Cross-validation for Least-Squares (Adaptive) Elastic Net Estimates

Perform (repeated) K-fold cross-validation for elnet().

Usage

elnet_cv(
  x,
  y,
  lambda,
  cv_k,
  cv_repl = 1,
  cv_type = "naive",
  cv_metric = c("rmspe", "tau_size", "mape", "auroc"),
  fit_all = TRUE,
  cl = NULL,
  ...
)

Arguments

x

n by p matrix of numeric predictors.

y

vector of response values of length n. For binary classification, y should be a factor with 2 levels.

lambda

optional user-supplied sequence of penalization levels. If given and not NULL, nlambda and lambda_min_ratio are ignored.

cv_k

number of folds per cross-validation.

cv_repl

number of cross-validation replications.

cv_type

what kind of cross-validation should be performed: robust information sharing (ris) or standard (naive) CV.

cv_metric

only for cv_type='naive'. Either a string specifying the performance metric to use, or a function to evaluate prediction errors in a single CV replication. If a function, the number of arguments define the data the function receives. If the function takes a single argument, it is called with a single numeric vector of prediction errors. If the function takes two or more arguments, it is called with the predicted values as first argument and the true values as second argument. The function must always return a single numeric value quantifying the prediction performance. The order of the given values corresponds to the order in the input data.

fit_all

only for cv_type='naive'. If TRUE, fit the model for all penalization levels. Can also be any combination of "min" and "{x}-se", in which case only models at the penalization level with smallest average CV accuracy, or within {x} standard errors, respectively. Setting fit_all to FALSE is equivalent to "min". Applies to all alpha value.

cl

a parallel cluster. Can only be used in combination with ncores = 1.

...

Arguments passed on to elnet

alpha: elastic net penalty mixing parameter with \(0 \le \alpha \le 1\). alpha = 1 is the LASSO penalty, and alpha = 0 the Ridge penalty. Can be a vector of several values, but alpha = 0 cannot be mixed with other values.
nlambda: number of penalization levels.
lambda_min_ratio: Smallest value of the penalization level as a fraction of the largest level (i.e., the smallest value for which all coefficients are zero). The default depends on the sample size relative to the number of variables and alpha. If more observations than variables are available, the default is 1e-3 * alpha, otherwise 1e-2 * alpha.
penalty_loadings: a vector of positive penalty loadings (a.k.a. weights) for different penalization of each coefficient.
standardize: standardize variables to have unit variance. Coefficients are always returned in original scale.
weights: a vector of positive observation weights.
intercept: include an intercept in the model.
sparse: use sparse coefficient vectors.
en_algorithm_opts: options for the EN algorithm. See en_algorithm_options for details.
eps: numerical tolerance.

Value

a list-like object with the same components as returned by elnet(), plus the following:

cvres: data frame of average cross-validated performance.

Details

The built-in CV metrics are

"tau_size": \(\tau\)-size of the prediction error, computed by tau_size() (default).
"mape": Median absolute prediction error.
"rmspe": Root mean squared prediction error.
"auroc": Area under the receiver operator characteristic curve (actually 1 - AUROC). Only sensible for binary responses.

Examples

# Compute the LS-EN regularization path for Freeny's revenue data
# (see ?freeny)
data(freeny)
x <- as.matrix(freeny[ , 2:5])

regpath <- elnet(x, freeny$y, alpha = c(0.5, 0.75))
plot(regpath)

plot(regpath, alpha = 0.75)


# Extract the coefficients at a certain penalization level
coef(regpath, lambda = regpath$lambda[[1]][[5]],
     alpha = 0.75)
#>           (Intercept) lag.quarterly.revenue           price.index 
#>              9.306304              0.000000              0.000000 
#>          income.level      market.potential 
#>              0.000000              0.000000 

# What penalization level leads to good prediction performance?
set.seed(123)
cv_results <- elnet_cv(x, freeny$y, alpha = c(0.5, 0.75),
                       cv_repl = 10, cv_k = 4,
                       cv_measure = "tau")
plot(cv_results, se_mult = 1.5)

plot(cv_results, se_mult = 1.5, what = "coef.path")



# Extract the coefficients at the penalization level with
# smallest prediction error ...
summary(cv_results)
#> EN fit with prediction performance estimated by replications of 4-fold 
#> cross-validation.
#> 
#> 4 out of 4 predictors have non-zero coefficients:
#> 
#>               Estimate
#> (Intercept) -9.6491805
#> X1           0.1899399
#> X2          -0.6858733
#> X3           0.7075924
#> X4           1.2247539
#> ---
#> 
#> Hyper-parameters: lambda=0.003787891, alpha=0.5
coef(cv_results)
#>           (Intercept) lag.quarterly.revenue           price.index 
#>            -9.6491805             0.1899399            -0.6858733 
#>          income.level      market.potential 
#>             0.7075924             1.2247539 
# ... or at the penalization level with prediction error
# statistically indistinguishable from the minimum.
summary(cv_results, lambda = "1.5-se")
#> EN fit with prediction performance estimated by replications of 4-fold 
#> cross-validation.
#> 
#> 4 out of 4 predictors have non-zero coefficients:
#> 
#>               Estimate
#> (Intercept) -9.5875726
#> X1           0.2270959
#> X2          -0.6216322
#> X3           0.6519060
#> X4           1.1972788
#> ---
#> 
#> Hyper-parameters: lambda=0.01303176, alpha=0.5
coef(cv_results, lambda = "1.5-se")
#>           (Intercept) lag.quarterly.revenue           price.index 
#>            -9.5875726             0.2270959            -0.6216322 
#>          income.level      market.potential 
#>             0.6519060             1.1972788