Formula: PRSS(beta, lambda, alpha) = RSS(beta) + lambda [ (1 - alpha)/2 * sum_j b_j^2
alpha * sum_j |b_j| ]
[ (1 - alpha)/2 * sum_j b_j^2
alpha * sum_j |b_j| ]: R:%20(1%20-%20alpha)/2%20*%20sum_j%20b_j%5E2%0A+%20alpha%20*%20sum_j%20%7Cb_j%7C%20
Arguments
- X
Design matrix, one record per row.
- y
Response vector of length n.
- beta
Coefficient vector.
- lam
Regularization parameter lambda; must be non-negative.
- alpha
Mixing parameter in [0, 1]: 0 is the ridge penalty, 1 is the lasso penalty.
- add_intercept
Treat the first entry of beta as an unpenalized intercept and prepend a column of ones to X.
References
Montesinos Lopez, Montesinos Lopez and Crossa (2022), Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, doi:10.1007/978-3-030-89010-0. Chapter 3, Sect. 3.6.2 p. 94 describes the elastic net as the combination of the ridge and lasso penalties implemented by glmnet, and Chapter 7 p. 230 defines its mixing parameter alpha: alpha = 0 gives the ridge penalty, alpha = 1 the lasso, and values in between the elastic net. The book names those two endpoints but does not print the mixed objective itself; the (1 - alpha)/2 and alpha weights are the glmnet parameterization of Friedman, Hastie and Tibshirani (2010), Regularization Paths for Generalized Linear Models via Coordinate Descent, Journal of Statistical Software 33(1):1-22, doi:10.18637/jss.v033.i01, which is the package the book cites for this penalty. Both chapters read from the PDFs; the mixing weights come from the cited paper, and the endpoints agree with the book: alpha = 0 reproduces MVSML Sect. 3.6.1 up to the factor 1/2 on the ridge term, alpha = 1 reproduces Sect. 3.6.2 exactly.
Examples
set.seed(1)
X <- matrix(rnorm(20), 10, 2)
y <- rnorm(10)
Enetobj(X, y, beta = c(0, 0.5, -0.3), lam = 0.1, alpha = 0.5)
#> $prss
#> [1] 19.70967
#>
#> $rss
#> [1] 19.66117
#>
#> $penalty
#> [1] 0.0485
#>
#> $l1
#> [1] 0.8
#>
#> $l2
#> [1] 0.34
#>
#> $lambda
#> [1] 0.1
#>
#> $alpha
#> [1] 0.5
#>
#> $n
#> [1] 10
#>
#> $p
#> [1] 3
#>
#> $method
#> [1] "Elastic net penalized RSS, MVSML Sect. 3.6.2 / Chap. 7"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"