

• ws: A $(B\times p)$-dimensional matrix with each row corresponding to the SEPaLS direction estimated on each bootstrap sample.

• cor: The correlation of each estimate direction on the Out-Of-Bag (OOB) sample with the response.

See Also

SEPaLS

Examples

set.seed(5)
n <- 3000
p <- 10
X <- matrix(rnorm(n*p),n,p)
beta <- c(5:1,rep(0,p-5)) ; beta <- beta/sqrt(sum(beta^2))
Y <- (X%*%beta)^3 + rnorm(n)
boot.sepals_Laplace <- bootstrap.SEPaLS(X,Y,yn=1,type="Laplace",lambda=0.01,
B=100)
boxplot(boot.sepals_Laplace$ws);abline(h=0,col="red",lty=2)

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Maximum Likelihood estimator

Description

Maximum Likelihood estimator

Usage

maximum_Likelihood_SEPaLS(X, Y, yn)

Arguments

X	$(n\times p)$-dimensional matrix of the covariates.
Y	$(n)$-dimensional vector of the response.
yn	the quantile corresponding to the lowest values of $Y$s to put in the tail.

Value

The maximum likelihood estimator.

Examples

n <- 3000
p <- 10
X <- matrix(rnorm(n*p),n,p)
beta <- c(5:1,rep(0,p-5)) ; beta <- beta/sqrt(sum(beta^2))
Y <- X%*%beta + rnorm(n,sd=1/3)
estimators <- do.call(rbind,lapply(seq(0,1,length.out=100),function(pp){
  yn <- quantile(Y,probs = pp)
  maximum_Likelihood_SEPaLS(X,Y,yn)
}))
matplot(estimators,type="l",lty=1,col=c(rep(2,5),rep(1,p-5)))
abline(h=beta/sqrt(sum(beta^2)),col=c(rep(2,5),rep(1,p-5)))

---

The RICA dataset describing the production of carrots (open field) (in quintals) from 2000 to 2015.

Description

A subset of data from the 'agreste' French governmental website <https://agreste.agriculture.gouv.fr/agreste-web/servicon/I.2/listeTypeServicon/>.

Usage

data(ricaCarrots)

Format

'ricaCarrots'

A List of 3 objects:

Y

a vector. The production of carrots (open field) (in quintals) for 598 French farms.

X

a matrix. The 259 covariates describing the same 598 French farms.

description

a matrix. Description of the 259 covariates.

Source

<https://agreste.agriculture.gouv.fr/agreste-web/servicon/I.2/listeTypeServicon/>

---

Function to estimate SEPaLS estimators

Description

Function to estimate SEPaLS estimators

Usage

SEPaLS(
  X,
  Y,
  yn,
  type = c("vMF", "Laplace"),
  mu0 = NULL,
  kappa0 = NULL,
  lambda = NULL
)

Arguments

X	$(n\times p)$-dimensional matrix of the covariates.
Y	$(n)$-dimensional vector of the response.
yn	$y_n$ the quantile correponding to lowest values of $Y$s to put in th e tail.
type	character, wether vMF for von Mises-Fisher prior or Laplace for Laplace prior. See details.
mu0	$\mu_0$, unitary $(p)$-dimensional vector. The direction parameter for the vMF prior.
kappa0	$\kappa_0$, positive. The concentration parameter for the vMF prior.
lambda	$\lambda$, positive. The concentration parameter for the Laplace prior.

Details

The SEPaLS estimators are built depending on the value given to type:

• vMF: then the estimator is proportional to

  $\hat{\beta}_{ml}(y_n) + \kappa_0\mu_0,$

  where $\hat{\beta}_{ml}(y_n)$ is the EPLS estimator, which coincides with the maximum-likelihood estimator of SEPaLS for a threshold $y_n$.

• Laplace: then the estimator is proportional to

  $S_\lambda\left(\hat{\beta}_{ml}(y_n)\right),$

  where $S_\lambda$ is the soft-thresholding operator of threshold $\lambda$.

Value

A SEPaLS estimator

See Also

bootstrap.SEPaLS

Examples

set.seed(1)
n <- 3000
p <- 10
X <- matrix(rnorm(n*p),n,p)
beta <- c(5:1,rep(0,p-5)) ; beta <- beta/sqrt(sum(beta^2))
Y <- (X%*%beta)^3 + rnorm(n,sd=1/3)
mu0 <- rnorm(p) ; mu0 <- mu0/sqrt(sum(mu0^2))
sepals_vMF <- SEPaLS(X,Y,yn=1,type="vMF",mu0=mu0,kappa0=1)
sepals_Laplace <- SEPaLS(X,Y,yn=1,type="Laplace",lambda=0.01)

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