Package 'roclab'

Generic function for AUC

Description

Compute AUC (Area Under the ROC Curve) for a fitted model. Dispatches to class-specific methods such as auc.roclearn.

Usage

auc(object, ...)

Arguments

object	A fitted model object.
...	Additional arguments passed to methods.

Value

Numeric scalar: estimated AUC.

---

Compute AUC for a fitted kernel model

Description

Estimate the AUC (Area Under the ROC Curve) for a fitted kernel model on new data.

Usage

## S3 method for class 'kroclearn'
auc(object, newdata, y, ...)

Arguments

object	A fitted model object of class "kroclearn" (kernel model).
newdata	A matrix or data.frame of test predictors. Must have the same structure as the training data (categorical variables are dummy-aligned automatically).
y	Response vector of test labels ({-1, 1} or convertible).
...	Not used.

Value

A numeric scalar giving the estimated AUC.

Examples

set.seed(123)

n_train <- 100
r_train <- sqrt(runif(n_train, 0.05, 1))
theta_train <- runif(n_train, 0, 2*pi)
X_train <- cbind(r_train * cos(theta_train), r_train * sin(theta_train))
y_train <- ifelse(r_train < 0.5, 1, -1)

n_test <- 10
r_test <- sqrt(runif(n_test, 0.05, 1))
theta_test <- runif(n_test, 0, 2*pi)
X_test <- cbind(r_test * cos(theta_test), r_test * sin(theta_test))
y_test <- ifelse(r_test < 0.5, 1, -1)

fit <- kroclearn(X_train, y_train, lambda = 0.1,
  kernel = "radial", approx=TRUE)

auc(fit, X_test, y_test)

---

Compute AUC for a fitted linear model

Description

Estimate the AUC (Area Under the ROC Curve) for a fitted linear model on new data.

Usage

## S3 method for class 'roclearn'
auc(object, newdata, y, ...)

Arguments

object	A fitted model object of class "roclearn" (linear model).
newdata	A matrix or data.frame of test predictors. Must have the same structure as the training data (categorical variables are dummy-aligned automatically).
y	Response vector of test labels ({-1, 1} or convertible).
...	Not used.

Value

A numeric scalar giving the estimated AUC.

Examples

set.seed(123)

n_train <- 100
n_pos <- round(0.2 * n_train)
n_neg <- n_train - n_pos

X_train <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y_train <- c(rep(-1, n_neg), rep(1, n_pos))

n_test <- 10
n_pos_test <- round(0.2 * n_test)
n_neg_test <- n_test - n_pos_test

X_test <- rbind(
  matrix(rnorm(2 * n_neg_test, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos_test, mean =  1), ncol = 2)
)
y_test <- c(rep(-1, n_neg_test), rep(1, n_pos_test))

fit <- roclearn(X_train, y_train, lambda = 0.1, approx=TRUE)

auc(fit, X_test, y_test)

---

Cross-validation for kernel models

Description

Perform k-fold cross-validation over a sequence of $\lambda$ values and select the optimal model based on AUC.

Usage

cv.kroclearn(
  X,
  y,
  lambda.vec = NULL,
  lambda.length = 30,
  kernel = "radial",
  param.kernel = NULL,
  loss = "hinge",
  approx = NULL,
  intercept = TRUE,
  nfolds = 10,
  target.perf = list(),
  param.convergence = list()
)

Arguments

X	Predictor matrix or data.frame (categorical variables are automatically one-hot encoded).
y	Response vector with class labels in {-1, 1}. Labels given as {0, 1} or as a two-level factor/character are automatically converted to this format.
lambda.vec	Optional numeric vector of regularization parameters (lambda values). If NULL (default), a decreasing sequence is generated automatically.
lambda.length	Number of $\lambda$ values to generate if lambda.vec is NULL. Default is 30.
kernel	Kernel type: "radial" (default), "polynomial", "linear", or "laplace".
param.kernel	Kernel-specific parameter: $\sigma$ for "radial" and "laplace" kernels (default $1/p$, where $p$ is the number of predictors after preprocessing, i.e., after categorical variables are one-hot encoded). Degree for "polynomial" kernel (default 2). Ignored for "linear" kernel.
loss	Surrogate loss function type. One of: "hinge" (default), "hinge2" (squared hinge), "logistic", or "exponential".
approx	Logical; enables a scalable approximation to accelerate training. The default is TRUE when nrow(X) >= 1000, and FALSE otherwise. For details about how approximation is applied, see the details section of the kroclearn function.
intercept	Logical; include an intercept in the model (default TRUE).
nfolds	Number of cross-validation folds (default 10).
target.perf	List with target sensitivity and specificity used when estimating the intercept (defaults to 0.9 each).
param.convergence	List of convergence controls (e.g., maxiter, eps). Default is list(maxiter = 5e4, eps = 1e-4).

