| Title: | Robust Structured Regression via the L2 Criterion |
|---|---|
| Description: | An implementation of a computational framework for performing robust structured regression with the L2 criterion from Chi and Chi (2021+). Improvements using the majorization-minimization (MM) principle from Liu, Chi, and Lange (2022+) added in Version 2.0. |
| Authors: | Xiaoqian Liu [aut, ctb], Jocelyn Chi [aut, cre], Lisa Lin [ctb], Kenneth Lange [aut], Eric Chi [aut] |
| Maintainer: | Jocelyn Chi <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.0 |
| Built: | 2025-02-14 04:26:35 UTC |
| Source: | https://github.com/cran/L2E |
Data from an Italian bank on 1,949 customers. The response y is the amount of money made over a year. The 13 covariates are possible macroservices the customers can sign up for.
A data frame with 1949 rows and 14 variables
Marco Riani, Andrea Cerioli, and Anthony C. Atkinson (2014). Monitoring robust regression. Electronic Journal of Statistics, Volume 8, 646-677.
CV_L2E_sparse_dist performs k-fold cross-validation for robust sparse regression under the L2 criterion with
distance penalty
CV_L2E_sparse_dist( y, X, beta0, tau0, kSeq, rhoSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )CV_L2E_sparse_dist( y, X, beta0, tau0, kSeq, rhoSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )
y |
Response vector |
X |
Design matrix |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
kSeq |
A sequence of tuning parameter k, the number of nonzero entries in the estimated coefficients |
rhoSeq |
A sequence of tuning parameter rho, can be omitted |
nfolds |
The number of cross-validation folds. Default is 5. |
seed |
Users can set the seed of the random number generator to obtain reproducible results. |
method |
Median or mean to compute the objective |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
trace |
Whether to trace the progress of the cross-validation |
Returns a list object containing the mean and standard error of the cross-validation error (vectors) – CVE and CVSE – for each value of k, the index of the k value with the minimum CVE and the k value itself (scalars), the index of the k value with the 1SE CVE and the k value itself (scalars), the sequence of rho and k used in the regression (vectors), and a vector listing which fold each element of y was assigned to
## Completes in 15 seconds set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data k <- c(6,5,4) # (not run) # cv <- CV_L2E_sparse_dist(y=y, X=X, kSeq=k, nfolds=2, seed=1234) # (k_min <- cv$k.min) ## selected number of nonzero entries # sol <- L2E_sparse_dist(y=y, X=X, kSeq=k_min) # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] # (not run) # cv <- CV_L2E_sparse_dist(y=y, X=X, kSeq=k, nfolds=2, seed=1234) # (k_min <- cv$k.min) ## selected number of nonzero entries # sol <- L2E_sparse_dist(y=y, X=X, kSeq=k_min) # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)## Completes in 15 seconds set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data k <- c(6,5,4) # (not run) # cv <- CV_L2E_sparse_dist(y=y, X=X, kSeq=k, nfolds=2, seed=1234) # (k_min <- cv$k.min) ## selected number of nonzero entries # sol <- L2E_sparse_dist(y=y, X=X, kSeq=k_min) # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] # (not run) # cv <- CV_L2E_sparse_dist(y=y, X=X, kSeq=k, nfolds=2, seed=1234) # (k_min <- cv$k.min) ## selected number of nonzero entries # sol <- L2E_sparse_dist(y=y, X=X, kSeq=k_min) # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
CV_L2E_sparse_ncv performs k-fold cross-validation for robust sparse regression under the L2 criterion.
Available penalties include lasso, MCP and SCAD.
