Date: 2026-05-31

# Introduction to pye and covYI estimators

#### 2026-05-31

    library(pye)

## 1. Package Introduction and Objective

The pye package provides an advanced framework for simultaneous variable selection and prediction within low- and high-dimensional binary classification contexts. Its core methodology focuses on maximizing the penalized Youden index function \(F_{Y=0}(\theta) - F_{Y=1}(\theta) - \Phi(\theta)\) with respect of the parameter vector \(\theta\), where \(F_{Y=y}(\theta)\) represents the distribution function of the feature combination for class \(Y=y\) and \(\Phi(\theta)\) is a sparsity-inducing penalty term. For the Penalized Youden Index Estimator (pye) - https://doi.org/10.1016/j.chemolab.2023.104786 -, \(\theta\) corresponds to the coefficients of a linear combination of biomarkers \(\beta\) together with diagnostic cut-off point \(c\). For the Covariate-Adjusted Youden Index Estimator (covYI), the biomarker score is a single known marker, and \(\theta\) denotes the coefficients of the linear combination of covariate that define the diagnostic cut-off point \(c\). pye is particularly suited for applications in medical diagnostics, where identifying a subset of relevant biomarkers is crucial for effective disease classification. covYI extends this framework by allowing the diagnostic cut-off to depend on patient covariates, thereby enhancing the accuracy of the biomarkers in heterogeneous populations.

The package implements two primary methodologies:

Penalized Youden Index Estimator (pye): The base estimator combining the Youden index function with sparsity-inducing penalization techniques (Lasso, SCAD, MCP) for simultaneous biomarker selection.

Covariate-Adjusted Youden Index Estimator (covYI): An extension of pye that allows the optimal diagnostic cut-off to be a linear function of patient covariates, improving the classification accuracy of the biomarker score.

## 3. Example Workflow

This section outlines a typical workflow. In practice, the user must define the specific functions for the gradient (`delta_fx`) and the proximal operator (`proxx`) based on their chosen penalty (e.g., SCAD, L1) and data structure.

    # Load the package
    library(pye)

    # 1. Simulate data for the example
    set.seed(123) # Always good to set a seed for reproducibility in examples
    cols <- 200
    cols_cov <- 20
    max_rho <- 0.2
    rows_train <- 200
    sim_data <- create_sample_with_covariates(rows_train = rows_train,
                                              cols = cols,
                                              cols_cov = cols_cov,
                                              max_rho = max_rho,
                                              seed = 1)
    df <- sim_data$train_df_scaled
    X <- sim_data$X
    y <- sim_data$y
    C <- sim_data$C
    regressors_betas <- sim_data$nregressors # True betas for evaluation
    regressors_gammas <- sim_data$ncovariates # True gammas for evaluation

    # 2. Set cross-validation parameters
    penalty <- "SCAD" # Penalty for betas in pye estimation
    penalty_g <- "L12" # Penalty for gammas in covYI estimation
    trend <- "monotone" # Trend for the KS estimation
    alpha <- 0.5
    c_function_of_covariates <- TRUE # Use covariates for 'c' estimation
    used_cores <- 1 # For this example, no parallelization
    max_iter <- 10 # Keep iterations low for a quick example run
    n_folds <- 3

    # 3. Calibrate lambda_max and lambda_min for betas (pye estimation)
    lambda_seq <- create_lambda(n = 4, lmax = 1.5, lmin = 0.1)
    lambda_seq <- as.numeric(formatC(lambda_seq, format = "e", digits = 9))

    # 4. Calibrate tau_max and tau_min for gammas (covYI estimation), if
    tau_seq <- create_lambda(n = 4, lmax = 0.15, lmin = 0.05)
    tau_seq <- as.numeric(formatC(tau_seq, format = "e", digits = 9))

