Package {rbridge}

Fit a bridge coefficient path

Description

Fit a bridge-penalized least-squares path using local quadratic approximation.

Usage

bridge(
  X,
  y,
  q = 1,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  lambda,
  eta = 1e-07,
  converge = 10^10
)

Arguments

Details

Predictors are centered and divided by their centered root-mean-square (RMS) deviations, using the observation count n as the divisor. The response is centered. The intercept is reconstructed after fitting, and returned coefficients are expressed on the original predictor scale. The generated penalties are logarithmically spaced between an internally computed starting value and the absolute endpoint lambda.min.

Let Z be the n by p centered, RMS-standardized design, y_c the centered response vector of length n, and b the p-dimensional standardized slope vector. For exponent q \in (0,2] and penalty weight \lambda \ge 0, Equation (1) defines the penalized least-squares target:

\min_{b} \left\{\|y_c-Zb\|_2^2+\lambda\sum_{j=1}^{p}|b_j|^q\right\}. \qquad (1)

In Equation (1), \|\cdot\|_2^2 is the squared Euclidean norm, so its first term is the unnormalized residual sum of squares. The index j runs over the predictors; the second term is the bridge penalty, with \lambda controlling its weight relative to residual error.

The fitting routine applies the article's LQA updates, coefficient elimination and stopping rules. The criterion is nonconvex when q < 1, so the iterative result is not a guarantee of a global minimum. The getting-started vignette explains the penalty geometry and the fixed-LQA restriction correction.

Value

A list of class bridge with components:

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

References

Yuzbasi, B., Arashi, M. and Akdeniz, F. (2021). Penalized regression via the restricted bridge estimator. Soft Computing, 25, 8401–8416. doi:10.1007/s00500-021-05763-9.

See Also

cv.bridge, coef.bridge, predict.bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
# L1 and L2 penalty forms on the same observations.
model1 <- bridge(X, y, q = 1, nlambda = 5)
model2 <- bridge(X, y, q = 2, nlambda = 5)
dim(model1$betas)
# Extract one fitted column and predict five observations.
selected <- model1$lambda[2]
coef(model1, s = selected)
predict(model1, newx = X[1:5, , drop = FALSE], s = selected)

Extract coefficients from a fitted bridge object

Description

Extract original-scale intercept and slope coefficients from a fitted bridge object.

Usage

## S3 method for class 'bridge'
coef(object, s = c("lambda.1se", "lambda.min"), ...)

Arguments

Details

The returned matrix retains its dimensions for a single selected penalty.

Value

Numeric matrix with p + 1 rows and one column per selected penalty, even when only one penalty is selected. The first row is (Intercept); subsequent rows are original-scale slopes in the training predictor order.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.bridge, bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
model1 <- bridge(X, y, q = 1, nlambda = 5)
# A single selected penalty still gives a coefficient matrix.
coefficients <- coef(model1, s = model1$lambda[2])
coefficients
dim(coefficients)
# Omitting s extracts the full fitted coefficient path.
dim(coef(model1))

Extract coefficients from a fitted cv.bridge object

Description

Extract original-scale intercept and slope coefficients from a fitted cross-validated cv.bridge object.

Usage

## S3 method for class 'cv.bridge'
coef(object, s = c("lambda.1se", "lambda.min"), ...)

Arguments

Details

The returned matrix retains its dimensions for a single selected penalty. The defaults differ between coef() and predict(): coefficient extraction uses lambda.1se, whereas response prediction uses lambda.min. Specify the same s in both calls when comparing coefficients and their predictions.

Value

Numeric matrix with p + 1 rows and one column per selected penalty, even when only one penalty is selected. The first row is (Intercept); subsequent rows are original-scale slopes in the training predictor order.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.cv.bridge, cv.bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.bridge(X, y, q = 1, nlambda = 5,
                     nfolds = 3, num_threads = 1)
# A single selected penalty still gives a coefficient matrix.
coefficients <- coef(model1, s = "lambda.min")
coefficients
dim(coefficients)
# The default uses the historical error-bar selector.
coef(model1)

Extract coefficients from a fitted cv.rbridge object

Description

Extract original-scale intercept and slope coefficients from a fitted cross-validated cv.rbridge object.