CV_L2E_sparse_ncv( y, X, beta0, tau0, lambdaSeq, penalty = "MCP", nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )CV_L2E_sparse_ncv( y, X, beta0, tau0, lambdaSeq, penalty = "MCP", nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )
y |
Response vector |
X |
Design matrix |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
lambdaSeq |
A decreasing sequence of tuning parameter lambda, can be omitted |
penalty |
Available penalties include lasso, MCP and SCAD. |
nfolds |
The number of cross-validation folds. Default is 5. |
seed |
Users can set the seed of the random number generator to obtain reproducible results. |
method |
Median or mean to compute the objective |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
trace |
Whether to trace the progress of the cross-validation |
Returns a list object containing the mean and standard error of the cross-validation error – CVE and CVSE – for each value of k (vectors), the index of the lambda with the minimum CVE and the lambda value itself (scalars), the index of the lambda value with the 1SE CVE and the lambda value itself (scalars), the sequence of lambda used in the regression (vector), and a vector listing which fold each element of y was assigned to
## Completes in 20 seconds set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data lambda <- 10^seq(-1, -2, length.out=20) # (not run) # cv <- CV_L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD", seed=1234, nfolds=2) # (lambda_min <- cv$lambda.min) # sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda_min, penalty="SCAD") # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] # (not run) # cv <- CV_L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD", seed=1234, nfolds=2) # (lambda_min <- cv$lambda.min) # sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda_min, penalty="SCAD") # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)## Completes in 20 seconds set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data lambda <- 10^seq(-1, -2, length.out=20) # (not run) # cv <- CV_L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD", seed=1234, nfolds=2) # (lambda_min <- cv$lambda.min) # sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda_min, penalty="SCAD") # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] # (not run) # cv <- CV_L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD", seed=1234, nfolds=2) # (lambda_min <- cv$lambda.min) # sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda_min, penalty="SCAD") # r <- y - X %*% sol$Beta # ix <- which(abs(r) > 3/sol$Tau) # l2e_fit <- X %*% sol$Beta # plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) # points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
CV_L2E_TF_dist performs k-fold cross-validation for robust trend filtering regression under the L2 criterion with distance penalty
CV_L2E_TF_dist( y, X, beta0, tau0, D, kSeq, rhoSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )CV_L2E_TF_dist( y, X, beta0, tau0, D, kSeq, rhoSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )
y |
Response vector |
X |
Design matrix. Default is the identity matrix. |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
D |
The fusion matrix |
kSeq |
A sequence of tuning parameter k, the number of nonzero entries in Dbeta |
rhoSeq |
A sequence of tuning parameter rho, can be omitted |
nfolds |
The number of cross-validation folds. Default is 5. |
seed |
Users can set the seed of the random number generator to obtain reproducible results. |
method |
Median or mean to calculate the objective value |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
trace |
Whether to trace the progress of the cross-validation |
Returns a list object containing the mean and standard error of the cross-validation error – CVE and CVSE – for each value of k (vectors), the index of the k value with the minimum CVE and the k value itself (scalars), the index of the k value with the 1SE CVE and the k value itself (scalars), the sequence of rho and k used in the regression (vectors), and a vector listing which fold each element of y was assigned to
## Completes in 20 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) k <- c(4,3,2) rho <- 10^8 # (not run) # cv <- CV_L2E_TF_dist(y=y0, D=D, kSeq=k, rhoSeq=rho, nfolds=2, seed=1234) # (k_min <- cv$k.min) # sol <- L2E_TF_dist(y=y0, D=D, kSeq=k_min, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3) ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # cv <- CV_L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho, nfolds=2, seed=1234) # (k_min <- cv$k.min) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k_min, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3)## Completes in 20 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) k <- c(4,3,2) rho <- 10^8 # (not run) # cv <- CV_L2E_TF_dist(y=y0, D=D, kSeq=k, rhoSeq=rho, nfolds=2, seed=1234) # (k_min <- cv$k.min) # sol <- L2E_TF_dist(y=y0, D=D, kSeq=k_min, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3) ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # cv <- CV_L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho, nfolds=2, seed=1234) # (k_min <- cv$k.min) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k_min, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3)
CV_L2E_TF_lasso performs k-fold cross-validation for robust trend filtering regression under the L2 criterion with the Lasso penalty
