Type: Package
Title: Restricted Bridge Estimation
Version: 1.0.2
Date: 2020-02-29
Author: Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi <b.yzb@hotmail.com>
Description: Bridge Regression estimation with linear restrictions defined in Yuzbasi et al. (2019) <doi:10.48550/arXiv.1910.03660>. Special cases of this approach fit the restricted LASSO, restricted RIDGE and restricted Elastic Net estimators.
License: GPL-3
Imports: Rcpp, Matrix, dplyr, methods
Suggests: utils, stats, testthat
Encoding: UTF-8
LinkingTo: Rcpp, RcppArmadillo
RoxygenNote: 6.1.1
NeedsCompilation: yes
Packaged: 2020-02-29 09:26:16 UTC; bahadiryuzbasi
Repository: CRAN
Date/Publication: 2020-02-29 11:40:03 UTC

Fit a Bridge Estimation

Description

Fit a bridge penalized maximum likelihood. It is computed the regularization path which is consisted of lasso or ridge penalty at the a grid values for lambda

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

X

Design matrix.

y

Response vector.

q

is the degree of norm which includes ridge regression with q=2 and lasso estimates with q=1 as special cases

lambda.min

The smallest value for lambda if n>p is 0.001 and 0.05 otherwise.

nlambda

The number of lambda values - default is 100

lambda

A user supplied lambda sequence. By default, the program compute a squence of values the length of nlambda.

eta

is a preselected small positive threshold value. It is deleted jth variable to make the algorithm stable and also is excluded jth variable from the final model. Default is 1e-07.

converge

is the value of converge. Defaults is 10^10. In each iteration, it is calculated by sum of square the change in linear predictor for each coefficient. The algorithm iterates until converge > eta.

Details

Computes bridge estimation

Value

An object of class rbridge, a list with entries

betas

Coefficients computed over the path of lambda

lambda

The lambda values which is given at the function

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

cv.bridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

model1 <- bridge(X, y, q = 1)
print(model1)

model2 <- bridge(X, y, q = 2)
print(model2)

Extract coefficients from a 'bridge' object

Description

Extract coefficients from a 'bridge' object.

Usage

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

Arguments

object

A 'bridge' object.

s

Value(s) of the penalty parameter lambda at which predictions are required.

...

Additional arguments for compatibility.

Value

A vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.bridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

model1 <- bridge(X, y, q = 1)
coef(model1,s='lambda.min')

Extract coefficients from a 'cv.bridge' object

Description

Extract coefficients from a 'cv.bridge' object.

Usage

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

Arguments

object

A 'cv.bridge' object.

s

Value(s) of the penalty parameter lambda at which predictions are required.

...

Additional arguments for compatibility.

Value

A vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.cv.rbridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

model1 <- cv.bridge(X, y, q = 1)
coef(model1,s='lambda.min')

Extract coefficients from a 'cv.rbridge' object

Description

Extract coefficients from a 'cv.rbridge' object.

Usage

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

Arguments

object

A 'cv.rbridge' object.

s

Value(s) of the penalty parameter lambda at which predictions are required.

...

Additional arguments for compatibility.

Value

A vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.cv.rbridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)

### Restricted Matrix and vector
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- cv.rbridge(X, y, q = 1, R1.mat, r1.vec)
coef(model1,s='lambda.min')

Extract coefficients from a 'rbridge' object

Description

Makes predictions from a cross-validated 'rbridge' model

Usage

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

Arguments

object

A 'rbridge' object.

s

Value(s) of the penalty parameter lambda at which predictions are required.

...

Additional arguments for compatibility.