Usage

## S3 method for class 'cv.rbridge'
coef(object, s = c("lambda.1se", "lambda.min"), ...)

Arguments

Details

The returned matrix retains its dimensions for a single selected penalty. The defaults differ between coef() and predict(): coefficient extraction uses lambda.1se, whereas response prediction uses lambda.min. Specify the same s in both calls when comparing coefficients and their predictions. Returned slopes are on the original predictor scale; the linear restrictions concern standardized slopes. See rbridge for converting scales before checking restriction residuals.

Value

Numeric matrix with p + 1 rows and one column per selected penalty, even when only one penalty is selected. The first row is (Intercept); subsequent rows are original-scale slopes in the training predictor order.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.cv.rbridge, cv.rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5,
                     nfolds = 3, num_threads = 1)
# A single selected penalty still gives a coefficient matrix.
coefficients <- coef(model1, s = "lambda.min")
coefficients
dim(coefficients)
# The default uses the historical error-bar selector.
coef(model1)

Extract coefficients from a fitted rbridge object

Description

Extract original-scale intercept and slope coefficients from a fitted rbridge object.

Usage

## S3 method for class 'rbridge'
coef(object, s = c("lambda.1se", "lambda.min"), ...)

Arguments

Details

The returned matrix retains its dimensions for a single selected penalty. Returned slopes are on the original predictor scale; the linear restrictions concern standardized slopes. See rbridge for converting scales before checking restriction residuals.

Value

Numeric matrix with p + 1 rows and one column per selected penalty, even when only one penalty is selected. The first row is (Intercept); subsequent rows are original-scale slopes in the training predictor order.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.rbridge, rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5)
# A single selected penalty still gives a coefficient matrix.
coefficients <- coef(model1, s = model1$lambda[2])
coefficients
dim(coefficients)
# Omitting s extracts the full fitted coefficient path.
dim(coef(model1))

Cross-validation for bridge regression

Description

Evaluate a fitted penalty path using contiguous validation folds and return the full-data coefficients together with two penalty selectors.

Usage

cv.bridge(
  X,
  y,
  q,
  lambda,
  nfolds = 10,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  eta = 1e-07,
  converge = 10^10,
  num_threads = 10
)

Arguments

Details

Fold assignment follows the supplied row order. The nfolds validation sets are contiguous blocks; their sizes can differ by one. Centering and RMS scaling use all observations before fold allocation, and each fold fit also reconstructs its training-set intercept. These historical preprocessing and error-bar conventions are retained. The reported validation error aggregates held-out predictions at each fitted penalty, and coefficients in betas come from a final fit to all observations.

For one fixed penalty, let K be nfolds, n the total observation count, I_k the validation indices in fold k, and e_i the held-out response residual for observation i. Equation (1) defines a fold's error contribution d_k, the pooled validation MSE, and the historical error-bar width:

\begin{aligned} d_k&=\frac{1}{n}\sum_{i\in I_k}e_i^2,\\ \mathrm{cve}&=\sum_{k=1}^{K}d_k,\\ \mathrm{cvse}&=\sqrt{\frac{1}{K(n-1)} \sum_{k=1}^{K}(d_k-\mathrm{cve})^2}. \qquad (1) \end{aligned}

The folds partition all observations, so the sum in Equation (1) weights each held-out observation equally. The final quantity supplies the plotted bar width and the historical lambda.1se threshold. Its centering uses the pooled cve, not the average fold contribution, and it is not a conventional foldwise standard error.

predict() defaults to the minimum-error penalty lambda.min. coef() defaults to lambda.1se. Supply s explicitly to use the same penalty for both operations. Use plot() to inspect the curve, error bars and selected penalties.

Value

A list of class cv.bridge with components:

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

bridge, plot.cv.bridge, coef.cv.bridge, predict.cv.bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
# Three contiguous folds and one worker keep this demonstration small.
model1 <- cv.bridge(X, y, q = 1,
                     nlambda = 5, nfolds = 3, num_threads = 1)
model2 <- cv.bridge(X, y, q = 2,
                     nlambda = 5, nfolds = 3, num_threads = 1)
# Inspect both selectors and explicitly choose the minimum-error fit.
c(minimum_error = model1$lambda.min, historical_1se = model1$lambda.1se)
coef(model1, s = "lambda.min")
predict(model1, newx = X[1:5, , drop = FALSE], s = "lambda.min")
plot(model1)

Cross-validation for restricted bridge regression

Description

Evaluate a fitted penalty path using contiguous validation folds and return the full-data coefficients together with two penalty selectors.