    # 5. Run the cross-validation
    pye_cv_result <- pye_KS_compute_cv(
      n_folds = n_folds,
      df = df,
      X = X,
      y = y,
      C = C,
      lambda = lambda_seq,
      tau = tau_seq,
      trace = 1, # Show final results
      alpha = alpha,
      penalty = penalty,
      regressors_betas = regressors_betas,
      regressors_gammas = regressors_gammas,
      seed = 1,
      used_cores = used_cores,
      trend = trend,
      max_iter = max_iter,
      c_function_of_covariates = c_function_of_covariates,
      measure_to_select_lambda = "ccr",
      penalty_g = penalty_g,
      trend_g = trend,
      max_iter_g = max_iter
    )
    #> Starting CV with the following estimation/convergence parameters:
    #> max_iter: 10
    #> min_alpha: 1e-10
    #> convergence_error: 1e-07
    #> stepsizeShrink: 0.8
    #> c_function_of_covariates: FALSE
    #> ----------------------------------------------------------------
    #> |       starting with the 1 -th fold for the CV of  lambda       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ;
    #> total iters: 10 ; Backtraking iters: 2801 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1439 ; youden_index: 0.4301 ; sensitivity: 0.8169 ; specificity: 0.8871 ; geometric_mean: 0.8513 ; fdr: 0.1077 ; mcc: 0.7026 ; auc: 0.9339 ; corrclass: 0.8496 ;
    #> TP: 58 ; TN: 55 ; FP: 7 ; FN: 13 ; betas:
    #>      norm2     norm51     norm52     norm85    norm101    norm102    norm151
    #> -8.883e-03  3.638e-02 -5.276e-02  4.132e-05  4.879e-02 -8.119e-03  3.583e-02
    #>          c
    #>  4.209e-03
    #> Estimation time: 0.4234 mins
    #>
    #>
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ;
    #> total iters: 7 ; Backtraking iters: 1072 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.4096 ; youden_index: 0.8015 ; sensitivity: 0.9577 ; specificity: 1 ; geometric_mean: 0.9786 ; fdr: 0 ; mcc: 0.9558 ; auc: 0.992 ; corrclass: 0.9774 ;
    #> TP: 68 ; TN: 62 ; FP: 0 ; FN: 3 ; betas:
    #>     norm2     norm7    norm12    norm19    norm35    norm36    norm39    norm48
    #> -0.038653  0.001023 -0.001766  0.002524  0.003457  0.002034 -0.001358 -0.018633
    #>    norm51    norm52
    #>  0.072787 -0.106127
    #>  [ reached getOption("max.print") -- omitted 28 entries ]
    #> Estimation time: 0.3206 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ;
    #> total iters: 10 ; Backtraking iters: 958 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.6462 ; youden_index: 0.9474 ; sensitivity: 1 ; specificity: 1 ; geometric_mean: 1 ; fdr: 0 ; mcc: 1 ; auc: 1 ; corrclass: 1 ;
    #> TP: 71 ; TN: 62 ; FP: 0 ; FN: 0 ; betas:
    #>      norm2      norm7     norm19     norm23     norm33     norm38     norm39
    #> -5.493e-02  2.611e-02  8.883e-04 -1.700e-02 -9.422e-12 -1.228e-04 -2.071e-02
    #>     norm48     norm51     norm52
    #> -4.322e-02  8.497e-02 -1.301e-01
    #>  [ reached getOption("max.print") -- omitted 40 entries ]
    #> Estimation time: 0.3424 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ;
    #> total iters: 10 ; Backtraking iters: 336 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.7432 ; youden_index: 0.9736 ; sensitivity: 1 ; specificity: 1 ; geometric_mean: 1 ; fdr: 0 ; mcc: 1 ; auc: 1 ; corrclass: 1 ;
    #> TP: 71 ; TN: 62 ; FP: 0 ; FN: 0 ; betas:
    #>    norm2    norm7   norm23   norm39   norm48   norm51   norm52   norm55
    #> -0.09533  0.05259 -0.03704 -0.03610 -0.10607  0.20820 -0.30154 -0.05035
    #>   norm56   norm60
    #>  0.13052 -0.09013
    #>  [ reached getOption("max.print") -- omitted 22 entries ]
    #> Estimation time: 0.08017 mins
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
    #> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.04325 ; youden_index: 0.3656 ; sensitivity: 0.8056 ; specificity: 0.7419 ; geometric_mean: 0.7731 ; fdr: 0.2162 ; mcc: 0.549 ; auc: 0.8656 ; corrclass: 0.7761
    #> TP: 29 ; TN: 23 ; FP: 8 ; FN: 7 ;  betas:
    #>      norm2     norm51     norm52     norm85    norm101    norm102    norm151
    #> -8.883e-03  3.638e-02 -5.276e-02  4.132e-05  4.879e-02 -8.119e-03  3.583e-02
    #>          c
    #>  4.209e-03
    #> Cross-validation time: 0.4234 ; Number of iterations: 10
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
    #> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.09793 ; youden_index: 0.4041 ; sensitivity: 0.8333 ; specificity: 0.8065 ; geometric_mean: 0.8198 ; fdr: 0.1667 ; mcc: 0.6398 ; auc: 0.9041 ; corrclass: 0.8209
    #> TP: 30 ; TN: 25 ; FP: 6 ; FN: 6 ;  betas:
    #>      norm2      norm7     norm12     norm19     norm35     norm36     norm39
    #> -0.0386530  0.0010231 -0.0017658  0.0025245  0.0034575  0.0020343 -0.0013577
    #>     norm48     norm51     norm52     norm56     norm60     norm62     norm63
    #> -0.0186333  0.0727869 -0.1061266  0.0239432 -0.0183848 -0.0065960  0.0041911
    #>     norm68     norm81     norm85     norm87     norm90    norm100    norm101
    #> -0.0137400 -0.0002867  0.0176195  0.0006448 -0.0001538  0.0008860  0.0861813
    #>    norm102    norm119    norm123    norm128    norm134    norm136    norm137
    #> -0.0615628 -0.0096232  0.0073251  0.0046528  0.0085929  0.0011488 -0.0046215
    #>    norm151    norm164    norm171    norm172    norm175    norm182    norm189
    #>  0.0672071  0.0024349  0.0257507  0.0144692 -0.0041685  0.0022548 -0.0011191
    #>    norm194    norm199          c
    #>  0.0002176 -0.0082597  0.0014823
    #> Cross-validation time: 0.3206 ; Number of iterations: 7
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
    #> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.08839 ; youden_index: 0.3056 ; sensitivity: 0.75 ; specificity: 0.6774 ; geometric_mean: 0.7128 ; fdr: 0.2703 ; mcc: 0.4286 ; auc: 0.8056 ; corrclass: 0.7164
    #> TP: 27 ; TN: 21 ; FP: 10 ; FN: 9 ;  betas:
    #>      norm2      norm7     norm19     norm23     norm33     norm38     norm39
    #> -5.493e-02  2.611e-02  8.883e-04 -1.700e-02 -9.422e-12 -1.228e-04 -2.071e-02
    #>     norm48     norm51     norm52     norm54     norm55     norm56     norm60
    #> -4.322e-02  8.497e-02 -1.301e-01 -1.487e-03 -2.805e-02  4.575e-02 -4.297e-02
    #>     norm62     norm63     norm68     norm78     norm81     norm84     norm85
    #> -2.932e-02  6.070e-03 -3.597e-03  2.328e-03 -2.429e-03 -1.606e-03  4.013e-02
    #>     norm87     norm90     norm93     norm99    norm100    norm101    norm102
    #>  3.208e-02 -3.774e-03  4.728e-03 -2.031e-03  3.562e-04  1.080e-01 -1.071e-01
    #>    norm107    norm116    norm118    norm119    norm123    norm128    norm134
    #> -9.170e-04 -2.780e-03 -3.537e-03 -1.569e-02  2.667e-02  3.345e-02  2.909e-02
    #>    norm137    norm146    norm148    norm151    norm152    norm160    norm164
    #> -6.717e-03  2.481e-12 -6.616e-03  8.708e-02 -6.304e-04  9.685e-04  1.148e-03
    #>    norm171    norm172    norm182    norm184    norm189    norm196    norm199
    #>  3.961e-02  3.715e-02  2.537e-02  2.590e-02 -7.419e-03  3.037e-03 -2.769e-02
    #>          c
    #> -7.046e-03
    #> Cross-validation time: 0.3424 ; Number of iterations: 10
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
    #> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1735 ; youden_index: 0.3038 ; sensitivity: 0.75 ; specificity: 0.6452 ; geometric_mean: 0.6956 ; fdr: 0.2895 ; mcc: 0.3977 ; auc: 0.8038 ; corrclass: 0.7015
    #> TP: 27 ; TN: 20 ; FP: 11 ; FN: 9 ;  betas:
    #>     norm2     norm7    norm23    norm39    norm48    norm51    norm52    norm55
    #> -0.095328  0.052588 -0.037036 -0.036104 -0.106066  0.208202 -0.301543 -0.050351
    #>    norm56    norm60    norm62    norm68    norm78    norm81    norm85    norm87
    #>  0.130519 -0.090131 -0.031066 -0.010497  0.006443 -0.028826  0.101753  0.054334
    #>   norm101   norm102   norm118   norm119   norm123   norm128   norm134   norm137
    #>  0.243413 -0.236360 -0.004181 -0.008083  0.049554  0.087110  0.036275 -0.011315
    #>   norm148   norm151   norm171   norm172   norm182   norm184   norm199         c
    #> -0.020250  0.210508  0.087033  0.052670  0.033252  0.030829 -0.045795 -0.005458
    #> Cross-validation time: 0.08017 ; Number of iterations: 10
    #>
    #>
    #> ----------------------------------------------------------------
    #> |       starting with the 2 -th fold for the CV of  lambda       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ;
    #> total iters: 9 ; Backtraking iters: 2063 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1274 ; youden_index: 0.3514 ; sensitivity: 0.7183 ; specificity: 0.8387 ; geometric_mean: 0.7762 ; fdr: 0.1639 ; mcc: 0.5577 ; auc: 0.8682 ; corrclass: 0.7744 ;