Value

Among a matrix with predictions, a vector non-zero indexing or a vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

predict.rbridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)

### Restricted Matrix and vector
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- rbridge(X, y, q = 1, R1.mat, r1.vec)
coef(model1,s='lambda.min')

Cross-validation for bridge

Description

Does k-fold cross-validation for bridge, produces a plot, and returns a value for lambda

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

X

X matrix as in bridge.

y

response y as in bridge.

q

is the degree of norm which includes ridge regression with q=2 and lasso estimates with q=1 as special cases

lambda

lambda sequence; default is NULL. It is given by user or cv.rbridge chooses its own sequence.

nfolds

number of folds - default is 10.

lambda.min

The smallest value for lambda if n>p is 0.001 and 0.05 otherwise.

nlambda

The number of lambda values - default is 100

eta

is a preselected small positive threshold value. It is deleted jth variable to make the algorithm stable and also is excluded jth variable from the final model. Default is 1e-07.

converge

is the value of converge. Defaults is 10^10. In each iteration, it is calculated by sum of square the change in linear predictor for each coefficient. The algorithm iterates until converge > eta.

num_threads

Number of threads used for parallel computation over the folds,

Details

Computes bridge

Value

An object of class rbridge, a list with entries

cve

the mean cross-validated error.

cvse

estimate of standard error of cvm.

cvup

upper curve = cvm+cvsd.

cvlo

lower curve = cvm-cvsd.

lambda

the values of lambda used in the fits

nz

number of non-zero coefficients at each lambda.

betas

estimated coefficient at each lambda.

lambda.min

value of lambda that gives minimum cve

lambda.1se

largest value of lambda such that error is within 1 standard error of the minimum

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

bridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 
model1 <- cv.bridge(X, y, q = 1)
print(model1)
coef(model1,s='lambda.min')
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")

######## Model 2 
model2 <- cv.bridge(X, y, q = 2)
print(model2)
coef(model2,s='lambda.min')
predict(model2,newx=X[1:5,], s="lambda.min", type="response")
predict(model2, s="lambda.min",type="coefficient")

Cross-validation for rbridge

Description

Does k-fold cross-validation for rbridge, produces a plot, and returns a value for lambda

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

X

X matrix as in rbridge.

y

response y as in rbridge.

q

is the degree of norm which includes ridge regression with q=2 and lasso estimates with q=1 as special cases

R

is m by p (m<p) matrix of constants.

r

is a m-vector of known prespecified constants. If it is given true restriction, then

r - R\beta = 0.

Values for r should be given as a matrix. See "Examples".

lambda

lambda sequence; default is NULL. It is given by user or cv.rbridge chooses its own sequence.

nfolds

number of folds - default is 10.

lambda.min

The smallest value for lambda if n>p is 0.001 and 0.05 otherwise.

nlambda

The number of lambda values - default is 100

eta

is a preselected small positive threshold value. It is deleted jth variable to make the algorithm stable and also is excluded jth variable from the final model. Default is 1e-07.

converge

is the value of converge. Defaults is 10^10. In each iteration, it is calculated by sum of square the change in linear predictor for each coefficient. The algorithm iterates until converge > eta.

num_threads

Number of threads used for parallel computation over the folds,

Details

Computes cv.rbridge

Value

An object of class rbridge, a list with entries

cve

the mean cross-validated error.

cvse

estimate of standard error of cvm.

cvup

upper curve = cvm+cvsd.

cvlo

lower curve = cvm-cvsd.

lambda

the values of lambda used in the fits

nz

number of non-zero coefficients at each lambda.

betas

estimated coefficient at each lambda.

lambda.min

value of lambda that gives minimum cve

lambda.1se

largest value of lambda such that error is within 1 standard error of the minimum

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

rbridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
p.active <- which(beta != 0)

### Restricted Matrix and vector
### Res 1
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)
### Res 2
c2 <- c(-1,1,0,0,1,0,0,0)
R2.mat <- matrix(c2,nrow = 1, ncol = p)
r2.vec <- matrix(c(0.5),nrow = 1, ncol = 1)
### Res 3
R3.mat <- t(matrix(c(c1,c2),nrow = p, ncol = 2))
r3.vec <- matrix(c(6.5,0.5),nrow = 2, ncol = 1)
### Res 4
R4.mat = diag(1,p,p)[-p.active,]
r4.vec <- matrix(rep(0,p-length(p.active)),nrow = p-length(p.active), ncol = 1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- cv.rbridge(X, y, q = 1, R1.mat, r1.vec)
print(model1)
coef(model1,s='lambda.min')
coef(model1,s='lambda.1se')
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")
predict(model1, s="lambda.1se",type="coefficient")