Usage

cv.rbridge(
  X,
  y,
  q,
  R,
  r,
  lambda,
  nfolds = 10,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  eta = 1e-07,
  converge = 10^10,
  num_threads = 10
)

Arguments

Details

Fold assignment follows the supplied row order. The nfolds validation sets are contiguous blocks; their sizes can differ by one. Centering and RMS scaling use all observations before fold allocation, and each fold fit also reconstructs its training-set intercept. These historical preprocessing and error-bar conventions are retained. The reported validation error aggregates held-out predictions at each fitted penalty, and coefficients in betas come from a final fit to all observations.

For one fixed penalty, let K be nfolds, n the total observation count, I_k the validation indices in fold k, and e_i the held-out response residual for observation i. Equation (1) defines a fold's error contribution d_k, the pooled validation MSE, and the historical error-bar width:

\begin{aligned} d_k&=\frac{1}{n}\sum_{i\in I_k}e_i^2,\\ \mathrm{cve}&=\sum_{k=1}^{K}d_k,\\ \mathrm{cvse}&=\sqrt{\frac{1}{K(n-1)} \sum_{k=1}^{K}(d_k-\mathrm{cve})^2}. \qquad (1) \end{aligned}

The folds partition all observations, so the sum in Equation (1) weights each held-out observation equally. The final quantity supplies the plotted bar width and the historical lambda.1se threshold. Its centering uses the pooled cve, not the average fold contribution, and it is not a conventional foldwise standard error.

predict() defaults to the minimum-error penalty lambda.min. coef() defaults to lambda.1se. Supply s explicitly to use the same penalty for both operations. Use plot() to inspect the curve, error bars and selected penalties.

Restrictions act on standardized slopes. To check a fitted original-scale slope vector, multiply it elementwise by the centered RMS deviations of the predictors before applying R. The resulting restriction residual is the fitted linear combination minus r; it should be examined with a numerical tolerance, particularly after coefficient thresholding. For example, a zero restriction on one slope has the same meaning on either scale, whereas a nonzero sum or contrast depends on the scales.

Value

A list of class cv.rbridge with components:

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

rbridge, plot.cv.rbridge, coef.cv.rbridge, predict.cv.rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)
# A second standardized-slope contrast: -b[1] + b[2] + b[5] = 0.5.
R2.mat <- matrix(c(-1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r2.vec <- matrix(0.5, nrow = 1, ncol = 1)
# Impose both independent restrictions together.
R3.mat <- rbind(R1.mat, R2.mat)
r3.vec <- matrix(c(6.5, 0.5), ncol = 1)
# Alternatively, set the five inactive generating slopes to zero.
p.active <- which(beta != 0)
R4.mat <- diag(p)[-p.active, , drop = FALSE]
r4.vec <- matrix(rep(0, p - length(p.active)), ncol = 1)

# Five penalties keep this help example fast; the default is 100.
# Three contiguous folds and one worker keep this demonstration small.
model1 <- cv.rbridge(X, y, q = 1, R = R1.mat, r = r1.vec,
                     nlambda = 5, nfolds = 3, num_threads = 1)
model2 <- cv.rbridge(X, y, q = 1, R = R2.mat, r = r2.vec,
                     nlambda = 5, nfolds = 3, num_threads = 1)
model3 <- cv.rbridge(X, y, q = 1, R = R3.mat, r = r3.vec,
                     nlambda = 5, nfolds = 3, num_threads = 1)
model4 <- cv.rbridge(X, y, q = 1, R = R4.mat, r = r4.vec,
                     nlambda = 5, nfolds = 3, num_threads = 1)
# Inspect both selectors and explicitly choose the minimum-error fit.
c(minimum_error = model1$lambda.min, historical_1se = model1$lambda.1se)
coef(model1, s = "lambda.min")
predict(model1, newx = X[1:5, , drop = FALSE], s = "lambda.min")
plot(model1)
# Check the restriction after converting the returned slopes.
x_centered <- sweep(X, 2, colMeans(X), "-")
x_rms <- sqrt(colMeans(x_centered^2))
standardized_slopes <- coef(model1, s = "lambda.min")[-1, 1] * x_rms
as.numeric(R1.mat %*% standardized_slopes - r1.vec)

Plot the bridge cross-validation curve

Description

Display validation mean squared error over the fitted penalty grid, with historical error bars, counts of nonzero slopes and both selected penalties.