CV_L2E_TF_lasso( y, X, beta0, tau0, D, lambdaSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )CV_L2E_TF_lasso( y, X, beta0, tau0, D, lambdaSeq, nfolds = 5, seed = 1234, method = "median", max_iter = 100, tol = 1e-04, trace = TRUE )
y |
Response vector |
X |
Design matrix. Default is the identity matrix. |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
D |
The fusion matrix |
lambdaSeq |
A decreasing sequence of tuning parameter lambda, can be omitted |
nfolds |
The number of cross-validation folds. Default is 5. |
seed |
Users can set the seed of the random number generator to obtain reproducible results. |
method |
Median or mean to calculate the objective value |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
trace |
Whether to trace the progress of the cross-validation |
Returns a list object containing the mean and standard error of the cross-validation error – CVE and CVSE – for each value of k (vectors), the index of the lambda with the minimum CVE and the lambda value itself (scalars), the index of the lambda value with the 1SE CVE and the lambda value itself (scalars), the sequence of lambda used in the regression (vector), and a vector listing which fold each element of y was assigned to
## Completes in 30 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) lambda <- 10^seq(-1, -2, length.out=20) # (not run) # cv <- CV_L2E_TF_lasso(y=y0, D=D, lambdaSeq=lambda, nfolds=2, seed=1234) # (lambda_min <- cv$lambda.min) # sol <- L2E_TF_lasso(y=y0, D=D, lambdaSeq=lambda_min) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3) ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # cv <- CV_L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda, nfolds=2, seed=1234) # (lambda_min <- cv$lambda.min) # sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda_min) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3)## Completes in 30 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) lambda <- 10^seq(-1, -2, length.out=20) # (not run) # cv <- CV_L2E_TF_lasso(y=y0, D=D, lambdaSeq=lambda, nfolds=2, seed=1234) # (lambda_min <- cv$lambda.min) # sol <- L2E_TF_lasso(y=y0, D=D, lambdaSeq=lambda_min) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3) ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # cv <- CV_L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda, nfolds=2, seed=1234) # (lambda_min <- cv$lambda.min) # sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda_min) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta, col='blue', lwd=3)
The L2E package is an R implementation of a user-friendly computational framework for performing a wide variety of robust structured regression methods via the L2 criterion.
Please refer to the vignette for examples of how to use this package.
L2E_convex performs convex regression under the L2 criterion. Available methods include proximal gradient descent (PG) and majorization-minimization (MM).
L2E_convex( y, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_convex( y, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
method |
Available methods include PG and MM. MM by default. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta (vector) and tau or eta (vector)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau) * rnorm(n) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y ## Least Squares method cvx <- fitted(cobs::conreg(y, convex=TRUE)) ## MM method sol_mm <- L2E_convex(y, b, tau) ## PG method sol_pg <- L2E_convex(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, cvx, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG ## Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y cvx <- fitted(cobs::conreg(y, convex=TRUE)) sol_mm <- L2E_convex(y, b, tau) sol_pg <- L2E_convex(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, cvx, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PGset.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau) * rnorm(n) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y ## Least Squares method cvx <- fitted(cobs::conreg(y, convex=TRUE)) ## MM method sol_mm <- L2E_convex(y, b, tau) ## PG method sol_pg <- L2E_convex(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, cvx, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG ## Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y cvx <- fitted(cobs::conreg(y, convex=TRUE)) sol_mm <- L2E_convex(y, b, tau) sol_pg <- L2E_convex(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, cvx, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG
L2E_isotonic performs isotonic regression under the L2 criterion. Available methods include proximal gradient descent (PG) and majorization-minimization (MM).