    #> TP: 51 ; TN: 52 ; FP: 10 ; FN: 20 ; betas:
    #>      norm2     norm51     norm52    norm101    norm102          c
    #> -3.042e-02  2.970e-03 -8.125e-02  3.468e-02 -1.046e-11  9.826e-03
    #> Estimation time: 0.4589 mins
    #>
    #>
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ;
    #> total iters: 6 ; Backtraking iters: 655 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.379 ; youden_index: 0.7511 ; sensitivity: 0.9718 ; specificity: 0.9355 ; geometric_mean: 0.9535 ; fdr: 0.05479 ; mcc: 0.9096 ; auc: 0.993 ; corrclass: 0.9549 ;
    #> TP: 69 ; TN: 58 ; FP: 4 ; FN: 2 ; betas:
    #>     norm2     norm7    norm12    norm22    norm34    norm36    norm40    norm51
    #> -0.078068  0.001017 -0.030436 -0.008491  0.002881  0.012551 -0.001568  0.039504
    #>    norm52    norm55
    #> -0.112806 -0.012067
    #>  [ reached getOption("max.print") -- omitted 34 entries ]
    #> Estimation time: 0.2256 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ;
    #> total iters: 10 ; Backtraking iters: 398 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.621 ; youden_index: 0.9351 ; sensitivity: 0.9859 ; specificity: 1 ; geometric_mean: 0.9929 ; fdr: 0 ; mcc: 0.985 ; auc: 0.9986 ; corrclass: 0.9925 ;
    #> TP: 70 ; TN: 62 ; FP: 0 ; FN: 1 ; betas:
    #>      norm2      norm4      norm7     norm12     norm16     norm19     norm22
    #> -0.0982438 -0.0045076  0.0012349 -0.0413095  0.0022233  0.0023343 -0.0271606
    #>     norm29     norm32     norm34
    #> -0.0006758 -0.0021577  0.0088901
    #>  [ reached getOption("max.print") -- omitted 61 entries ]
    #> Estimation time: 0.08021 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ;
    #> total iters: 10 ; Backtraking iters: 383 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.7666 ; youden_index: 0.9071 ; sensitivity: 0.9859 ; specificity: 0.9516 ; geometric_mean: 0.9686 ; fdr: 0.0411 ; mcc: 0.9399 ; auc: 0.9984 ; corrclass: 0.9699 ;
    #> TP: 70 ; TN: 59 ; FP: 3 ; FN: 1 ; betas:
    #>     norm2    norm12    norm16    norm19    norm22    norm28    norm29    norm32
    #> -0.093286 -0.025668  0.013286  0.004502 -0.025337 -0.002432 -0.007680 -0.005990
    #>    norm34    norm35
    #>  0.010927  0.001249
    #>  [ reached getOption("max.print") -- omitted 57 entries ]
    #> Estimation time: 0.1002 mins
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
    #> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.09457 ; youden_index: 0.3342 ; sensitivity: 0.75 ; specificity: 0.6129 ; geometric_mean: 0.678 ; fdr: 0.3077 ; mcc: 0.3669 ; auc: 0.8342 ; corrclass: 0.6866
    #> TP: 27 ; TN: 19 ; FP: 12 ; FN: 9 ;  betas:
    #>      norm2     norm51     norm52    norm101    norm102          c
    #> -3.042e-02  2.970e-03 -8.125e-02  3.468e-02 -1.046e-11  9.826e-03
    #> Cross-validation time: 0.4589 ; Number of iterations: 9
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
    #> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1824 ; youden_index: 0.4534 ; sensitivity: 0.8889 ; specificity: 0.7419 ; geometric_mean: 0.8121 ; fdr: 0.2 ; mcc: 0.6413 ; auc: 0.9534 ; corrclass: 0.8209
    #> TP: 32 ; TN: 23 ; FP: 8 ; FN: 4 ;  betas:
    #>      norm2      norm7     norm12     norm22     norm34     norm36     norm40
    #> -0.0780677  0.0010172 -0.0304361 -0.0084905  0.0028808  0.0125506 -0.0015678
    #>     norm51     norm52     norm55     norm60     norm66     norm69     norm71
    #>  0.0395042 -0.1128065 -0.0120667 -0.0008384 -0.0122452 -0.0002343 -0.0014696
    #>     norm80     norm81     norm83     norm85     norm99    norm100    norm101
    #>  0.0029695 -0.0023213  0.0030099  0.0273645 -0.0028010  0.0107344  0.0811139
    #>    norm102    norm107    norm112    norm113    norm117    norm119    norm120
    #> -0.0535526 -0.0027352 -0.0056054 -0.0052368  0.0001407 -0.0130925  0.0021955
    #>    norm128    norm130    norm133    norm142    norm146    norm151    norm157
    #>  0.0015812  0.0085726  0.0038849 -0.0098571  0.0002589  0.0061427 -0.0076841
    #>    norm165    norm170    norm173    norm175    norm182    norm186    norm192
    #>  0.0053785  0.0006581 -0.0121030 -0.0093119  0.0021588  0.0094583 -0.0057368
    #>    norm195          c
    #> -0.0019014 -0.0045452
    #> Cross-validation time: 0.2256 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
    #> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1879 ; youden_index: 0.3934 ; sensitivity: 0.9167 ; specificity: 0.6452 ; geometric_mean: 0.769 ; fdr: 0.25 ; mcc: 0.59 ; auc: 0.8934 ; corrclass: 0.791
    #> TP: 33 ; TN: 20 ; FP: 11 ; FN: 3 ;  betas:
    #>      norm2      norm4      norm7     norm12     norm16     norm19     norm22
    #> -0.0982438 -0.0045076  0.0012349 -0.0413095  0.0022233  0.0023343 -0.0271606
    #>     norm29     norm32     norm34     norm35     norm36     norm49     norm50
    #> -0.0006758 -0.0021577  0.0088901  0.0008342  0.0316300 -0.0018156 -0.0015810
    #>     norm51     norm52     norm55     norm57     norm60     norm65     norm66
    #>  0.0575839 -0.1567682 -0.0256998 -0.0023048 -0.0011210  0.0001039 -0.0334586
    #>     norm67     norm70     norm71     norm74     norm79     norm81     norm82
    #>  0.0059217 -0.0003351 -0.0253818  0.0010431  0.0032396 -0.0024270  0.0064319
    #>     norm83     norm85     norm90     norm94     norm95     norm99    norm100
    #>  0.0184821  0.0377789 -0.0022337 -0.0202300 -0.0034231 -0.0240638  0.0342187
    #>    norm101    norm102    norm107    norm112    norm113    norm115    norm117
    #>  0.1148352 -0.0696042 -0.0195901 -0.0179246 -0.0343204  0.0003692  0.0115940
    #>    norm119    norm120    norm121    norm126    norm128    norm130    norm132
    #> -0.0210717  0.0054719  0.0016188 -0.0005131  0.0290092  0.0219371 -0.0021867
    #>    norm133    norm136    norm137    norm142    norm148    norm151    norm152
    #>  0.0138209  0.0014166 -0.0007836 -0.0462914 -0.0034694  0.0194033 -0.0147033
    #>    norm154    norm157    norm162    norm164    norm170    norm172    norm173
    #>  0.0022122 -0.0046509  0.0002648 -0.0015111  0.0020626  0.0001302 -0.0320901
    #>    norm175    norm185    norm186    norm187    norm192    norm195    norm200
    #> -0.0303693 -0.0002396  0.0154924 -0.0023889 -0.0269424 -0.0079648  0.0103961
    #>          c
    #> -0.0014563
    #> Cross-validation time: 0.08021 ; Number of iterations: 10
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
    #> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.4388 ; youden_index: 0.4104 ; sensitivity: 0.8889 ; specificity: 0.7419 ; geometric_mean: 0.8121 ; fdr: 0.2 ; mcc: 0.6413 ; auc: 0.9104 ; corrclass: 0.8209
    #> TP: 32 ; TN: 23 ; FP: 8 ; FN: 4 ;  betas:
    #>      norm2     norm12     norm16     norm19     norm22     norm28     norm29
    #> -9.329e-02 -2.567e-02  1.329e-02  4.502e-03 -2.534e-02 -2.432e-03 -7.680e-03
    #>     norm32     norm34     norm35     norm36     norm37     norm48     norm49
    #> -5.990e-03  1.093e-02  1.249e-03  3.461e-02  1.279e-04 -2.407e-02 -5.492e-05
    #>     norm50     norm51     norm52     norm55     norm57     norm60     norm65
    #> -3.872e-03  6.988e-02 -2.110e-01 -3.880e-03 -1.767e-02 -7.679e-03  1.189e-03
    #>     norm66     norm67     norm71     norm80     norm81     norm84     norm85
    #> -2.292e-02  2.852e-02 -5.040e-02  1.528e-02 -2.665e-03 -9.797e-04  1.847e-02
    #>     norm94     norm95     norm99    norm100    norm101    norm102    norm107
    #> -1.589e-02 -1.477e-03 -4.190e-02  2.180e-02  1.462e-01 -7.730e-02 -4.294e-03
    #>    norm112    norm113    norm115    norm117    norm119    norm121    norm128
    #> -1.747e-02 -1.561e-02  7.649e-03  1.396e-02 -4.997e-03  4.669e-03  3.167e-02
    #>    norm130    norm131    norm132    norm133    norm142    norm143    norm146
    #>  1.202e-02  3.601e-02 -8.618e-03  1.524e-02 -3.889e-02  4.398e-03  9.943e-03
    #>    norm149    norm150    norm151    norm154    norm162    norm163    norm165
    #> -8.861e-03  4.281e-03  3.562e-02  1.441e-02  1.907e-03 -2.118e-03  2.326e-02
    #>    norm168    norm169    norm173    norm175    norm176    norm182    norm187
    #>  8.125e-03  2.277e-03 -1.079e-02 -2.696e-02 -1.122e-02  2.526e-02 -1.777e-03
    #>    norm192    norm195    norm200          c
    #> -1.272e-02 -7.356e-04  7.703e-03  3.162e-03
    #> Cross-validation time: 0.1002 ; Number of iterations: 10
    #>
    #>
    #> ----------------------------------------------------------------
    #> |       starting with the 3 -th fold for the CV of  lambda       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ;