Usage

## S3 method for class 'cv.bridge'
plot(x, sign.lambda = 1, ...)

Arguments

Details

Points show x$cve; bars span x$cvlo to x$cvup. The bars use the historical cvse calculation described in cv.bridge, rather than conventional foldwise standard errors. The top axis shows full-data nonzero-slope counts, excluding the intercept. Vertical dotted lines mark lambda.min and lambda.1se; the lines coincide when both selectors choose the same fitted penalty. The curve is a diagnostic for the supplied grid and folds; its shape depends on the data and need not be U-shaped.

Value

Draws the plot and returns NULL invisibly.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

cv.bridge, coef.cv.bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.bridge(X, y, q = 1, nlambda = 5,
                     nfolds = 3, num_threads = 1)
plot(model1, main = "Cross-validation over the fitted penalty grid")
# Reverse the log-penalty axis without refitting the model.
plot(model1, sign.lambda = -1)

Plot the restricted bridge cross-validation curve

Description

Display validation mean squared error over the fitted penalty grid, with historical error bars, counts of nonzero slopes and both selected penalties.

Usage

## S3 method for class 'cv.rbridge'
plot(x, sign.lambda = 1, ...)

Arguments

Details

Points show x$cve; bars span x$cvlo to x$cvup. The bars use the historical cvse calculation described in cv.rbridge, rather than conventional foldwise standard errors. The top axis shows full-data nonzero-slope counts, excluding the intercept. Vertical dotted lines mark lambda.min and lambda.1se; the lines coincide when both selectors choose the same fitted penalty. The curve is a diagnostic for the supplied grid and folds; its shape depends on the data and need not be U-shaped.

Value

Draws the plot and returns NULL invisibly.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

cv.rbridge, coef.cv.rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5,
                     nfolds = 3, num_threads = 1)
plot(model1, main = "Cross-validation over the fitted penalty grid")
# Reverse the log-penalty axis without refitting the model.
plot(model1, sign.lambda = -1)

Predict from a fitted bridge object

Description

Compute response predictions, extract coefficients, or identify nonzero predictors from a fitted bridge object.

Usage

## S3 method for class 'bridge'
predict(
  object,
  newx,
  s = c("lambda.min", "lambda.1se"),
  type = c("response", "nonzero", "coefficients"),
  ...
)

Arguments

Details

Prediction uses the fitted intercept and original-scale slopes. Columns of newx are matched by position, not reordered by their names. For type = "nonzero", all fitted penalties are inspected and s does not restrict the returned indices.

Value

For type = "response", a numeric vector for one selected penalty or a matrix with one row per new observation and one column per selected penalty. For type = "coefficients", a numeric matrix with p + 1 rows, including the intercept, even for one penalty. For type = "nonzero", a pathwise collection of 1-based predictor indices excluding the intercept. Its historical container can simplify depending on the support pattern and number of predictors.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.bridge, bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
model1 <- bridge(X, y, q = 1, nlambda = 5)
# Predict five observations at one explicitly selected penalty.
predict(model1, newx = X[1:5, , drop = FALSE], s = model1$lambda[2])
# Extract coefficients using the complete type name.
predict(model1, s = model1$lambda[2], type = "coefficients")
# Predictor indices for every fitted penalty; the intercept is excluded.
predict(model1, type = "nonzero")

Predict from a fitted cv.bridge object

Description

Compute response predictions, extract coefficients, or identify nonzero predictors from a fitted cv.bridge object.

Usage

## S3 method for class 'cv.bridge'
predict(
  object,
  newx,
  s = c("lambda.min", "lambda.1se"),
  type = c("response", "nonzero", "coefficients"),
  ...
)

Arguments

Details

Prediction uses the fitted intercept and original-scale slopes. Columns of newx are matched by position, not reordered by their names. For type = "nonzero", all fitted penalties are inspected and s does not restrict the returned indices. The defaults differ between coef() and predict(): coefficient extraction uses lambda.1se, whereas response prediction uses lambda.min. Specify the same s in both calls when comparing coefficients and their predictions.