L2E_isotonic( y, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_isotonic( y, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
method |
Available methods include PG and MM. MM by default. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta (vector) and tau or eta (vector)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau) * rnorm(n) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y ## Least Squares method iso <- isotone::gpava(1:n, y)$x ## MM method sol_mm <- L2E_isotonic(y, b, tau) ## PG method sol_pg <- L2E_isotonic(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, iso, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG ## Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y iso <- isotone::gpava(1:n, y)$x sol_mm <- L2E_isotonic(y, b, tau) sol_pg <- L2E_isotonic(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, iso, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PGset.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau) * rnorm(n) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y ## Least Squares method iso <- isotone::gpava(1:n, y)$x ## MM method sol_mm <- L2E_isotonic(y, b, tau) ## PG method sol_pg <- L2E_isotonic(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, iso, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG ## Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y iso <- isotone::gpava(1:n, y)$x sol_mm <- L2E_isotonic(y, b, tau) sol_pg <- L2E_isotonic(y, b, tau, method='PG') plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, iso, col='blue', lwd=3) ## LS lines(x, sol_mm$beta, col='red', lwd=3) ## MM lines(x, sol_pg$beta, col='dark green', lwd=3) ## PG
L2E_multivariate performs multivariate regression under the L2 criterion. Available methods include proximal gradient descent (PG) and majorization-minimization (MM).
L2E_multivariate( y, X, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_multivariate( y, X, beta, tau, method = "MM", max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
method |
Available methods include PG and MM. MM by default. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta (vector) and tau or eta (vector)
# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tau <- 1/mad(y) b <- matrix(0, 14, 1) # MM method sol_mm <- L2E_multivariate(y, X0, b, tau) r_mm <- y - X0 %*% sol_mm$beta ix_mm <- which(abs(r_mm) > 3/sol_mm$tau) l2e_fit_mm <- X0 %*% sol_mm$beta # PG method sol_pg <- L2E_multivariate(y, X0, b, tau, method="PG") r_pg <- y - X0 %*% sol_pg$beta ix_pg <- which(abs(r_pg) > 3/sol_pg$tau) l2e_fit_pg <- X0 %*% sol_pg$beta plot(y, l2e_fit_mm, ylab='Predicted values', main='MM', pch=16, cex=0.8) # MM points(y[ix_mm], l2e_fit_mm[ix_mm], pch=16, col='blue', cex=0.8) # MM plot(y, l2e_fit_pg, ylab='Predicted values', main='PG', pch=16, cex=0.8) # PG points(y[ix_pg], l2e_fit_pg[ix_pg], pch=16, col='blue', cex=0.8) # PG# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tau <- 1/mad(y) b <- matrix(0, 14, 1) # MM method sol_mm <- L2E_multivariate(y, X0, b, tau) r_mm <- y - X0 %*% sol_mm$beta ix_mm <- which(abs(r_mm) > 3/sol_mm$tau) l2e_fit_mm <- X0 %*% sol_mm$beta # PG method sol_pg <- L2E_multivariate(y, X0, b, tau, method="PG") r_pg <- y - X0 %*% sol_pg$beta ix_pg <- which(abs(r_pg) > 3/sol_pg$tau) l2e_fit_pg <- X0 %*% sol_pg$beta plot(y, l2e_fit_mm, ylab='Predicted values', main='MM', pch=16, cex=0.8) # MM points(y[ix_mm], l2e_fit_mm[ix_mm], pch=16, col='blue', cex=0.8) # MM plot(y, l2e_fit_pg, ylab='Predicted values', main='PG', pch=16, cex=0.8) # PG points(y[ix_pg], l2e_fit_pg[ix_pg], pch=16, col='blue', cex=0.8) # PG
l2e_regression performs L2E multivariate regression via block coordinate descent
with proximal gradient for updating both beta and tau.
l2e_regression(y, X, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE)l2e_regression(y, X, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE)
y |
Response vector |
X |
Design matrix |
b |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and tau (vectors)
# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tauinit <- 1/mad(y) binit <- matrix(0, 14, 1) sol <- l2e_regression(y, X0, binit, tauinit) r <- y - X0 %*% sol$beta ix <- which(abs(r) > 3/sol$tau) l2e_fit <- X0 %*% sol$beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tauinit <- 1/mad(y) binit <- matrix(0, 14, 1) sol <- l2e_regression(y, X0, binit, tauinit) r <- y - X0 %*% sol$beta ix <- which(abs(r) > 3/sol$tau) l2e_fit <- X0 %*% sol$beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
l2e_regression_convex performs L2E convex regression via block coordinate descent
with proximal gradient for updating both beta and tau.
