optPenalty.aLOOCV {rags2ridges}

Select optimal penalty parameter by approximate leave-one-out cross-validation

Description

Function that selects the optimal penalty parameter for the ridgeP call by usage of approximate leave-one-out cross-validation. Its output includes (a.o.) the precision matrix under the optimal value of the penalty parameter.

Usage

optPenalty.aLOOCV(
  Y,
  lambdaMin,
  lambdaMax,
  step,
  type = "Alt",
  cor = FALSE,
  target = default.target(covML(Y)),
  output = "light",
  graph = TRUE,
  verbose = TRUE
)

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.

step

An integer determining the number of steps in moving through the grid [lambdaMin, lambdaMax].

type

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

cor

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

target

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

output

A character indicating if the output is either heavy or light. Must be one of: "all", "light".

graph

A logical indicating if the grid search for the optimal penalty parameter should be visualized.

verbose

A logical indicating if information on progress should be printed on screen.

Details

The function calculates an approximate leave-one-out cross-validated (aLOOCV) negative log-likelihood score (using a regularized ridge estimator for the precision matrix) for each value of the penalty parameter contained in the search grid. The utilized aLOOCV score was proposed by Lian (2011) and Vujacic et al. (2014). The aLOOCV negative log-likeliho od score is computationally more efficient than its non-approximate counterpart (see optPenalty.LOOCV). For details on the aLOOCV negative log-likelihood score see Lian (2011) and Vujacic et al (2014). For scalar matrix targets (see default.target) the complete solution path of the alternative Type I and II ridge estimators (see ridgeP) depends on only 1 eigendecomposition and 1 matrix inversion, making the determination of the optimal penalty value particularly efficient (see van Wieringen and Peeters, 2015).

The value of the penalty parameter that achieves the lowest aLOOCV negative log-likelihood score is deemed optimal. The penalty parameter must be positive such that lambdaMin must be a positive scalar. The maximum allowable value of lambdaMax depends on the type of ridge estimator employed. For details on the type of ridge estimator one may use (one of: "Alt", "ArchI", "ArchII") see ridgeP. The ouput consists of an object of class list (see below). When output = "light" (default) only the optLambda and optPrec elements of the list are given.

Value

An object of class list:

optLambda

A numeric giving the optimal value of the penalty parameter.

optPrec

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

lambdas

A numeric vector representing all values of the penalty parameter for which approximate cross-validation was performed; Only given when output = "all".

aLOOCVs

A numeric vector representing the approximate cross-validated negative log-likelihoods for each value of the penalty parameter given in lambdas; Only given when output = "all".

Note

When cor = TRUE correlation matrices are used in the computation of the approximate (cross-validated) negative log-likelihood score, i.e., the 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.

Author(s)

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

References

Lian, H. (2011). Shrinkage tuning parameter selection in precision matrices estimation. Journal of Statistical Planning and Inference, 141: 2839-2848.

van Wieringen, W.N. & Peeters, C.F.W. (2016). Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data, Computational Statistics & Data Analysis, vol. 103: 284-303. Also available as arXiv:1403.0904v3 [stat.ME].

Vujacic, I., Abbruzzo, A., and Wit, E.C. (2014). A computationally fast alternative to cross-validation in penalized Gaussian graphical models. arXiv: 1309.6216v2 [stat.ME].

See Also

ridgeP, optPenalty.LOOCV, optPenalty.LOOCVauto,
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
OPT  <- optPenalty.aLOOCV(X, lambdaMin = .001, lambdaMax = 30, step = 400); 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.aLOOCV(X, lambdaMin = .001, lambdaMax = 30,
                          step = 400, cor = TRUE,
                          target = default.target(covML(X, cor = TRUE))); OPT
OPT$optLambda	# Optimal penalty
OPT$optPrec	  # Regularized precision under optimal penalty