rags2ridges (version 1.2)

optPenalty.aLOOCV: Select optimal penalty parameter by approximate leave-one-out cross-validation

Description

Function that selects the optimal penalty parameter for the ridgeS 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", target = 
diag(1/diag(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".

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 intermediate output should be printed on screen.

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".

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-likelihood score is computationally more efficient than its non-approximate counterpart (see optPenaltyCV). For details on the aLOOCV negative log-likelihood score see Lian (2011) and Vujacic et al (2014). 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 ridgeS. 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.

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. and Peeters, C.F.W. (2014). Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data. arXiv:1403.0904 [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

ridgeS, optPenaltyCV

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 = 25, step = 100); OPT	
OPT$optLambda	# Optimal penalty
OPT$optPrec	# Regularized precision under optimal penalty