l2e_regression_convex(y, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE)l2e_regression_convex(y, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE)
y |
Response vector |
b |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and tau (vectors)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau) * rnorm(n) ## Clean data example plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3) ## Contaminated data example ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3)set.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau) * rnorm(n) ## Clean data example plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3) ## Contaminated data example ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3)
l2e_regression_convex_MM performs L2E convex regression via block coordinate descent
with MM for updating beta and modified Newton for updating tau.
l2e_regression_convex_MM( y, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_convex_MM( y, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
response |
beta |
initial vector of regression coefficients |
tau |
initial precision estimate |
max_iter |
maximum number of iterations |
tol |
relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and eta (vectors)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau)*rnorm(n) ## Clean plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='red', lwd=3) ## Contaminated ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='red', lwd=3)set.seed(12345) n <- 200 tau <- 1 x <- seq(-2, 2, length.out=n) f <- x^4 + x y <- f + (1/tau)*rnorm(n) ## Clean plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='red', lwd=3) ## Contaminated ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_convex_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) cvx <- fitted(cobs::conreg(y, convex=TRUE)) lines(x, cvx, col='blue', lwd=3) lines(x, sol$beta, col='red', lwd=3)
l2e_regression_isotonic performs L2E isotonic regression via block coordinate descent
with proximal gradient for updating both beta and tau.
l2e_regression_isotonic( y, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_isotonic( y, b, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
b |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and tau (vectors)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau)*rnorm(n) # Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3) # Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3)set.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau)*rnorm(n) # Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3) # Contaminated Data ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x, sol$beta, col='dark green', lwd=3)
l2e_regression_isotonic_MM performs L2E isotonic regression via block coordinate descent
with MM for updating beta and modified Newton for updating tau.
l2e_regression_isotonic_MM( y, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_isotonic_MM( y, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and eta (vectors)
set.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau)*rnorm(n) ## Clean plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x,sol$beta,col='red',lwd=3) ## Contaminated ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic_MM(y,b,tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x,sol$beta,col='red',lwd=3)set.seed(12345) n <- 200 tau <- 1 x <- seq(-2.5, 2.5, length.out=n) f <- x^3 y <- f + (1/tau)*rnorm(n) ## Clean plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic_MM(y, b, tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x,sol$beta,col='red',lwd=3) ## Contaminated ix <- 0:9 y[45 + ix] <- 14 + rnorm(10) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) tau <- 1 b <- y sol <- l2e_regression_isotonic_MM(y,b,tau) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) iso <- isotone::gpava(1:n, y)$x lines(x, iso, col='blue', lwd=3) lines(x,sol$beta,col='red',lwd=3)
l2e_regression_MM performs L2E multivariate regression via block coordinate descent
with MM for updating beta and modified Newton for updating tau.
l2e_regression_MM( y, X, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_MM( y, X, beta, tau, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
response |
X |
Design matrix |
beta |
initial vector of regression coefficients |
tau |
initial precision estimate |
max_iter |
maximum number of iterations |
tol |
relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (vector) and tau (scalar), the number of outer block descent iterations until convergence (scalar), and the number of inner iterations per outer iteration for updating beta and eta (vectors)
# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tau <- 1/mad(y) b <- matrix(0, 14, 1) sol <- l2e_regression_MM(y, X0, b, tau) r <- y - X0 %*% sol$beta ix <- which(abs(r) > 3/sol$tau) l2e_fit <- X0 %*% sol$beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)# Bank data example y <- bank$y X <- as.matrix(bank[,1:13]) X0 <- as.matrix(cbind(rep(1,length(y)), X)) tau <- 1/mad(y) b <- matrix(0, 14, 1) sol <- l2e_regression_MM(y, X0, b, tau) r <- y - X0 %*% sol$beta ix <- which(abs(r) > 3/sol$tau) l2e_fit <- X0 %*% sol$beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
l2e_regression_sparse_dist performs robust sparse regression under the L2 criterion with the distance penalty
l2e_regression_sparse_dist( y, X, beta, tau, k, rho = 1, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_sparse_dist( y, X, beta, tau, k, rho = 1, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
k |
The number of nonzero entries in the estimated coefficients |
rho |
The parameter in the proximal distance algorithm |
stepsize |
The stepsize parameter for the MM algorithm (0, 1) |
sigma |
The halving parameter sigma (0, 1) |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
l2e_regression_sparse_ncv performs robust sparse regression under the L2 criterion. Available penalties include lasso, MCP and SCAD.