Value

For type = "response", a numeric vector for one selected penalty or a matrix with one row per new observation and one column per selected penalty. For type = "coefficients", a numeric matrix with p + 1 rows, including the intercept, even for one penalty. For type = "nonzero", a pathwise collection of 1-based predictor indices excluding the intercept. Its historical container can simplify depending on the support pattern and number of predictors.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.cv.bridge, cv.bridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.bridge(X, y, q = 1, nlambda = 5,
                     nfolds = 3, num_threads = 1)
# Predict five observations at one explicitly selected penalty.
predict(model1, newx = X[1:5, , drop = FALSE], s = "lambda.min")
# Extract coefficients using the complete type name.
predict(model1, s = "lambda.min", type = "coefficients")
# Predictor indices for every fitted penalty; the intercept is excluded.
predict(model1, type = "nonzero")

Predict from a fitted cv.rbridge object

Description

Compute response predictions, extract coefficients, or identify nonzero predictors from a fitted cv.rbridge object.

Usage

## S3 method for class 'cv.rbridge'
predict(
  object,
  newx,
  s = c("lambda.min", "lambda.1se"),
  type = c("response", "nonzero", "coefficients"),
  ...
)

Arguments

Details

Prediction uses the fitted intercept and original-scale slopes. Columns of newx are matched by position, not reordered by their names. For type = "nonzero", all fitted penalties are inspected and s does not restrict the returned indices. The defaults differ between coef() and predict(): coefficient extraction uses lambda.1se, whereas response prediction uses lambda.min. Specify the same s in both calls when comparing coefficients and their predictions. Returned slopes are on the original predictor scale; the linear restrictions concern standardized slopes. See rbridge for converting scales before checking restriction residuals.

Value

For type = "response", a numeric vector for one selected penalty or a matrix with one row per new observation and one column per selected penalty. For type = "coefficients", a numeric matrix with p + 1 rows, including the intercept, even for one penalty. For type = "nonzero", a pathwise collection of 1-based predictor indices excluding the intercept. Its historical container can simplify depending on the support pattern and number of predictors.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.cv.rbridge, cv.rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- cv.rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5,
                     nfolds = 3, num_threads = 1)
# Predict five observations at one explicitly selected penalty.
predict(model1, newx = X[1:5, , drop = FALSE], s = "lambda.min")
# Extract coefficients using the complete type name.
predict(model1, s = "lambda.min", type = "coefficients")
# Predictor indices for every fitted penalty; the intercept is excluded.
predict(model1, type = "nonzero")

Predict from a fitted rbridge object

Description

Compute response predictions, extract coefficients, or identify nonzero predictors from a fitted rbridge object.

Usage

## S3 method for class 'rbridge'
predict(
  object,
  newx,
  s = c("lambda.min", "lambda.1se"),
  type = c("response", "nonzero", "coefficients"),
  ...
)

Arguments

Details

Prediction uses the fitted intercept and original-scale slopes. Columns of newx are matched by position, not reordered by their names. For type = "nonzero", all fitted penalties are inspected and s does not restrict the returned indices. Returned slopes are on the original predictor scale; the linear restrictions concern standardized slopes. See rbridge for converting scales before checking restriction residuals.

Value

For type = "response", a numeric vector for one selected penalty or a matrix with one row per new observation and one column per selected penalty. For type = "coefficients", a numeric matrix with p + 1 rows, including the intercept, even for one penalty. For type = "nonzero", a pathwise collection of 1-based predictor indices excluding the intercept. Its historical container can simplify depending on the support pattern and number of predictors.

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.rbridge, rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5)
# Predict five observations at one explicitly selected penalty.
predict(model1, newx = X[1:5, , drop = FALSE], s = model1$lambda[2])
# Extract coefficients using the complete type name.
predict(model1, s = model1$lambda[2], type = "coefficients")
# Predictor indices for every fitted penalty; the intercept is excluded.
predict(model1, type = "nonzero")

Fit a restricted bridge coefficient path

Description

Fit a bridge-penalized least-squares path subject to user-supplied linear restrictions on standardized slopes, using local quadratic approximation.

Usage

rbridge(
  X,
  y,
  q = 1,
  R,
  r,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  lambda,
  eta = 1e-07,
  converge = 10^10
)

Arguments

Details

Predictors are centered and divided by their centered root-mean-square (RMS) deviations, using the observation count n as the divisor. The response is centered. The intercept is reconstructed after fitting, and returned coefficients are expressed on the original predictor scale. The generated penalties are logarithmically spaced between an internally computed starting value and the absolute endpoint lambda.min. Restrictions act on standardized slopes. To check a fitted original-scale slope vector, multiply it elementwise by the centered RMS deviations of the predictors before applying R. The resulting restriction residual is the fitted linear combination minus r; it should be examined with a numerical tolerance, particularly after coefficient thresholding. For example, a zero restriction on one slope has the same meaning on either scale, whereas a nonzero sum or contrast depends on the scales.