Value

An object of class "cv.kroclearn" with:

• optimal.lambda — selected $\lambda$.

• optimal.fit — model trained at optimal.lambda.

• lambda.vec — grid of penalty values considered.

• auc.mean, auc.sd — mean and sd of cross-validated AUC.

• auc.result — fold-by-lambda AUC matrix.

• time.mean, time.sd — mean and sd of training time.

• time.result — fold-by-lambda training time matrix.

• nfolds, loss, kernel — settings.

See Also

kroclearn

Examples

set.seed(123)
n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

cvfit <- cv.kroclearn(
  X, y,
  lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  kernel = "radial",
  approx=TRUE, nfolds = 2
)
cvfit$optimal.lambda

---

Cross-validation for linear models

Description

Perform k-fold cross-validation over a sequence of $\lambda$ values and select the optimal model based on AUC.

Usage

cv.roclearn(
  X,
  y,
  lambda.vec = NULL,
  lambda.length = 30,
  penalty = "ridge",
  param.penalty = NULL,
  loss = "hinge",
  approx = NULL,
  intercept = TRUE,
  nfolds = 10,
  target.perf = list(),
  param.convergence = list()
)

Arguments

X	Predictor matrix or data.frame (categorical variables are automatically one-hot encoded).
y	Response vector with class labels in {-1, 1}. Labels given as {0, 1} or as a two-level factor/character are automatically converted to this format.
lambda.vec	Optional numeric vector of regularization parameters (lambda values). If NULL (default), a decreasing sequence is generated automatically.
lambda.length	Number of $\lambda$ values to generate if lambda.vec is NULL. Default is 30.
penalty	Regularization penalty type: "ridge" (default), "lasso", "elastic", "alasso", "scad", or "mcp".
param.penalty	Penalty-specific parameter: Ignored for "ridge" and "lasso". Mixing parameter $\alpha \in (0,1)$ for "elastic". Default is 0.5. Adaptive weight exponent $\gamma > 0$ for "alasso". Default is 1. Tuning parameter (default 3.7) for "scad" and "mcp".
loss	Surrogate loss function type. One of: "hinge" (default), "hinge2" (squared hinge), "logistic", or "exponential".
approx	Logical; enables a scalable approximation to accelerate training. The default is TRUE when nrow(X) >= 1000, and FALSE otherwise. For details about how approximation is applied, see the details section of the roclearn function.
intercept	Logical; include an intercept in the model (default TRUE).
nfolds	Number of cross-validation folds (default 10).
target.perf	List with target sensitivity and specificity used when estimating the intercept (defaults to 0.9 each).
param.convergence	List of convergence controls (e.g., maxiter, eps). Default is list(maxiter = 5e4, eps = 1e-4).

Value

An object of class "cv.roclearn" with:

• optimal.lambda — selected $\lambda$.

• optimal.fit — model refit on the full data at optimal.lambda.

• lambda.vec — grid of penalty values considered.

• auc.mean, auc.sd — mean and sd of cross-validated AUC.

• auc.result — fold-by-lambda AUC matrix.

• time.mean, time.sd — mean and sd of training time.

• time.result — fold-by-lambda training time matrix.

• nfolds, loss, penalty — settings.

See Also

roclearn

Examples

set.seed(123)
n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos

X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

cvfit <- cv.roclearn(
  X, y,
  lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  approx=TRUE, nfolds = 2
)
cvfit$optimal.lambda

---

Fit a kernel model

Description

Fit a kernel model

Usage

kroclearn(
  X,
  y,
  lambda,
  kernel = "radial",
  param.kernel = NULL,
  loss = "hinge",
  approx = NULL,
  intercept = TRUE,
  target.perf = list(),
  param.convergence = list()
)

