optPenalty.kCVauto {rags2ridges}

Automatic search for optimal penalty parameter

Description

Function that performs an 'automatic' search for the optimal penalty parameter for the ridgeP call by employing Brent's method to the calculation of a cross-validated negative log-likelihood score.

Usage

optPenalty.kCVauto(
  Y,
  lambdaMin,
  lambdaMax,
  lambdaInit = (lambdaMin + lambdaMax)/2,
  fold = nrow(Y),
  cor = FALSE,
  target = default.target(covML(Y)),
  type = "Alt"
)

Arguments

Y

Data matrix. Variables assumed to be represented by columns.

lambdaMin

A numeric giving the minimum value for the penalty parameter.

lambdaMax

A numeric giving the maximum value for the penalty parameter.

lambdaInit

A numeric giving the initial (starting) value for the penalty parameter.

fold

A numeric or integer specifying the number of folds to apply in the cross-validation.

cor

A logical indicating if the evaluation of the LOOCV score should be performed on the correlation scale.

target

A target matrix (in precision terms) for Type I ridge estimators.

type

A character indicating the type of ridge estimator to be used. Must be one of: "Alt", "ArchI", "ArchII".

Details

The function determines the optimal value of the penalty parameter by application of the Brent algorithm (1971) to the K-fold cross-validated negative log-likelihood score (using a regularized ridge estimator for the precision matrix). The search for the optimal value is automatic in the sense that in order to invoke the root-finding abilities of the Brent method, only a minimum value and a maximum value for the penalty parameter need to be specified as well as a starting penalty value. The value at which the K-fold cross-validated negative log-likelihood score is minimized is deemed optimal. The function employs the Brent algorithm as implemented in the optim function.

Value

An object of class list:

optLambda

A numeric giving the optimal value for the penalty parameter.

optPrec

A matrix representing the precision matrix of the chosen type (see ridgeP) under the optimal value of the penalty parameter.

Note

When cor = TRUE correlation matrices are used in the computation of the (cross-validated) negative log-likelihood score, i.e., the K-fold sample covariance matrix is a matrix on the correlation scale. When performing evaluation on the correlation scale the data are assumed to be standardized. If cor = TRUE and one wishes to used the default target specification one may consider using target = default.target(covML(Y, cor = TRUE)). This gives a default target under the assumption of standardized data.

Under the default setting of the fold-argument, fold = nrow(Y), one performes leave-one-out cross-validation.

Author(s)

Wessel N. van Wieringen, Carel F.W. Peeters <[e-mail]>

References

Brent, R.P. (1971). An Algorithm with Guaranteed Convergence for Finding a Zero of a Function. Computer Journal 14: 422-425.

See Also

GGMblockNullPenalty, GGMblockTest, ridgeP, optPenalty.aLOOCV, optPenalty.kCV,
default.target, covML

Examples

## Obtain some (high-dimensional) data
p = 25
n = 10
set.seed(333)
X = matrix(rnorm(n*p), nrow = n, ncol = p)
colnames(X)[1:25] = letters[1:25]

## Obtain regularized precision under optimal penalty using K = n
OPT <- optPenalty.kCVauto(X, lambdaMin = .001, lambdaMax = 30); OPT
OPT$optLambda # Optimal penalty
OPT$optPrec   # Regularized precision under optimal penalty

## Another example with standardized data
X <- scale(X, center = TRUE, scale = TRUE)
OPT <- optPenalty.kCVauto(X, lambdaMin = .001, lambdaMax = 30, cor = TRUE,
                          target = default.target(covML(X, cor = TRUE))); OPT
OPT$optLambda # Optimal penalty
OPT$optPrec   # Regularized precision under optimal penalty

## Another example using K = 5
OPT <- optPenalty.kCVauto(X, lambdaMin = .001, lambdaMax = 30, fold = 5); OPT
OPT$optLambda # Optimal penalty
OPT$optPrec   # Regularized precision under optimal penalty