######## Model 2 based on second restrictions
model2 <- cv.rbridge(X, y, q = 1, R2.mat, r2.vec)
print(model2)
coef(model2,s='lambda.min')
coef(model2,s='lambda.1se')
predict(model2,newx=X[1:5,], s="lambda.min", type="response")
predict(model2, s="lambda.min",type="coefficient")
predict(model2, s="lambda.1se",type="coefficient")

######## Model 3 based on third restrictions
model3 <- cv.rbridge(X, y, q = 1, R3.mat, r3.vec)
print(model3)
coef(model3,s='lambda.min')
coef(model3,s='lambda.1se')
predict(model3,newx=X[1:5,], s="lambda.min", type="response")
predict(model3, s="lambda.min",type="coefficient")
predict(model3, s="lambda.1se",type="coefficient")

######## Model 4 based on fourth restrictions
model4 <- cv.rbridge(X, y, q = 1, R4.mat, r4.vec)
print(model4)
coef(model4,s='lambda.min')
coef(model4,s='lambda.1se')
predict(model4,newx=X[1:5,], s="lambda.min", type="response")
predict(model4, s="lambda.min",type="coefficient")
predict(model4, s="lambda.1se",type="coefficient")

Plot a 'cv.bridge' object function

Description

Plots the cross-validation curve, and upper and lower standard deviation curves, as a function of the lambda values used.

Usage

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

Arguments

x

Design matrix.

sign.lambda

Either plot against log(lambda) (default) or its negative if sign.lambda=-1.

...

Other graphical parameters to plot

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

Plot a 'cv.rbridge' object function

Description

Plots the cross-validation curve, and upper and lower standard deviation curves, as a function of the lambda values used.

Usage

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

Arguments

x

Design matrix.

sign.lambda

Either plot against log(lambda) (default) or its negative if sign.lambda=-1.

...

Other graphical parameters to plot

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

Make predictions from a 'bridge' object

Description

Makes predictions from a cross-validated 'bridge' model

Usage

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

Arguments

object

A 'bridge' object.

newx

Matrix of new values for x at which predictions are to be made.

s

Value(s) of the penalty parameter lambda at which predictions are required.

type

It should one of "response", "nonzero" or "coefficients". The "response" is for predicted values, the "nonzero" is for exacting non-zero coefficients and the "coefficients" is for the estimated coefficients.

...

Additional arguments for compatibility.

Value

Among a matrix with predictions, a vector non-zero indexing or a vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.bridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

model1 <- bridge(X, y, q = 1)
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")

Make predictions from a 'cv.bridge' object

Description

Makes predictions from a cross-validated 'cv.bridge' model

Usage

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

Arguments

object

A 'cv.bridge' object.

newx

Matrix of new values for x at which predictions are to be made.

s

Value(s) of the penalty parameter lambda at which predictions are required.

type

It should one of "response", "nonzero" or "coefficients". The "response" is for predicted values, the "nonzero" is for exacting non-zero coefficients and the "coefficients" is for the estimated coefficients.

...

Additional arguments for compatibility.

Value

Among a matrix with predictions, a vector non-zero indexing or a vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.cv.bridge

Examples

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),n,p)
y = X%*%beta + rnorm(n) 

model1 <- cv.bridge(X, y, q = 1)
coef(model1,s='lambda.min')
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")

Make predictions from a 'cv.rbridge' object

Description

Makes predictions from a cross-validated 'cv.rbridge' model

Usage

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

Arguments

object

A 'cv.rbridge' object.

newx

Matrix of new values for x at which predictions are to be made.

s

Value(s) of the penalty parameter lambda at which predictions are required.

type

It should one of "response", "nonzero" or "coefficients". The "response" is for predicted values, the "nonzero" is for exacting non-zero coefficients and the "coefficients" is for the estimated coefficients.

...

Additional arguments for compatibility.

