rsc_cv {RSC}

Optimal threshold selection for the RSC estimator

Description

Perform cross-validation to select an adaptive optimal threshold for the RSC estimator proposed in Serra et al. (2018).

Usage

  rsc_cv(x, cv.type = "kfold", R = 10, K = 10, threshold = seq(0.05, 0.95, by = 0.025),
         even.correction = FALSE, na.rm = FALSE, ncores = NULL, monitor = TRUE) 

Arguments

x

A matrix or a data.frame. Rows of x correspond to sample units and columns correspond to variables. Categorical variables are not allowed.

cv.type

A character string indicating the cross-validation algorithm. Possible values are "kfold" for repeated K-fold cross-validation, and "random" for random cross-validation (see Details).

R

An integer corresponding to the number of repeated foldings when cv.type = "kfold". When cv.type = "random" R defines the number of random splits (see Details).

K

An integer corresponding to the number of folds in K-fold cross-validation. Therefore this argument is not relevant when cv.type = "random".

threshold

A sequence of reals taken onto the interval (0,1) defining the threshold values at which the loss is estimated.

even.correction

A logical value. It sets the parameter even.correction in each of the underlying RMAD computations (see Details in rmad).

na.rm

A logical value, it defines the treatment of missing values in each of the underlying RMAD computations (see Details).

ncores

An integer value defining the number of cores used for parallel computing. When ncores=NULL (default), the number r of available cores is detected, and (r-1) of them are used (see Details).

monitor

A logical value. If TRUE progress messages are printed on screen.

Details

The rsc_cv function performs cross-validation to estimate the expected Frobenius loss proposed in Bickel and Levina (2008). The original contribution of Bickel and Levina (2008), and its extension in Serra et al. (2018), is based on a random cross-validation algorithm where the training/test size depends on the sample size n. The latter is implemented selecting cv.type = "ramdom", and fixing an appropriate number R of random train/test splits. R should be as large as possible, but in practice this impacts the computing time strongly for high-dimensional data sets.

Although Serra et al. (2018) showed that the random cross-validation of Bickel and Levina (2008) works well for the RSC estimator, subsequent experiments suggested that repeated K-fold cross-validation on average produces better results. Repeated K-fold cross-validation is implemented with the default cv.type = "kfold". In this case K defines the number of folds, while R defines the number of times that the K-fold cross-validation is repeated with R independent shuffles of the original data. Selecting R=1 and K=10 one performs the standard 10-fold cross-validation. Ten replicates (R=10) of the K-fold cross-validation are generally sufficient to obtain reasonable estimates of the underlying loss, but for extremely high-dimensional data R may be varied to speed up calculations.

On multi-core hardware the cross-validation is executed in parallel setting ncores. The parallelism is implemented on the total number of data splits, that is R for the random cross-validation, and R*K for the repeated K-fold cross-validation. The software is optimized so that generally the total computing time scales almost linearly with the number of available computer cores (ncores).

For both the random and the K-fold cross-validation it is computed the normalized version of the expected squared Frobenius loss proposed in Bickel and Levina (2008). The normalization is such that the squared Frobenius norm of the identity matrix equals to 1 whatever is its dimension.

Two optimal threshold selection types are reported with flags (see Value section below): "minimum" and "minimum1se". The flag "minimum" denotes the threshold value that minimizes the average loss. The flag "minimum1se" implements the so called 1-SE rule: this is the maximum threshold value such that the corresponding average loss is within 1-standard-error with respect to the threshold that minimizes the average loss (that is the one corresponding to the "minimum" flag).

Since unbiased standard errors for the K-fold cross-validation are impossible to compute (see Bengio and Grandvalet, 2004), when cv.type="kfold" the reported standard errors have to be considered as a downward biased approximation.

Value

An S3 object of class 'cv_rsc' with the following components:

rmadvec

A vector containing the lower triangle of the underlying RMAD matrix.

varnames

A character vector if variable names are available for the input data set x. Otherwise this is NULL.

loss

A data.frame reporting cross-validation estimates. Columns of loss are as follows: loss$Threshold is the threshold value; loss$Average is averaged loss; loss$SE is the standard error for the average loss; loss$Flag="minimum" denotes the threshold achieving the minimum average loss; loss$Flag = "*" denotes threshold values such that the average loss is within 1-standard-error with respect to the "minimum" solution.

minimum

A numeric value. This is the minimum of the average loss. This corresponds to the flag "minimum" in the loss component above (see Details).

minimum1se

A numeric value. This is the largest threshold such that the corresponding flag = "*". In practice this selects the optimal threshold based on the 1-SE rule discussed in the Details Section above.

References

Bengio, Y., and Grandvalet, Y. (2004). No unbiased estimator of the variance of k-fold cross-validation. Journal of Machine Learning Research, 5(Sep), 1089-1105.

Bickel, P. J., and Levina, E. (2008). Covariance regularization by thresholding. The Annals of Statistics, 36(6), 2577-2604. doi:10.1214/08-AOS600

Serra, A., Coretto, P., Fratello, M., and Tagliaferri, R. (2018). Robust and sparsecorrelation matrix estimation for the analysis of high-dimensional genomics data. Bioinformatics, 34(4), 625-634. doi:10.1093/bioinformatics/btx642

See Also

rsc, plot.rsc_cv

Examples

## simulate a random sample from a multivariate Cauchy distribution
## note: example in high-dimension are obtained increasing p
set.seed(1)
n   <- 100  # sample size
p   <- 10   # dimension
dat <- matrix(rt(n*p, df = 1), nrow = n, ncol = p)
colnames(dat) <- paste0("Var", 1:p)

   
## perform 10-fold cross-validation repeated R=10 times
## note: for multi-core machines experiment with 'ncores'
set.seed(2)
a <- rsc_cv(x = dat, R = 10, K = 10, ncores = 1)
a

   
## threshold selection: note that here, knowing the sampling designs,
## we would like to threshold any correlation larger than zero in
## absolute value
## 
a$minimum        ## "minimum"    flagged solution 
a$minimum1se     ## "minimum1se" flagged solution

## plot the cross-validation estimates
plot(a)

## to obtain the RSC matrix we pass 'a' to the rsc() function
b <- rsc(cv = a, threshold = "minimum")
b

d <- rsc(cv = a, threshold = "minimum1se")
d

## since the object 'a' stores the RMAD underlying estimator, we can
## apply thresholding at any level without re-estimating the RMAD 
## matrix
e <- rsc(cv = a, threshold = 0.5)
e