Let Z be the n by p centered, RMS-standardized design, y_c the centered response vector of length n, and b the p-dimensional standardized slope vector. For exponent q \in (0,2] and penalty weight \lambda \ge 0, Equation (1) defines the penalized least-squares target:

\min_{b:\,Rb=r} \left\{\|y_c-Zb\|_2^2+\lambda\sum_{j=1}^{p}|b_j|^q\right\}. \qquad (1)

In Equation (1), \|\cdot\|_2^2 is the squared Euclidean norm, so its first term is the unnormalized residual sum of squares. The index j runs over the predictors; the second term is the bridge penalty, with \lambda controlling its weight relative to residual error. Here R is an m by p matrix with independent rows and r is the m-dimensional target vector. The condition Rb=r restricts the standardized slopes and excludes the intercept. The fitting routine applies the article's LQA updates, coefficient elimination and stopping rules. The criterion is nonconvex when q < 1, so the iterative result is not a guarantee of a global minimum. The getting-started vignette explains the penalty geometry and the fixed-LQA restriction correction.

Value

A list of class rbridge with components:

Author(s)

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

References

Yuzbasi, B., Arashi, M. and Akdeniz, F. (2021). Penalized regression via the restricted bridge estimator. Soft Computing, 25, 8401–8416. doi:10.1007/s00500-021-05763-9.

See Also

cv.rbridge, coef.rbridge, predict.rbridge

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))
# A user-specified sum of standardized slopes: b[1] + b[2] + b[5] = 6.5.
# This target is illustrative; it is not asserted to equal the generating sum.
R1.mat <- matrix(c(1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r1.vec <- matrix(6.5, nrow = 1, ncol = 1)
# A second standardized-slope contrast: -b[1] + b[2] + b[5] = 0.5.
R2.mat <- matrix(c(-1, 1, 0, 0, 1, 0, 0, 0), nrow = 1)
r2.vec <- matrix(0.5, nrow = 1, ncol = 1)
# Impose both independent restrictions together.
R3.mat <- rbind(R1.mat, R2.mat)
r3.vec <- matrix(c(6.5, 0.5), ncol = 1)
# Alternatively, set the five inactive generating slopes to zero.
p.active <- which(beta != 0)
R4.mat <- diag(p)[-p.active, , drop = FALSE]
r4.vec <- matrix(rep(0, p - length(p.active)), ncol = 1)

# Five penalties keep this help example fast; the default is 100.
model1 <- rbridge(X, y, q = 1, R = R1.mat, r = r1.vec, nlambda = 5)
model2 <- rbridge(X, y, q = 1, R = R2.mat, r = r2.vec, nlambda = 5)
model3 <- rbridge(X, y, q = 1, R = R3.mat, r = r3.vec, nlambda = 5)
model4 <- rbridge(X, y, q = 1, R = R4.mat, r = r4.vec, nlambda = 5)
# Rows include the intercept; columns correspond to fitted penalties.
dim(model1$betas)
# Extract the second fitted penalty exactly; no interpolation is performed.
selected <- model1$lambda[2]
coefficients <- coef(model1, s = selected)
coefficients
predict(model1, newx = X[1:5, , drop = FALSE], s = selected)
# Convert returned slopes to standardized slopes before checking the target.
x_centered <- sweep(X, 2, colMeans(X), "-")
x_rms <- sqrt(colMeans(x_centered^2))
standardized_slopes <- coefficients[-1, 1] * x_rms
as.numeric(R1.mat %*% standardized_slopes - r1.vec)

Compiled engine capabilities

Description

Report whether this installed package was compiled with OpenMP support. This reports a build capability, not a measured speedup or the number of threads used in a particular fit. CV uses at most one worker per fold.

Usage

rbridge_info()

Details

Without OpenMP, CV runs sequentially for any num_threads. With OpenMP, num_threads requests fold-level workers. Set BLAS threads to one before starting R to avoid nested parallelism.

Value

A list with three components:

Examples

rbridge_info()