    #> total iters: 10 ; Backtraking iters: 2466 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.141 ; youden_index: 0.4049 ; sensitivity: 0.8472 ; specificity: 0.8226 ; geometric_mean: 0.8348 ; fdr: 0.1528 ; mcc: 0.6698 ; auc: 0.914 ; corrclass: 0.8358 ;
    #> TP: 61 ; TN: 51 ; FP: 11 ; FN: 11 ; betas:
    #>    norm2   norm51   norm52  norm101        c
    #> -0.02326  0.02758 -0.07538  0.04966  0.00851
    #> Estimation time: 0.5881 mins
    #>
    #>
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ;
    #> total iters: 6 ; Backtraking iters: 1126 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3879 ; youden_index: 0.7759 ; sensitivity: 1 ; specificity: 0.9839 ; geometric_mean: 0.9919 ; fdr: 0.0137 ; mcc: 0.9851 ; auc: 0.9913 ; corrclass: 0.9925 ;
    #> TP: 72 ; TN: 61 ; FP: 1 ; FN: 0 ; betas:
    #>      norm2      norm9     norm12     norm28     norm43     norm49     norm51
    #> -0.0864853  0.0017694 -0.0093260 -0.0013788 -0.0023924 -0.0069509  0.0716253
    #>     norm52     norm54     norm63
    #> -0.1217726  0.0005595  0.0015980
    #>  [ reached getOption("max.print") -- omitted 21 entries ]
    #> Estimation time: 0.3577 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ;
    #> total iters: 10 ; Backtraking iters: 397 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.6223 ; youden_index: 0.9441 ; sensitivity: 1 ; specificity: 0.9839 ; geometric_mean: 0.9919 ; fdr: 0.0137 ; mcc: 0.9851 ; auc: 1 ; corrclass: 0.9925 ;
    #> TP: 72 ; TN: 61 ; FP: 1 ; FN: 0 ; betas:
    #>      norm1      norm2      norm9     norm12     norm13     norm17     norm18
    #>  0.0179363 -0.1487871  0.0030999 -0.0156110 -0.0046801 -0.0009403  0.0002065
    #>     norm24     norm27     norm28
    #> -0.0191010 -0.0001254 -0.0004629
    #>  [ reached getOption("max.print") -- omitted 50 entries ]
    #> Estimation time: 0.07991 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ;
    #> total iters: 10 ; Backtraking iters: 318 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.712 ; youden_index: 0.9873 ; sensitivity: 1 ; specificity: 1 ; geometric_mean: 1 ; fdr: 0 ; mcc: 1 ; auc: 1 ; corrclass: 1 ;
    #> TP: 72 ; TN: 62 ; FP: 0 ; FN: 0 ; betas:
    #>      norm1      norm2      norm4      norm7     norm12     norm13     norm21
    #>  0.1115717 -0.2509538  0.0003126  0.0046432 -0.0286906 -0.0639457 -0.0024715
    #>     norm24     norm28     norm31
    #> -0.0570286 -0.0399914 -0.0283538
    #>  [ reached getOption("max.print") -- omitted 55 entries ]
    #> Estimation time: 0.07492 mins
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
    #> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.09437 ; youden_index: 0.3765 ; sensitivity: 0.7143 ; specificity: 0.871 ; geometric_mean: 0.7887 ; fdr: 0.1379 ; mcc: 0.5885 ; auc: 0.8765 ; corrclass: 0.7879
    #> TP: 25 ; TN: 27 ; FP: 4 ; FN: 10 ;  betas:
    #>    norm2   norm51   norm52  norm101        c
    #> -0.02326  0.02758 -0.07538  0.04966  0.00851
    #> Cross-validation time: 0.5881 ; Number of iterations: 10
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
    #> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2544 ; youden_index: 0.4493 ; sensitivity: 0.8857 ; specificity: 0.8387 ; geometric_mean: 0.8619 ; fdr: 0.1389 ; mcc: 0.7261 ; auc: 0.9493 ; corrclass: 0.8636
    #> TP: 31 ; TN: 26 ; FP: 5 ; FN: 4 ;  betas:
    #>      norm2      norm9     norm12     norm28     norm43     norm49     norm51
    #> -0.0864853  0.0017694 -0.0093260 -0.0013788 -0.0023924 -0.0069509  0.0716253
    #>     norm52     norm54     norm63     norm66     norm85     norm97    norm101
    #> -0.1217726  0.0005595  0.0015980 -0.0079350  0.0222619 -0.0020575  0.0946004
    #>    norm102    norm112    norm119    norm122    norm124    norm125    norm141
    #> -0.0668193 -0.0115183 -0.0190459  0.0003671  0.0116841 -0.0063316  0.0063667
    #>    norm145    norm146    norm151    norm156    norm157    norm160    norm175
    #> -0.0086135  0.0041223  0.0301803 -0.0021094 -0.0002038 -0.0008814 -0.0105161
    #>    norm182    norm194          c
    #>  0.0213159  0.0072770  0.0053100
    #> Cross-validation time: 0.3577 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
    #> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3478 ; youden_index: 0.429 ; sensitivity: 0.9143 ; specificity: 0.8065 ; geometric_mean: 0.8587 ; fdr: 0.1579 ; mcc: 0.7278 ; auc: 0.929 ; corrclass: 0.8636
    #> TP: 32 ; TN: 25 ; FP: 6 ; FN: 3 ;  betas:
    #>      norm1      norm2      norm9     norm12     norm13     norm17     norm18
    #>  1.794e-02 -1.488e-01  3.100e-03 -1.561e-02 -4.680e-03 -9.403e-04  2.065e-04
    #>     norm24     norm27     norm28     norm31     norm43     norm48     norm49
    #> -1.910e-02 -1.254e-04 -4.629e-04 -8.200e-04 -1.864e-02 -3.414e-04 -2.465e-02
    #>     norm51     norm52     norm54     norm62     norm64     norm66     norm84
    #>  1.081e-01 -1.848e-01  1.163e-03 -4.066e-04 -1.047e-03 -1.791e-02  1.292e-03
    #>     norm85     norm87     norm88     norm97     norm98    norm101    norm102
    #>  5.810e-02  7.141e-03 -8.505e-04 -1.998e-02  6.098e-03  1.490e-01 -1.136e-01
    #>    norm105    norm112    norm115    norm119    norm122    norm123    norm124
    #> -2.670e-04 -3.564e-02  2.459e-03 -2.650e-02  1.697e-03  9.751e-05  2.206e-02
    #>    norm125    norm136    norm137    norm138    norm141    norm145    norm146
    #> -2.589e-02  3.045e-04 -1.435e-02  2.267e-04  2.590e-02 -1.969e-02  1.949e-02
    #>    norm147    norm151    norm152    norm156    norm159    norm160    norm162
    #> -7.455e-03  3.868e-02 -1.761e-02 -3.567e-04  1.061e-03 -1.876e-02  3.543e-03
    #>    norm164    norm168    norm171    norm175    norm181    norm182    norm183
    #>  1.335e-02  7.076e-05  1.625e-03 -3.516e-02  2.668e-03  3.568e-02  3.933e-04
    #>    norm184    norm190    norm194          c
    #>  8.390e-03  7.280e-05  3.768e-04 -6.697e-03
    #> Cross-validation time: 0.07991 ; Number of iterations: 10
    #>
    #>
    #> -> Results on the TEST SET of pye
    #> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
    #> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3318 ; youden_index: 0.3931 ; sensitivity: 0.8857 ; specificity: 0.7419 ; geometric_mean: 0.8106 ; fdr: 0.2051 ; mcc: 0.6371 ; auc: 0.8931 ; corrclass: 0.8182
    #> TP: 31 ; TN: 23 ; FP: 8 ; FN: 4 ;  betas:
    #>      norm1      norm2      norm4      norm7     norm12     norm13     norm21
    #>  1.116e-01 -2.510e-01  3.126e-04  4.643e-03 -2.869e-02 -6.395e-02 -2.471e-03
    #>     norm24     norm28     norm31     norm37     norm43     norm46     norm48
    #> -5.703e-02 -3.999e-02 -2.835e-02  7.803e-03 -4.974e-02  3.209e-03 -7.010e-03
    #>     norm49     norm50     norm51     norm52     norm54     norm57     norm66
    #> -5.573e-02  1.040e-02  2.144e-01 -3.096e-01  2.716e-03 -1.843e-04 -7.040e-02
    #>     norm67     norm70     norm79     norm81     norm84     norm85     norm86
    #>  7.129e-04 -1.112e-05  7.906e-03  1.408e-03  9.077e-03  1.416e-01 -1.631e-03
    #>     norm89     norm90     norm97     norm98    norm101    norm102    norm112
    #> -6.102e-03  3.190e-03 -7.917e-02  1.264e-02  2.600e-01 -1.861e-01 -6.506e-02
    #>    norm115    norm119    norm120    norm124    norm125    norm126    norm130
    #>  3.688e-02 -8.066e-02  3.564e-02  3.670e-02 -8.236e-02  5.415e-03  2.711e-02
    #>    norm137    norm141    norm145    norm146    norm147    norm151    norm152
    #> -2.414e-02  1.025e-03 -6.051e-03  7.258e-02 -4.856e-03  8.383e-02 -3.097e-03
    #>    norm157    norm160    norm162    norm164    norm168    norm175    norm176
    #> -4.484e-03 -7.040e-02  1.385e-03  7.926e-03  4.625e-03 -7.948e-02  9.018e-04
    #>    norm181    norm182    norm183    norm184    norm185    norm192    norm194
    #>  3.567e-02  9.738e-02  1.578e-03  4.849e-02  1.669e-03 -2.268e-03  1.413e-02
    #>    norm196          c
    #>  6.558e-04 -2.613e-02
    #> Cross-validation time: 0.07492 ; Number of iterations: 10
    #>
    #>
    #>
    #>  Starting the CV of tau using as lambda: 0.6082 , that is the best value of lambda as per: ccr
    #> ----------------------------------------------------------------
    #> |       starting with the 1 -th fold for the CV of  tau       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ;
    #> total iters: 7 ; Backtraking iters: 1072 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.4096 ; youden_index: 0.8015 ; sensitivity: 0.9577 ; specificity: 1 ; geometric_mean: 0.9786 ; fdr: 0 ; mcc: 0.9558 ; auc: 0.992 ; corrclass: 0.9774 ;
    #> TP: 68 ; TN: 62 ; FP: 0 ; FN: 3 ; betas:
    #>     norm2     norm7    norm12    norm19    norm35    norm36    norm39    norm48
    #> -0.038653  0.001023 -0.001766  0.002524  0.003457  0.002034 -0.001358 -0.018633
    #>    norm51    norm52
    #>  0.072787 -0.106127
    #>  [ reached getOption("max.print") -- omitted 28 entries ]
    #> Estimation time: 0.3292 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ; total iters: 8 ; Backtraking iters: 72 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8039 ; youden_index: 0.8327 ; aYI: 0 ; sensitivity: 0.9718 ; specificity: 1 ; geometric_mean: 0.9858 ; fdr: 0 ; mcc: 0.9703 ; auc: 0.992 ; aauc: 0 ; corrclass: 0.985 ;
    #> TP: 69 ; TN: 62 ; FP: 0 ; FN: 2 ; gammas:
    #> norm_cov6
    #>  -0.03678
    #> Estimation time: 0.01414 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ; total iters: 10 ; Backtraking iters: 90 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8132 ; youden_index: 0.8348 ; aYI: 0 ; sensitivity: 0.9718 ; specificity: 1 ; geometric_mean: 0.9858 ; fdr: 0 ; mcc: 0.9703 ; auc: 0.992 ; aauc: 0 ; corrclass: 0.985 ;
    #> TP: 69 ; TN: 62 ; FP: 0 ; FN: 2 ; gammas:
    #> norm_cov6
    #>  -0.04334
    #> Estimation time: 0.01545 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ; total iters: 8 ; Backtraking iters: 72 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8229 ; youden_index: 0.85 ; aYI: 0 ; sensitivity: 0.9718 ; specificity: 1 ; geometric_mean: 0.9858 ; fdr: 0 ; mcc: 0.9703 ; auc: 0.992 ; aauc: 0 ; corrclass: 0.985 ;
    #> TP: 69 ; TN: 62 ; FP: 0 ; FN: 2 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.04046    0.03042
    #> Estimation time: 0.01624 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 1 ; total iters: 10 ; Backtraking iters: 131 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8321 ; youden_index: 0.859 ; aYI: 0 ; sensitivity: 0.9859 ; specificity: 1 ; geometric_mean: 0.9929 ; fdr: 0 ; mcc: 0.985 ; auc: 0.992 ; aauc: 0 ; corrclass: 0.9925 ;
    #> TP: 70 ; TN: 62 ; FP: 0 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11 norm_cov16
    #>   -0.04665    0.03748   -0.01644
    #> Estimation time: 0.01961 mins
    #>
    #>
    #>
    #> Final results on TEST SET of covYI for the FOLD: 1 :
    #> With lambda: 0.6082 ,  penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
    #> pye_KS: 0.09793 ; youden_index: 0.4041 ; sensitivity: 0.8333 ; specificity: 0.8065 ; geometric_mean: 0.8198 ; fdr: 0.1667 ; mcc: 0.6398 ; auc: 0.9041 ; corrclass: 0.8209 ;
    #> TP: 30 ; TN: 25 ; FP: 6 ; FN: 6 ;  betas:
    #>      norm2      norm7     norm12     norm19     norm35     norm36     norm39
    #> -0.0386530  0.0010231 -0.0017658  0.0025245  0.0034575  0.0020343 -0.0013577
    #>     norm48     norm51     norm52     norm56     norm60     norm62     norm63
    #> -0.0186333  0.0727869 -0.1061266  0.0239432 -0.0183848 -0.0065960  0.0041911
    #>     norm68     norm81     norm85     norm87     norm90    norm100    norm101
    #> -0.0137400 -0.0002867  0.0176195  0.0006448 -0.0001538  0.0008860  0.0861813
    #>    norm102    norm119    norm123    norm128    norm134    norm136    norm137
    #> -0.0615628 -0.0096232  0.0073251  0.0046528  0.0085929  0.0011488 -0.0046215
    #>    norm151    norm164    norm171    norm172    norm175    norm182    norm189
    #>  0.0672071  0.0024349  0.0257507  0.0144692 -0.0041685  0.0022548 -0.0011191
    #>    norm194    norm199          c
    #>  0.0002176 -0.0082597  0.0014823
    #> Cross-validation time: 0.3292 ; Number of iterations: 7
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.15 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5248 ; youden_index: 0.5833 ; aYI: 0 ; sensitivity: 0.8889 ; specificity: 0.871 ; geometric_mean: 0.8799 ; fdr: 0.1111 ; mcc: 0.7599 ; auc: 0.9041 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 32 ; TN: 27 ; FP: 4 ; FN: 4 ; gammas:
    #> norm_cov6
    #>  -0.03678
    #> Cross-validation time: 0.01414 ; Number of iterations: 8
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.104 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5382 ; youden_index: 0.5833 ; aYI: 0 ; sensitivity: 0.8889 ; specificity: 0.871 ; geometric_mean: 0.8799 ; fdr: 0.1111 ; mcc: 0.7599 ; auc: 0.9041 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 32 ; TN: 27 ; FP: 4 ; FN: 4 ; gammas:
    #> norm_cov6
    #>  -0.04334
    #> Cross-validation time: 0.01545 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5638 ; youden_index: 0.5833 ; aYI: 0 ; sensitivity: 0.8889 ; specificity: 0.871 ; geometric_mean: 0.8799 ; fdr: 0.1111 ; mcc: 0.7599 ; auc: 0.9041 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 32 ; TN: 27 ; FP: 4 ; FN: 4 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.04046    0.03042
    #> Cross-validation time: 0.01624 ; Number of iterations: 8
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.05 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5644 ; youden_index: 0.5833 ; aYI: 0 ; sensitivity: 0.8889 ; specificity: 0.871 ; geometric_mean: 0.8799 ; fdr: 0.1111 ; mcc: 0.7599 ; auc: 0.9041 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 32 ; TN: 27 ; FP: 4 ; FN: 4 ; gammas:
    #>  norm_cov6 norm_cov11 norm_cov16
    #>   -0.04665    0.03748   -0.01644
    #> Cross-validation time: 0.01961 ; Number of iterations: 10
    #>
    #>
    #>
    #> ----------------------------------------------------------------
    #> |       starting with the 2 -th fold for the CV of  tau       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ;
    #> total iters: 6 ; Backtraking iters: 655 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.379 ; youden_index: 0.7511 ; sensitivity: 0.9718 ; specificity: 0.9355 ; geometric_mean: 0.9535 ; fdr: 0.05479 ; mcc: 0.9096 ; auc: 0.993 ; corrclass: 0.9549 ;
    #> TP: 69 ; TN: 58 ; FP: 4 ; FN: 2 ; betas:
    #>     norm2     norm7    norm12    norm22    norm34    norm36    norm40    norm51
    #> -0.078068  0.001017 -0.030436 -0.008491  0.002881  0.012551 -0.001568  0.039504
    #>    norm52    norm55
    #> -0.112806 -0.012067
    #>  [ reached getOption("max.print") -- omitted 34 entries ]
    #> Estimation time: 0.2483 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ; total iters: 10 ; Backtraking iters: 169 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7602 ; youden_index: 0.7895 ; aYI: 0 ; sensitivity: 0.9859 ; specificity: 0.9516 ; geometric_mean: 0.9686 ; fdr: 0.0411 ; mcc: 0.9399 ; auc: 0.993 ; aauc: 0 ; corrclass: 0.9699 ;
    #> TP: 70 ; TN: 59 ; FP: 3 ; FN: 1 ; gammas:
    #> norm_cov6
    #>  -0.03815
    #> Estimation time: 0.02088 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ; total iters: 10 ; Backtraking iters: 151 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7719 ; youden_index: 0.8101 ; aYI: 0 ; sensitivity: 1 ; specificity: 0.9355 ; geometric_mean: 0.9672 ; fdr: 0.05333 ; mcc: 0.9411 ; auc: 0.993 ; aauc: 0 ; corrclass: 0.9699 ;
    #> TP: 71 ; TN: 58 ; FP: 4 ; FN: 0 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03445    0.03306
    #> Estimation time: 0.01732 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ; total iters: 10 ; Backtraking iters: 144 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7839 ; youden_index: 0.8116 ; aYI: 0 ; sensitivity: 1 ; specificity: 0.9355 ; geometric_mean: 0.9672 ; fdr: 0.05333 ; mcc: 0.9411 ; auc: 0.993 ; aauc: 0 ; corrclass: 0.9699 ;
    #> TP: 71 ; TN: 58 ; FP: 4 ; FN: 0 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03705    0.03708
    #> Estimation time: 0.01738 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 2 ; total iters: 10 ; Backtraking iters: 134 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7937 ; youden_index: 0.8206 ; aYI: 0 ; sensitivity: 0.9718 ; specificity: 0.9355 ; geometric_mean: 0.9535 ; fdr: 0.05479 ; mcc: 0.9096 ; auc: 0.993 ; aauc: 0 ; corrclass: 0.9549 ;
    #> TP: 69 ; TN: 58 ; FP: 4 ; FN: 2 ; gammas:
    #>  norm_cov6  norm_cov7 norm_cov11
    #>   -0.03667    0.01900    0.04409
    #> Estimation time: 0.01821 mins
    #>
    #>
    #>
    #> Final results on TEST SET of covYI for the FOLD: 2 :
    #> With lambda: 0.6082 ,  penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
    #> pye_KS: 0.1824 ; youden_index: 0.4534 ; sensitivity: 0.8889 ; specificity: 0.7419 ; geometric_mean: 0.8121 ; fdr: 0.2 ; mcc: 0.6413 ; auc: 0.9534 ; corrclass: 0.8209 ;
    #> TP: 32 ; TN: 23 ; FP: 8 ; FN: 4 ;  betas:
    #>      norm2      norm7     norm12     norm22     norm34     norm36     norm40
    #> -0.0780677  0.0010172 -0.0304361 -0.0084905  0.0028808  0.0125506 -0.0015678
    #>     norm51     norm52     norm55     norm60     norm66     norm69     norm71