Arguments

X	Predictor matrix or data.frame (categorical variables are automatically one-hot encoded).
y	Response vector with class labels in {-1, 1}. Labels given as {0, 1} or as a two-level factor/character are automatically converted to this format.
lambda	Positive scalar regularization parameter.
kernel	Kernel type: "radial" (default), "polynomial", "linear", or "laplace".
param.kernel	Kernel-specific parameter: $\sigma$ for "radial" and "laplace" kernels (default $1/p$, where $p$ is the number of predictors after preprocessing, i.e., after categorical variables are one-hot encoded). Degree for "polynomial" kernel (default 2). Ignored for "linear" kernel.
loss	Surrogate loss function type. One of: "hinge" (default), "hinge2" (squared hinge), "logistic", or "exponential".
approx	Logical; enables a scalable approximation to accelerate training. The default is TRUE when nrow(X) >= 1000, and FALSE otherwise. For details about how approximation is applied, see the details section.
intercept	Logical; include an intercept in the model (default TRUE).
target.perf	List with target sensitivity and specificity used when estimating the intercept (defaults to 0.9 each).
param.convergence	List of convergence controls (e.g., maxiter, eps). Default is list(maxiter = 5e4, eps = 1e-4).

Details

For large-scale data, the model is computationally prohibitive because its loss is a U-statistic involving a double summation. To reduce this burden, the package adopts an efficient algorithm based on an incomplete U-statistic, which approximates the loss with a single summation. In kernel models, a Nyström low-rank approximation is further applied to efficiently compute the kernel matrix. These approximations substantially reduce computational cost and accelerate training, while maintaining accuracy, making the model feasible for large-scale datasets. This option is available when @param approx = TRUE.

Value

An object of class "kroclearn", a list containing:

• theta.hat — estimated dual coefficient vector.

• intercept — fitted intercept (if applicable).

• lambda, kernel, param.kernel, loss.

• approx, B (number of sampled pairs if approximation used).

• time — training time (seconds).

• nobs, p — number of observations and predictors.

• converged, n.iter — convergence information.

• kfunc — kernel function object.

• nystrom — low rank kernel approximation details (if used).

• X — training data (post-preprocessing).

• preprocessing — details on categorical variables, removed columns, and column names.

• call — the function call.

Examples

set.seed(123)
n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

fit <- kroclearn(X, y, lambda = 0.1, kernel = "radial", approx=TRUE)

---

Plot Receiver Operating Characteristic (ROC) curve using ggroc

Description

Draws an ROC curve based on decision values. There is an option to display the AUC in the plot title and to print the ROC summary object.

Usage

plot_roc(
  y_true,
  y_score,
  col = "blue",
  size = 1.2,
  title = TRUE,
  summary = FALSE,
  ...
)

Arguments

y_true	Response vector with class labels in {-1, 1}. Labels given as {0, 1} or as a two-level factor/character are automatically converted to this format.
y_score	Numeric vector of predicted scores or decision values.
col	Line color.
size	Line width.
title	Logical; if TRUE, displays AUC in the plot title.
summary	Logical; if TRUE, prints the ROC object summary.
...	Additional arguments passed to pROC::ggroc().

Value

A ggplot object of the ROC curve.

Examples

set.seed(123)
n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos
X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

fit <- roclearn(X, y, lambda = 0.1, approx=TRUE)

y_score <- predict(fit, X, type = "response")

plot_roc(y, y_score)

---

Visualize Cross-Validation results for kernel models

Description

Produce a visualization of cross-validation results from a fitted cv.kroclearn object. The plot shows the mean AUC across regularization parameters $\lambda$, with error bars reflecting the cross-validation standard deviation. Optionally, the selected optimal $\lambda$ is highlighted with a dashed line and marker.

Usage

## S3 method for class 'cv.kroclearn'
plot(x, highlight = TRUE, ...)

Arguments

x	A cross-validation object of class "cv.kroclearn".
highlight	Logical; if TRUE, mark the selected optimal $\lambda$ with a vertical dashed line with a red point (default TRUE).
...	Additional arguments passed to underlying ggplot2 functions.

Details

This function is a method for the generic plot() function, designed specifically for cross-validation objects from cv.kroclearn. The x-axis is displayed on a log scale for $\lambda$, and the y-axis represents AUC values. Error bars show variability across folds. This is the kernel counterpart of plot.cv.roclearn.

Value

A ggplot2 object is returned and drawn to the current device.

Examples

set.seed(123)
n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

cvfit <- cv.kroclearn(
  X, y,
  lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  kernel = "radial",
  approx=TRUE, nfolds = 2
)
plot(cvfit)

---

Visualize Cross-Validation results for linear models

Description

Produce a visualization of cross-validation results from a fitted cv.roclearn object. The plot shows the mean AUC across regularization parameters $\lambda$, with error bars reflecting the cross-validation standard deviation. Optionally, the selected optimal $\lambda$ is highlighted with a dashed line and marker.