l2e_regression_sparse_ncv( y, X, beta, tau, lambda, penalty, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_sparse_ncv( y, X, beta, tau, lambda, penalty, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
lambda |
Tuning parameter |
penalty |
Available penalties include lasso, MCP and SCAD. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
l2e_regression_TF_dist performs robust trend filtering regression under the L2 criterion with distance penalty
l2e_regression_TF_dist( y, X, beta, tau, D, k, rho = 1, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_TF_dist( y, X, beta, tau, D, k, rho = 1, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
D |
The fusion matrix |
k |
The number of nonzero entries in D*beta |
rho |
The parameter in the proximal distance algorithm |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
l2e_regression_TF_lasso performs robust trend filtering regression under the L2 criterion with Lasso penalty
l2e_regression_TF_lasso( y, X, beta, tau, D, lambda = 1, max_iter = 100, tol = 1e-04, Show.Time = TRUE )l2e_regression_TF_lasso( y, X, beta, tau, D, lambda = 1, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
D |
The fusion matrix |
lambda |
The tuning parameter |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
L2E_sparse_dist computes the solution path of the robust sparse regression under the L2 criterion with distance penalty
L2E_sparse_dist( y, X, beta0, tau0, kSeq, rhoSeq, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_sparse_dist( y, X, beta0, tau0, kSeq, rhoSeq, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
kSeq |
A sequence of tuning parameter k, the number of nonzero entries in the estimated coefficients |
rhoSeq |
An increasing sequence of tuning parameter rho, can be omitted |
stepsize |
The stepsize parameter for the MM algorithm (0, 1) |
sigma |
The halving parameter sigma (0, 1) |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (matrix) and tau (vector) for each value of the tuning parameter k, the path of estimates for beta (list of matrices) and tau (matrix) for each value of rho, the run time (vector) for each k, and the sequence of rho and k used in the regression (vectors)
set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data k <- 5 sol <- L2E_sparse_dist(y=y, X=X, kSeq=k) r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] sol <- L2E_sparse_dist(y=y, X=X, kSeq=k) r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data k <- 5 sol <- L2E_sparse_dist(y=y, X=X, kSeq=k) r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] sol <- L2E_sparse_dist(y=y, X=X, kSeq=k) r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
L2E_sparse_ncv computes the solution path of robust sparse regression under the L2 criterion. Available penalties include lasso, MCP and SCAD.