    #>  0.0395042 -0.1128065 -0.0120667 -0.0008384 -0.0122452 -0.0002343 -0.0014696
    #>     norm80     norm81     norm83     norm85     norm99    norm100    norm101
    #>  0.0029695 -0.0023213  0.0030099  0.0273645 -0.0028010  0.0107344  0.0811139
    #>    norm102    norm107    norm112    norm113    norm117    norm119    norm120
    #> -0.0535526 -0.0027352 -0.0056054 -0.0052368  0.0001407 -0.0130925  0.0021955
    #>    norm128    norm130    norm133    norm142    norm146    norm151    norm157
    #>  0.0015812  0.0085726  0.0038849 -0.0098571  0.0002589  0.0061427 -0.0076841
    #>    norm165    norm170    norm173    norm175    norm182    norm186    norm192
    #>  0.0053785  0.0006581 -0.0121030 -0.0093119  0.0021588  0.0094583 -0.0057368
    #>    norm195          c
    #> -0.0019014 -0.0045452
    #> Cross-validation time: 0.2483 ; Number of iterations: 6
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.15 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5652 ; youden_index: 0.6389 ; aYI: 0 ; sensitivity: 0.9167 ; specificity: 0.8387 ; geometric_mean: 0.8768 ; fdr: 0.1316 ; mcc: 0.7602 ; auc: 0.9534 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 33 ; TN: 26 ; FP: 5 ; FN: 3 ; gammas:
    #> norm_cov6
    #>  -0.03815
    #> Cross-validation time: 0.02088 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.104 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5817 ; youden_index: 0.6389 ; aYI: 0 ; sensitivity: 0.9722 ; specificity: 0.871 ; geometric_mean: 0.9202 ; fdr: 0.1026 ; mcc: 0.8524 ; auc: 0.9534 ; aauc: 0 ; corrclass: 0.9254 ;
    #> TP: 35 ; TN: 27 ; FP: 4 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03445    0.03306
    #> Cross-validation time: 0.01732 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5957 ; youden_index: 0.6389 ; aYI: 0 ; sensitivity: 0.9722 ; specificity: 0.871 ; geometric_mean: 0.9202 ; fdr: 0.1026 ; mcc: 0.8524 ; auc: 0.9534 ; aauc: 0 ; corrclass: 0.9254 ;
    #> TP: 35 ; TN: 27 ; FP: 4 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03705    0.03708
    #> Cross-validation time: 0.01738 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.05 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.5948 ; youden_index: 0.6389 ; aYI: 0 ; sensitivity: 0.9167 ; specificity: 0.8387 ; geometric_mean: 0.8768 ; fdr: 0.1316 ; mcc: 0.7602 ; auc: 0.9534 ; aauc: 0 ; corrclass: 0.8806 ;
    #> TP: 33 ; TN: 26 ; FP: 5 ; FN: 3 ; gammas:
    #>  norm_cov6  norm_cov7 norm_cov11
    #>   -0.03667    0.01900    0.04409
    #> Cross-validation time: 0.01821 ; Number of iterations: 10
    #>
    #>
    #>
    #> ----------------------------------------------------------------
    #> |       starting with the 3 -th fold for the CV of  tau       |
    #> ----------------------------------------------------------------
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ;
    #> total iters: 6 ; Backtraking iters: 1126 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3879 ; youden_index: 0.7759 ; sensitivity: 1 ; specificity: 0.9839 ; geometric_mean: 0.9919 ; fdr: 0.0137 ; mcc: 0.9851 ; auc: 0.9913 ; corrclass: 0.9925 ;
    #> TP: 72 ; TN: 61 ; FP: 1 ; FN: 0 ; betas:
    #>      norm2      norm9     norm12     norm28     norm43     norm49     norm51
    #> -0.0864853  0.0017694 -0.0093260 -0.0013788 -0.0023924 -0.0069509  0.0716253
    #>     norm52     norm54     norm63
    #> -0.1217726  0.0005595  0.0015980
    #>  [ reached getOption("max.print") -- omitted 21 entries ]
    #> Estimation time: 0.5246 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ; total iters: 10 ; Backtraking iters: 598 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7806 ; youden_index: 0.8105 ; aYI: 0 ; sensitivity: 1 ; specificity: 0.9839 ; geometric_mean: 0.9919 ; fdr: 0.0137 ; mcc: 0.9851 ; auc: 0.9913 ; aauc: 0 ; corrclass: 0.9925 ;
    #> TP: 72 ; TN: 61 ; FP: 1 ; FN: 0 ; gammas:
    #> norm_cov11
    #>    0.03962
    #> Estimation time: 0.06074 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ; total iters: 9 ; Backtraking iters: 558 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7966 ; youden_index: 0.8361 ; aYI: 0 ; sensitivity: 0.9861 ; specificity: 0.9839 ; geometric_mean: 0.985 ; fdr: 0.01389 ; mcc: 0.97 ; auc: 0.9913 ; aauc: 0 ; corrclass: 0.9851 ;
    #> TP: 71 ; TN: 61 ; FP: 1 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03429    0.03806
    #> Estimation time: 0.07736 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ; total iters: 10 ; Backtraking iters: 558 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.809 ; youden_index: 0.8378 ; aYI: 0 ; sensitivity: 0.9861 ; specificity: 0.9839 ; geometric_mean: 0.985 ; fdr: 0.01389 ; mcc: 0.97 ; auc: 0.9913 ; aauc: 0 ; corrclass: 0.9851 ;
    #> TP: 71 ; TN: 61 ; FP: 1 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03747    0.04205
    #> Estimation time: 0.02821 mins
    #>
    #>
    #> mmAPG converged because the maximum number of iteration has been reached.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> fold = 3 ; total iters: 10 ; Backtraking iters: 571 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.818 ; youden_index: 0.8385 ; aYI: 0 ; sensitivity: 0.9861 ; specificity: 0.9839 ; geometric_mean: 0.985 ; fdr: 0.01389 ; mcc: 0.97 ; auc: 0.9913 ; aauc: 0 ; corrclass: 0.9851 ;
    #> TP: 71 ; TN: 61 ; FP: 1 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03959    0.04470
    #> Estimation time: 0.02355 mins
    #>
    #>
    #>
    #> Final results on TEST SET of covYI for the FOLD: 3 :
    #> With lambda: 0.6082 ,  penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
    #> pye_KS: 0.2544 ; youden_index: 0.4493 ; sensitivity: 0.8857 ; specificity: 0.8387 ; geometric_mean: 0.8619 ; fdr: 0.1389 ; mcc: 0.7261 ; auc: 0.9493 ; corrclass: 0.8636 ;
    #> TP: 31 ; TN: 26 ; FP: 5 ; FN: 4 ;  betas:
    #>      norm2      norm9     norm12     norm28     norm43     norm49     norm51
    #> -0.0864853  0.0017694 -0.0093260 -0.0013788 -0.0023924 -0.0069509  0.0716253
    #>     norm52     norm54     norm63     norm66     norm85     norm97    norm101
    #> -0.1217726  0.0005595  0.0015980 -0.0079350  0.0222619 -0.0020575  0.0946004
    #>    norm102    norm112    norm119    norm122    norm124    norm125    norm141
    #> -0.0668193 -0.0115183 -0.0190459  0.0003671  0.0116841 -0.0063316  0.0063667
    #>    norm145    norm146    norm151    norm156    norm157    norm160    norm175
    #> -0.0086135  0.0041223  0.0301803 -0.0021094 -0.0002038 -0.0008814 -0.0105161
    #>    norm182    norm194          c
    #>  0.0213159  0.0072770  0.0053100
    #> Cross-validation time: 0.5246 ; Number of iterations: 6
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.15 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.6481 ; youden_index: 0.6857 ; aYI: 0 ; sensitivity: 0.9143 ; specificity: 0.8065 ; geometric_mean: 0.8587 ; fdr: 0.1579 ; mcc: 0.7278 ; auc: 0.9493 ; aauc: 0 ; corrclass: 0.8636 ;
    #> TP: 32 ; TN: 25 ; FP: 6 ; FN: 3 ; gammas:
    #> norm_cov11
    #>    0.03962
    #> Cross-validation time: 0.06074 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.104 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.6912 ; youden_index: 0.6857 ; aYI: 0 ; sensitivity: 0.9714 ; specificity: 0.8387 ; geometric_mean: 0.9026 ; fdr: 0.1282 ; mcc: 0.8224 ; auc: 0.9493 ; aauc: 0 ; corrclass: 0.9091 ;
    #> TP: 34 ; TN: 26 ; FP: 5 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03429    0.03806
    #> Cross-validation time: 0.07736 ; Number of iterations: 9
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.7067 ; youden_index: 0.6857 ; aYI: 0 ; sensitivity: 0.9714 ; specificity: 0.8387 ; geometric_mean: 0.9026 ; fdr: 0.1282 ; mcc: 0.8224 ; auc: 0.9493 ; aauc: 0 ; corrclass: 0.9091 ;
    #> TP: 34 ; TN: 26 ; FP: 5 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03747    0.04205
    #> Cross-validation time: 0.02821 ; Number of iterations: 10
    #>
    #>
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> tau: 0.05 ; weight: 0.5 ; penalty: L12 ;  covYI_KS_L12 : 0.7177 ; youden_index: 0.6857 ; aYI: 0 ; sensitivity: 0.9714 ; specificity: 0.8387 ; geometric_mean: 0.9026 ; fdr: 0.1282 ; mcc: 0.8224 ; auc: 0.9493 ; aauc: 0 ; corrclass: 0.9091 ;
    #> TP: 34 ; TN: 26 ; FP: 5 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.03959    0.04470
    #> Cross-validation time: 0.02355 ; Number of iterations: 10
    #>
    #>
    #>
    #> ----------------------> END OF THE CROSS-VALIDATION OF THE PYE METHOD <-----------------
    #> -------------> For the whoole Cross-validation it took: 4.574 minutes <------------