Value

Among a matrix with predictions, a vector non-zero indexing or a vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

Bahadir Yuzbasi maintainer Baha

See Also

coef.cv.rbridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)

### Restricted Matrix and vector
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- cv.rbridge(X, y, q = 1, R1.mat, r1.vec)
coef(model1,s='lambda.min')
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")

Make predictions from a 'rbridge' object

Description

Makes predictions from a cross-validated 'rbridge' model

Usage

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

Arguments

object

A 'rbridge' object.

newx

Matrix of new values for x at which predictions are to be made.

s

Value(s) of the penalty parameter lambda at which predictions are required.

type

It should one of "response", "nonzero" or "coefficients". The "response" is for predicted values, the "nonzero" is for exacting non-zero coefficients and the "coefficients" is for the estimated coefficients.

...

Additional arguments for compatibility.

Value

Among a matrix with predictions, a vector non-zero indexing or a vector of coefficients

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

coef.cv.bridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)

### Restricted Matrix and vector
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- rbridge(X, y, q = 1, R1.mat, r1.vec)
predict(model1,newx=X[1:5,], s="lambda.min", type="response")
predict(model1, s="lambda.min",type="coefficient")

Fit a Restricted Bridge Estimation

Description

Fit a restricted linear model via bridge penalized maximum likelihood. It is computed the regularization path which is consisted of lasso or ridge penalty at the a grid values for lambda

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

X

Design matrix.

y

Response vector.

q

is the degree of norm which includes ridge regression with q=2 and lasso estimates with q=1 as special cases

R

is m by p (m<p) matrix of constants.

r

is a m-vector of known prespecified constants. If it is given true restriction, then

r - R\beta = 0.

Values for r should be given as a matrix. See "Examples".

lambda.min

The smallest value for lambda if n>p is 0.001 and 0.05 otherwise.

nlambda

The number of lambda values - default is 100

lambda

A user supplied lambda sequence. By default, the program compute a squence of values the length of nlambda.

eta

is a preselected small positive threshold value. It is deleted jth variable to make the algorithm stable and also is excluded jth variable from the final model. Default is 1e-07.

converge

is the value of converge. Defaults is 10^10. In each iteration, it is calculated by sum of square the change in linear predictor for each coefficient. The algorithm iterates until converge > eta.

Details

In order to couple the bridge estimator with the restriction R beta = r, we solve the following optimization problem

\min RSS w.r.t ||\beta||_q and R\beta = r.

Value

An object of class rbridge, a list with entries

betas

Coefficients computed over the path of lambda

lambda

The lambda values which is given at the function

Author(s)

Bahadir Yuzbasi, Mohammad Arashi and Fikri Akdeniz
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

See Also

cv.rbridge

Examples

set.seed(2019) 
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
p.active <- which(beta != 0)

### Restricted Matrix and vector
### Res 1
c1 <- c(1,1,0,0,1,0,0,0)
R1.mat <- matrix(c1,nrow = 1, ncol = p)
r1.vec <- as.matrix(c(6.5),1,1)
### Res 2
c2 <- c(-1,1,0,0,1,0,0,0)
R2.mat <- matrix(c2,nrow = 1, ncol = p)
r2.vec <- matrix(c(0.5),nrow = 1, ncol = 1)
### Res 3
R3.mat <- t(matrix(c(c1,c2),nrow = p, ncol = 2))
r3.vec <- matrix(c(6.5,0.5),nrow = 2, ncol = 1)
### Res 4
R4.mat = diag(1,p,p)[-p.active,]
r4.vec <- matrix(rep(0,p-length(p.active)),nrow = p-length(p.active), ncol = 1)

n = 100
X = matrix(rnorm(n*p),n,p)
y = X%*%beta + rnorm(n) 

######## Model 1 based on first restrictions
model1 <- rbridge(X, y, q = 1, R1.mat, r1.vec)
print(model1)

######## Model 2 based on second restrictions
model2 <- rbridge(X, y, q = 1, R2.mat, r2.vec)
print(model2)

######## Model 3 based on third restrictions
model3 <- rbridge(X, y, q = 1, R3.mat, r3.vec)
print(model3)

######## Model 4 based on fourth restrictions
model4 <- rbridge(X, y, q = 1, R4.mat, r4.vec)
print(model4)