Usage

## S3 method for class 'cv.roclearn'
plot(x, highlight = TRUE, ...)

Arguments

x	A cross-validation object of class "cv.roclearn".
highlight	Logical; if TRUE, mark the selected optimal $\lambda$ with a vertical dashed line with a red point (default TRUE).
...	Additional arguments passed to underlying ggplot2 functions.

Details

This function is a method for the generic plot() function, designed specifically for cross-validation objects from cv.roclearn. The x-axis is displayed on a log scale for $\lambda$, and the y-axis represents AUC values. Error bars show variability across folds.

Value

A ggplot2 object is returned and drawn to the current device.

Examples

set.seed(123)

n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos

X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

cvfit <- cv.roclearn(
  X, y,lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  approx=TRUE, nfolds = 2)

plot(cvfit)

---

Predictions from a fitted kernel model

Description

Generate predictions from a fitted kernel model.

Usage

## S3 method for class 'kroclearn'
predict(object, newdata, type = c("class", "response"), ...)

Arguments

object	A fitted model object of class "kroclearn" (kernel).
newdata	A data frame or matrix of predictors for which predictions are desired. Categorical variables are automatically dummy-encoded and aligned to the training structure.
type	Prediction type: "class" for {-1, 1} labels, or "response" for raw decision scores.
...	Not used.

Value

A numeric vector of predictions ({-1, 1}) if type = "class", or raw decision scores if type = "response".

See Also

kroclearn, cv.kroclearn

Examples

set.seed(123)
n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

fit <- kroclearn(X, y, lambda = 0.1, kernel = "radial", approx=TRUE)

# Predict classes {-1, 1}
predict(fit, X, type = "class")

# Predict decision scores
predict(fit, X, type = "response")

---

Predictions from a fitted linear model

Description

Generate predictions from a fitted linear model.

Usage

## S3 method for class 'roclearn'
predict(object, newdata, type = c("class", "response"), ...)

Arguments

object	A fitted model object of class "roclearn" (linear).
newdata	A data frame or matrix of predictors for which predictions are desired. Categorical variables are automatically dummy-encoded and aligned to the training structure.
type	Prediction type: "class" for {-1, 1} labels, or "response" for raw decision scores.
...	Not used.

Value

A numeric vector of predictions ({-1, 1}) if type = "class", or raw decision scores if type = "response".

See Also

roclearn, cv.roclearn

Examples

set.seed(123)
n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos
X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

fit <- roclearn(X, y, lambda = 0.1, approx=TRUE)

# Predict classes {-1, 1}
predict(fit, X, type = "class")

# Predict decision scores
predict(fit, X, type = "response")

---

Fit a linear model

Description

Fit a linear model

Usage

roclearn(
  X,
  y,
  lambda,
  penalty = "ridge",
  param.penalty = NULL,
  loss = "hinge",
  approx = NULL,
  intercept = TRUE,
  target.perf = list(),
  param.convergence = list()
)

Arguments

X	Predictor matrix or data.frame (categorical variables are automatically one-hot encoded).
y	Response vector with class labels in {-1, 1}. Labels given as {0, 1} or as a two-level factor/character are automatically converted to this format.
lambda	Positive scalar regularization parameter.
penalty	Regularization penalty type: "ridge" (default), "lasso", "elastic", "alasso", "scad", or "mcp".
param.penalty	Penalty-specific parameter: Ignored for "ridge" and "lasso". Mixing parameter $\alpha \in (0,1)$ for "elastic". Default is 0.5. Adaptive weight exponent $\gamma > 0$ for "alasso". Default is 1. Tuning parameter (default 3.7) for "scad" and "mcp".
loss	Surrogate loss function type. One of: "hinge" (default), "hinge2" (squared hinge), "logistic", or "exponential".
approx	Logical; enables a scalable approximation to accelerate training. The default is TRUE when nrow(X) >= 1000, and FALSE otherwise. For details about how approximation is applied, see the details section.
intercept	Logical; include an intercept in the model (default TRUE).
target.perf	List with target sensitivity and specificity used when estimating the intercept (defaults to 0.9 each).
param.convergence	List of convergence controls (e.g., maxiter, eps). Default is list(maxiter = 5e4, eps = 1e-4).