    # 6. Print results and access optimal lambda/tau
    cat("\nOptimal Lambda (based on CCR):", pye_cv_result$lambda_hat_ccr, "\n")
    #>
    #> Optimal Lambda (based on CCR): 0.6082
    if (c_function_of_covariates == TRUE) {
      cat("Optimal Tau (based on CCR):", pye_cv_result$tau_hat_ccr, "\n")
    }
    #> Optimal Tau (based on CCR): 0.104

    # You can access other results like:
    pye_cv_result$auc # AUC values
    #> $`lambda=0.6082201996`
    #> $`lambda=0.6082201996`$train
    #>        tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
    #> fold=1   0.9920           0.9920            0.9920   0.9920
    #> fold=2   0.9930           0.9930            0.9930   0.9930
    #> fold=3   0.9913           0.9913            0.9913   0.9913
    #>
    #> $`lambda=0.6082201996`$test
    #>        tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
    #> fold=1   0.9041           0.9041            0.9041   0.9041
    #> fold=2   0.9534           0.9534            0.9534   0.9534
    #> fold=3   0.9493           0.9493            0.9493   0.9493
    pye_cv_result$n_betas # Number of non-zero betas for each lambda
    #>        lambda=1.5 lambda=0.6082201996 lambda=0.2466212074 lambda=0.1
    #> fold=1          7                  37                  49         31
    #> fold=2          5                  43                  70         66
    #> fold=3          4                  30                  59         64
    pye_cv_result$n_gammas # Number of non-zero gammas for each tau
    #> $`lambda=0.6082201996`
    #>      tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
    #> [1,]        1                1                 2        3
    #> [2,]        1                2                 2        3
    #> [3,]        1                2                 2        2

    # 7. Compute the performance over 50 simulations with the optimal lambda and tau
    sim_result <- pye_KS_simulation_study(
      n = 5,
      df = df,
      X = X,
      y = y,
      C = C,
      lambda = pye_cv_result$lambda_hat_ccr,
      tau = pye_cv_result$tau_hat_ccr,
      trace = 1,
      penalty = penalty, # Options: "L12", "L1", "EN", "SCAD", "MCP"
      penalty_g = penalty_g, # Options: "L12", "L1", "EN", "SCAD", "MCP"
      used_cores = 1,
      c_function_of_covariates = c_function_of_covariates,
      max_iter = max_iter, # Reduced for a quick example
      max_iter_g = 20 # Reduced for a quick example
    )
    #> ------------------------------------------------------------------
    #> |         Starting simulation study with 5 simulations         |
    #> ------------------------------------------------------------------
    #> Running simulation in sequential mode (used_cores = 1).
    #>
    #> --------------------> experiment n 1 of 5 <----------------------
    #> lambda =  0.6082 ; weight: 0.5 ; penalty =  SCAD
    #> tau =  0.104 ; weight_g: 0.5 ; penalty_g =  L12
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 6 ; Backtraking iters: 622 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3929 ; youden_index: 0.7464 ; sensitivity: 0.9867 ; specificity: 0.9385 ; geometric_mean: 0.9623 ; fdr: 0.05128 ; mcc: 0.9289 ; auc: 0.9943 ; corrclass: 0.9643 ;
    #> TP: 74 ; TN: 61 ; FP: 4 ; FN: 1 ; betas:
    #>      norm2      norm7     norm12     norm13     norm31     norm34     norm36
    #> -6.171e-02  1.894e-02 -1.017e-02 -1.448e-02 -8.586e-03  1.890e-02  7.934e-06
    #>     norm38     norm46     norm48
    #> -4.686e-03  5.564e-03 -5.040e-03
    #>  [ reached getOption("max.print") -- omitted 23 entries ]
    #> Estimation time: 0.2849 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 10 ; Backtraking iters: 126 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.788 ; youden_index: 0.832 ; aYI: 0 ; sensitivity: 0.9733 ; specificity: 1 ; geometric_mean: 0.9866 ; fdr: 0 ; mcc: 0.9717 ; auc: 0.9943 ; aauc: 0 ; corrclass: 0.9857 ;
    #> TP: 73 ; TN: 65 ; FP: 0 ; FN: 2 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.05240    0.03755
    #> Estimation time: 0.03612 mins
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 1 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2049 ; youden_index: 0.4263 ; sensitivity: 0.8125 ; fdr: 0.1333 ; mcc: 0.6682 ; auc: 0.9263 ; corrclass: 0.8333
    #> TP: 26 ; TN: 24 ; FP: 4 ; FN: 6 ;  betas:
    #>      norm2      norm7     norm12     norm13     norm31     norm34     norm36
    #> -6.171e-02  1.894e-02 -1.017e-02 -1.448e-02 -8.586e-03  1.890e-02  7.934e-06
    #>     norm38     norm46     norm48     norm49     norm51     norm52     norm67
    #> -4.686e-03  5.564e-03 -5.040e-03 -4.949e-03  7.090e-02 -1.247e-01  1.852e-03
    #>     norm73     norm83     norm85     norm90    norm101    norm102    norm107
    #> -1.285e-03  2.340e-04  1.720e-03 -5.524e-12  8.768e-02 -6.054e-02 -1.284e-03
    #>    norm112    norm119    norm131    norm137    norm141    norm151    norm152
    #> -2.412e-04 -1.551e-02  1.130e-03 -6.090e-03  1.113e-02  3.110e-02 -1.291e-04
    #>    norm165    norm172    norm175    norm178          c
    #>  1.859e-03  2.484e-03 -7.443e-03 -9.801e-04 -2.534e-03
    #> Estimation time: 0.2849 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.6044 ; youden_index: 0.6562 ; aYI: 0 ; sensitivity: 0.9688 ; specificity: 0.8214 ; geometric_mean: 0.8921 ; fdr: 0.1389 ; mcc: 0.8047 ; auc: 0.9263 ; aauc: 0 ; corrclass: 0.9
    #> TP: 31 ; TN: 23 ; FP: 5 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.05240    0.03755
    #> Estimation time: 0.03612 ; Number of iterations: 10
    #>
    #>
    #>
    #> --------------------> experiment n 2 of 5 <----------------------
    #> lambda =  0.6082 ; weight: 0.5 ; penalty =  SCAD
    #> tau =  0.104 ; weight_g: 0.5 ; penalty_g =  L12
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 5 ; Backtraking iters: 558 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3938 ; youden_index: 0.7643 ; sensitivity: 0.9867 ; specificity: 0.9385 ; geometric_mean: 0.9623 ; fdr: 0.05128 ; mcc: 0.9289 ; auc: 0.9822 ; corrclass: 0.9643 ;
    #> TP: 74 ; TN: 61 ; FP: 4 ; FN: 1 ; betas:
    #>     norm2    norm16    norm31    norm39    norm51    norm52    norm85    norm90
    #> -0.089614  0.006572 -0.010083 -0.003420  0.061847 -0.124959  0.051063 -0.001076
    #>   norm101   norm102
    #>  0.101349 -0.081709
    #>  [ reached getOption("max.print") -- omitted 14 entries ]
    #> Estimation time: 0.3478 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 10 ; Backtraking iters: 290 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7853 ; youden_index: 0.8098 ; aYI: 0 ; sensitivity: 0.96 ; specificity: 0.9538 ; geometric_mean: 0.9569 ; fdr: 0.04 ; mcc: 0.9138 ; auc: 0.9822 ; aauc: 0 ; corrclass: 0.9571 ;
    #> TP: 72 ; TN: 62 ; FP: 3 ; FN: 3 ; gammas:
    #> norm_cov6
    #>   -0.0557
    #> Estimation time: 0.02286 mins
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 2 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1181 ; youden_index: 0.4252 ; sensitivity: 0.9375 ; fdr: 0.2683 ; mcc: 0.5841 ; auc: 0.9252 ; corrclass: 0.7833
    #> TP: 30 ; TN: 17 ; FP: 11 ; FN: 2 ;  betas:
    #>      norm2     norm16     norm31     norm39     norm51     norm52     norm85
    #> -8.961e-02  6.572e-03 -1.008e-02 -3.420e-03  6.185e-02 -1.250e-01  5.106e-02
    #>     norm90    norm101    norm102    norm119    norm132    norm136    norm141
    #> -1.076e-03  1.013e-01 -8.171e-02 -1.449e-02 -2.018e-03  6.930e-03  6.352e-03
    #>    norm146    norm151    norm156    norm172    norm175    norm181    norm194
    #>  7.656e-03  2.546e-03 -2.701e-03  7.449e-12 -8.860e-03  1.739e-03  3.254e-03
    #>    norm196    norm200          c
    #>  1.345e-02  7.543e-03 -1.314e-02
    #> Estimation time: 0.3478 ; Number of iterations: 5
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.5482 ; youden_index: 0.625 ; aYI: 0 ; sensitivity: 0.9062 ; specificity: 0.8571 ; geometric_mean: 0.8814 ; fdr: 0.1212 ; mcc: 0.7655 ; auc: 0.9252 ; aauc: 0 ; corrclass: 0.8833
    #> TP: 29 ; TN: 24 ; FP: 4 ; FN: 3 ; gammas:
    #> norm_cov6
    #>   -0.0557
    #> Estimation time: 0.02286 ; Number of iterations: 10
    #>
    #>
    #>
    #> --------------------> experiment n 3 of 5 <----------------------
    #> lambda =  0.6082 ; weight: 0.5 ; penalty =  SCAD
    #> tau =  0.104 ; weight_g: 0.5 ; penalty_g =  L12
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 6 ; Backtraking iters: 632 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3616 ; youden_index: 0.7458 ; sensitivity: 0.9867 ; specificity: 0.9231 ; geometric_mean: 0.9543 ; fdr: 0.06329 ; mcc: 0.915 ; auc: 0.9822 ; corrclass: 0.9571 ;
    #> TP: 74 ; TN: 60 ; FP: 5 ; FN: 1 ; betas:
    #>      norm2      norm7     norm12     norm31     norm36     norm51     norm52
    #> -0.0741990  0.0045660 -0.0058695 -0.0004951  0.0215791  0.0659637 -0.1098548
    #>     norm56     norm63     norm66
    #>  0.0045634  0.0024302 -0.0041567
    #>  [ reached getOption("max.print") -- omitted 27 entries ]
    #> Estimation time: 0.3407 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 12 ; Backtraking iters: 387 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7622 ; youden_index: 0.8005 ; aYI: 0 ; sensitivity: 0.9867 ; specificity: 0.9385 ; geometric_mean: 0.9623 ; fdr: 0.05128 ; mcc: 0.9289 ; auc: 0.9822 ; aauc: 0 ; corrclass: 0.9643 ;
    #> TP: 74 ; TN: 61 ; FP: 4 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.02938    0.03878
    #> Estimation time: 0.07971 mins
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 3 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2335 ; youden_index: 0.4643 ; sensitivity: 0.9062 ; fdr: 0.1212 ; mcc: 0.7655 ; auc: 0.9643 ; corrclass: 0.8833
    #> TP: 29 ; TN: 24 ; FP: 4 ; FN: 3 ;  betas:
    #>      norm2      norm7     norm12     norm31     norm36     norm51     norm52
    #> -7.420e-02  4.566e-03 -5.869e-03 -4.951e-04  2.158e-02  6.596e-02 -1.099e-01
    #>     norm56     norm63     norm66     norm85     norm99    norm100    norm101
    #>  4.563e-03  2.430e-03 -4.157e-03  1.709e-02 -5.406e-03  2.680e-03  8.912e-02
    #>    norm102    norm112    norm119    norm124    norm128    norm131    norm132
    #> -5.018e-02 -1.914e-02 -2.761e-02  4.996e-03  1.226e-03  2.301e-02 -1.864e-03
    #>    norm134    norm136    norm137    norm145    norm146    norm151    norm152
    #>  7.427e-04  4.293e-05 -1.768e-04 -1.438e-02  8.901e-03  2.447e-02 -1.352e-03
    #>    norm156    norm157    norm165    norm172    norm174    norm175    norm186
    #> -5.788e-04 -7.629e-04  2.289e-02  4.773e-03 -3.051e-03 -1.850e-03  1.478e-03
    #>    norm196          c
    #>  1.021e-02  4.527e-03
    #> Estimation time: 0.3407 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.6235 ; youden_index: 0.7188 ; aYI: 0 ; sensitivity: 0.9375 ; specificity: 0.8929 ; geometric_mean: 0.9149 ; fdr: 0.09091 ; mcc: 0.8327 ; auc: 0.9643 ; aauc: 0 ; corrclass: 0.9167
    #> TP: 30 ; TN: 25 ; FP: 3 ; FN: 2 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.02938    0.03878
    #> Estimation time: 0.07971 ; Number of iterations: 12
    #>
    #>
    #>
    #> --------------------> experiment n 4 of 5 <----------------------
    #> lambda =  0.6082 ; weight: 0.5 ; penalty =  SCAD
    #> tau =  0.104 ; weight_g: 0.5 ; penalty_g =  L12
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 6 ; Backtraking iters: 643 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3778 ; youden_index: 0.7141 ; sensitivity: 0.9467 ; specificity: 0.8923 ; geometric_mean: 0.9191 ; fdr: 0.08974 ; mcc: 0.8424 ; auc: 0.9793 ; corrclass: 0.9214 ;
    #> TP: 71 ; TN: 58 ; FP: 7 ; FN: 4 ; betas:
    #>      norm2      norm7      norm8      norm9     norm12     norm19     norm20
    #> -4.838e-02  5.157e-04 -5.046e-13  1.525e-02 -1.114e-02  5.464e-03 -4.653e-04
    #>     norm31     norm34     norm36
    #> -3.263e-13  3.260e-04  7.730e-13
    #>  [ reached getOption("max.print") -- omitted 32 entries ]
    #> Estimation time: 0.3649 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 8 ; Backtraking iters: 72 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7669 ; youden_index: 0.8168 ; aYI: 0 ; sensitivity: 0.9867 ; specificity: 0.9846 ; geometric_mean: 0.9856 ; fdr: 0.01333 ; mcc: 0.9713 ; auc: 0.9793 ; aauc: 0 ; corrclass: 0.9857 ;
    #> TP: 74 ; TN: 64 ; FP: 1 ; FN: 1 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.06183    0.05357
    #> Estimation time: 0.02 mins
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 4 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2706 ; youden_index: 0.4531 ; sensitivity: 0.9062 ; fdr: 0.09375 ; mcc: 0.7991 ; auc: 0.9531 ; corrclass: 0.9
    #> TP: 29 ; TN: 25 ; FP: 3 ; FN: 3 ;  betas:
    #>      norm2      norm7      norm8      norm9     norm12     norm19     norm20
    #> -4.838e-02  5.157e-04 -5.046e-13  1.525e-02 -1.114e-02  5.464e-03 -4.653e-04
    #>     norm31     norm34     norm36     norm48     norm51     norm52     norm68
    #> -3.263e-13  3.260e-04  7.730e-13 -3.976e-03  4.911e-02 -1.227e-01 -2.643e-03
    #>     norm81     norm83     norm85    norm100    norm101    norm102    norm112
    #> -3.130e-03  9.388e-13  1.203e-02  4.194e-03  1.083e-01 -5.192e-02 -3.767e-03
    #>    norm119    norm123    norm130    norm131    norm134    norm136    norm137
    #> -1.309e-02  1.715e-12  2.496e-03  8.372e-03  2.762e-13  1.550e-12 -5.094e-13
    #>    norm141    norm144    norm146    norm151    norm152    norm157    norm171
    #>  1.106e-03  7.063e-03  1.883e-03  3.251e-02 -5.847e-03 -1.098e-02  4.592e-03
    #>    norm172    norm174    norm175    norm189    norm196    norm200          c
    #>  7.661e-03 -5.244e-03 -3.170e-03 -2.170e-13  5.499e-03  4.073e-13  2.865e-03
    #> Estimation time: 0.3649 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.5851 ; youden_index: 0.6875 ; aYI: 0 ; sensitivity: 0.9375 ; specificity: 0.8571 ; geometric_mean: 0.8964 ; fdr: 0.1176 ; mcc: 0.8 ; auc: 0.9531 ; aauc: 0 ; corrclass: 0.9
    #> TP: 30 ; TN: 24 ; FP: 4 ; FN: 2 ; gammas:
    #>  norm_cov6 norm_cov11
    #>   -0.06183    0.05357
    #> Estimation time: 0.02 ; Number of iterations: 8
    #>
    #>
    #>
    #> --------------------> experiment n 5 of 5 <----------------------
    #> lambda =  0.6082 ; weight: 0.5 ; penalty =  SCAD
    #> tau =  0.104 ; weight_g: 0.5 ; penalty_g =  L12
    #> mmAPG converged because alpha is below the threshold min_alpha.
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 6 ; Backtraking iters: 921 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.4007 ; youden_index: 0.7513 ; sensitivity: 0.9733 ; specificity: 0.9231 ; geometric_mean: 0.9479 ; fdr: 0.0641 ; mcc: 0.9 ; auc: 0.9817 ; corrclass: 0.95 ;
    #> TP: 73 ; TN: 60 ; FP: 5 ; FN: 2 ; betas:
    #>      norm2      norm7     norm22     norm23     norm36     norm37     norm48
    #> -8.670e-02  1.183e-02 -5.194e-03 -1.992e-03  7.495e-03  1.700e-03 -7.397e-03
    #>     norm51     norm52     norm97
    #>  4.963e-02 -1.217e-01 -1.668e-11
    #>  [ reached getOption("max.print") -- omitted 19 entries ]
    #> Estimation time: 0.3835 mins
    #>
    #>
    #> mmAPG converged because the convergence error is below the threshold
    #> Final estimation:
    #> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
    #> total iters: 11 ; Backtraking iters: 227 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.761 ; youden_index: 0.7832 ; aYI: 0 ; sensitivity: 0.96 ; specificity: 0.9538 ; geometric_mean: 0.9569 ; fdr: 0.04 ; mcc: 0.9138 ; auc: 0.9817 ; aauc: 0 ; corrclass: 0.9571 ;
    #> TP: 72 ; TN: 62 ; FP: 3 ; FN: 3 ; gammas:
    #> norm_cov6
    #>  -0.04561
    #> Estimation time: 0.05806 mins
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 5 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1876 ; youden_index: 0.4208 ; sensitivity: 0.8125 ; fdr: 0.1875 ; mcc: 0.5982 ; auc: 0.9208 ; corrclass: 0.8
    #> TP: 26 ; TN: 22 ; FP: 6 ; FN: 6 ;  betas:
    #>      norm2      norm7     norm22     norm23     norm36     norm37     norm48
    #> -8.670e-02  1.183e-02 -5.194e-03 -1.992e-03  7.495e-03  1.700e-03 -7.397e-03
    #>     norm51     norm52     norm97    norm101    norm102    norm107    norm115
    #>  4.963e-02 -1.217e-01 -1.668e-11  1.068e-01 -8.346e-02 -3.201e-03  2.528e-03
    #>    norm119    norm124    norm128    norm130    norm134    norm136    norm145
    #> -7.072e-03  1.266e-12  1.053e-03  1.900e-03  8.589e-03  4.835e-03 -4.677e-03
    #>    norm151    norm157    norm161    norm169    norm172    norm175    norm182
    #>  4.135e-02 -3.391e-12  4.506e-03 -5.726e-12  1.547e-03 -3.504e-12  1.120e-02
    #>          c
    #>  7.528e-03
    #> Estimation time: 0.3835 ; Number of iterations: 6
    #>
    #>
    #> -> Results on the TEST SET
    #> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.628 ; youden_index: 0.5938 ; aYI: 0 ; sensitivity: 0.9375 ; specificity: 0.8214 ; geometric_mean: 0.8775 ; fdr: 0.1429 ; mcc: 0.768 ; auc: 0.9208 ; aauc: 0 ; corrclass: 0.8833
    #> TP: 30 ; TN: 23 ; FP: 5 ; FN: 2 ; gammas:
    #> norm_cov6
    #>  -0.04561
    #> Estimation time: 0.05806 ; Number of iterations: 11
    #>
    #>
    #> Total Simulation Time:  1.95 mins  mins.

    # 8. View summary of simulation results
    colMeans(sim_result$corrclass)
    #>  train   test
    #> 0.9514 0.8400
    colMeans(sim_result$corrclass_covYI)
    #>  train   test
    #> 0.9700 0.8967

### Interpretation

The estimated coefficients \(\hat{\beta}\) (typically element `x1` in the output list) provide the optimal linear combination of features. Coefficients successfully shrunk to zero by the penalty are those deemed irrelevant and are effectively excluded from the final diagnostic model.

## 3. References

1. The PYE estimation: Salaroli, C. J., & Pardo, M. C. (2023). PYE: A Penalized Youden Index Estimator for selecting and combining biomarkers in high-dimensional data. Chemometrics and Intelligent Laboratory Systems, 236, 104786. https://doi.org/10.1016/j.chemolab.2023.104786

2. The covYI Estimation: Salaroli, C. J., & Pardo, M. C. (2026). covYI: A Covariate-Adjusted Penalized Youden Index Estimator for Selecting and Combining Covariates in High-Dimensional Data. Submitted for publication, currently under review.