Details

For large-scale data, the model is computationally prohibitive because its loss is a U-statistic involving a double summation. To reduce this burden, the package adopts an efficient algorithm based on an incomplete U-statistic, which approximates the loss with a single summation. These approximations substantially reduce computational cost and accelerate training, while maintaining accuracy, making the model feasible for large-scale datasets. This option is available when approx = TRUE.

Value

An object of class "roclearn", a list containing:

• beta.hat — estimated coefficient vector.

• intercept — fitted intercept (if applicable).

• lambda, penalty, param.penalty, loss.

• approx, B (number of sampled pairs if approximation used).

• time — training time (seconds).

• nobs, p — number of observations and predictors.

• converged, n.iter — convergence information.

• preprocessing — details on categorical variables, removed columns, and column names.

• call — the function call.

Examples

set.seed(123)
n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos
X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

fit <- roclearn(X, y, lambda = 0.1, penalty = "ridge", approx=TRUE)

---

Summarize Cross-Validation results for kernel models

Description

Print a concise summary of cross-validation results for a kernel model.

Usage

## S3 method for class 'cv.kroclearn'
summary(object, ...)

Arguments

object	A fitted cross-validation object of class "cv.kroclearn" (kernel).
...	Not used.

Details

This is a method for the generic summary() function, applied to objects of class "cv.kroclearn". It prints training settings (loss, kernel type, number of folds, the set of candidate $\lambda$), the selected optimal $\lambda$, the corresponding mean and standard deviation of cross-validated AUC, and a truncated table of AUC results across candidate $\lambda$ values.

Value

Invisibly returns the input object, after printing a summary to the console.

Examples

set.seed(123)

n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

cvfit <- cv.kroclearn(
  X, y,
  lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  kernel = "radial",
  approx=TRUE, nfolds = 2
)

summary(cvfit)

---

Summarize Cross-Validation results for linear models

Description

Print a concise summary of cross-validation results for a linear model.

Usage

## S3 method for class 'cv.roclearn'
summary(object, ...)

Arguments

object	A fitted cross-validation object of class "cv.roclearn" (linear).
...	Not used.

Details

This is a method for the generic summary() function, applied to objects of class "cv.roclearn". It prints training settings (loss, penalty, number of folds, the set of candidate $\lambda$), the selected optimal $\lambda$, the corresponding mean and standard deviation of cross-validated AUC, and a truncated table of AUC results across candidate $\lambda$ values.

Value

Invisibly returns the input object, after printing a summary to the console.

Examples

set.seed(123)

n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos
X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

cvfit <- cv.roclearn(
  X, y,
  lambda.vec = exp(seq(log(0.01), log(5), length.out = 3)),
  approx=TRUE, nfolds = 2
)

summary(cvfit)

---

Summarize a fitted kernel model

Description

Display key information from a fitted "kroclearn" object, including: data dimensions, kernel specification, convergence status, training time, and leading coefficient estimates.

Usage

## S3 method for class 'kroclearn'
summary(object, ...)

Arguments

object	A fitted model of class "kroclearn".
...	Unused.

Value

Invisibly returns object after printing a formatted summary.

See Also

kroclearn, summary.roclearn, cv.kroclearn, cv.roclearn

Examples

set.seed(123)
n <- 100
r <- sqrt(runif(n, 0.05, 1))
theta <- runif(n, 0, 2*pi)
X <- cbind(r * cos(theta), r * sin(theta))
y <- ifelse(r < 0.5, 1, -1)

fit <- kroclearn(X, y, lambda = 0.1, kernel = "radial", approx=TRUE)
summary(fit)

---

Summarize a fitted linear model

Description

Display key information from a fitted "roclearn" object, including: data dimensions, model specification, convergence status, training time, and leading coefficient estimates.

Usage

## S3 method for class 'roclearn'
summary(object, ...)

Arguments

object	A fitted model of class "roclearn".
...	Unused.

Value

Invisibly returns object after printing a formatted summary.

See Also

roclearn, summary.kroclearn, cv.roclearn, cv.kroclearn

Examples

set.seed(123)
n <- 100
n_pos <- round(0.2 * n)
n_neg <- n - n_pos
X <- rbind(
  matrix(rnorm(2 * n_neg, mean = -1), ncol = 2),
  matrix(rnorm(2 * n_pos, mean =  1), ncol = 2)
)
y <- c(rep(-1, n_neg), rep(1, n_pos))

fit <- roclearn(X, y, lambda = 0.1, approx=TRUE)
summary(fit)

---