L2E_sparse_ncv( y, X, b, tau, lambdaSeq, penalty = "MCP", max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_sparse_ncv( y, X, b, tau, lambdaSeq, penalty = "MCP", max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix |
b |
Initial vector of regression coefficients, can be omitted |
tau |
Initial precision estimate, can be omitted |
lambdaSeq |
A decreasing sequence of values for the tuning parameter lambda, can be omitted |
penalty |
Available penalties include lasso, MCP and SCAD. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (matrix) and tau (vector) for each value of the tuning parameter lambda, the run time (vector) for each lambda, and the sequence of lambda used in the regression (vector)
set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data lambda <- 10^(-1) sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD") r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD") r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)set.seed(12345) n <- 100 tau <- 1 f <- matrix(c(rep(2,5), rep(0,45)), ncol = 1) X <- X0 <- matrix(rnorm(n*50), nrow = n) y <- y0 <- X0 %*% f + (1/tau)*rnorm(n) ## Clean Data lambda <- 10^(-1) sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD") r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8) ## Contaminated Data i <- 1:5 y[i] <- 2 + y0[i] X[i,] <- 2 + X0[i,] sol <- L2E_sparse_ncv(y=y, X=X, lambdaSeq=lambda, penalty="SCAD") r <- y - X %*% sol$Beta ix <- which(abs(r) > 3/sol$Tau) l2e_fit <- X %*% sol$Beta plot(y, l2e_fit, ylab='Predicted values', pch=16, cex=0.8) points(y[ix], l2e_fit[ix], pch=16, col='blue', cex=0.8)
L2E_TF_dist computes the solution path of the robust trend filtering regression under the L2 criterion with distance penalty
L2E_TF_dist( y, X, beta0, tau0, D, kSeq, rhoSeq, max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_TF_dist( y, X, beta0, tau0, D, kSeq, rhoSeq, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix. Default is the identity matrix. |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
D |
The fusion matrix |
kSeq |
A sequence of tuning parameter k, the number of nonzero entries in D*beta |
rhoSeq |
An increasing sequence of tuning parameter rho, can be omitted |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (matrix) and tau (vector) for each value of the tuning parameter k, the path of estimates for beta (list of matrices) and tau (matrix) for each value of rho, the run time (vector) for each k, and the sequence of rho and k used in the regression (vectors)
## Completes in 15 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) k <- c(4,3,2) rho <- 10^8 # (not run) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta[,3], col='blue', lwd=3) ## k=2 # lines(x, sol$Beta[,2], col='red', lwd=3) ## k=3 # lines(x, sol$Beta[,1], col='dark green', lwd=3) ## k=4 ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta[,3], col='blue', lwd=3) ## k=2 # lines(x, sol$Beta[,2], col='red', lwd=3) ## k=3 # lines(x, sol$Beta[,1], col='dark green', lwd=3) ## k=4## Completes in 15 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) k <- c(4,3,2) rho <- 10^8 # (not run) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta[,3], col='blue', lwd=3) ## k=2 # lines(x, sol$Beta[,2], col='red', lwd=3) ## k=3 # lines(x, sol$Beta[,1], col='dark green', lwd=3) ## k=4 ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) # (not run) # sol <- L2E_TF_dist(y=y, D=D, kSeq=k, rhoSeq=rho) # plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') # lines(x, f, lwd=3) # lines(x, sol$Beta[,3], col='blue', lwd=3) ## k=2 # lines(x, sol$Beta[,2], col='red', lwd=3) ## k=3 # lines(x, sol$Beta[,1], col='dark green', lwd=3) ## k=4
L2E_TF_lasso computes the solution path of the robust trend filtering regression under the L2 criterion with Lasso penalty
L2E_TF_lasso( y, X, beta0, tau0, D, lambdaSeq, max_iter = 100, tol = 1e-04, Show.Time = TRUE )L2E_TF_lasso( y, X, beta0, tau0, D, lambdaSeq, max_iter = 100, tol = 1e-04, Show.Time = TRUE )
y |
Response vector |
X |
Design matrix. Default is the identity matrix. |
beta0 |
Initial vector of regression coefficients, can be omitted |
tau0 |
Initial precision estimate, can be omitted |
D |
The fusion matrix |
lambdaSeq |
A decreasing sequence of values for the tuning parameter lambda, can be omitted |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Show.Time |
Report the computing time |
Returns a list object containing the estimates for beta (matrix) and tau (vector) for each value of the tuning parameter lambda, the run time (vector) for each lambda, and the sequence of lambda used in the regression (vector)
## Completes in 10 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) lambda <- 10^seq(-1, -2, length.out=20) sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, sol$Beta[,1], col='blue', lwd=3) ## 1st lambda ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, sol$Beta[,1], col='blue', lwd=3) ## 1st lambda## Completes in 10 seconds set.seed(12345) n <- 100 x <- 1:n f <- matrix(rep(c(-2,5,0,-10), each=n/4), ncol=1) y <- y0 <- f + rnorm(length(f)) ## Clean Data plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) D <- myGetDkn(1, n) lambda <- 10^seq(-1, -2, length.out=20) sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, sol$Beta[,1], col='blue', lwd=3) ## 1st lambda ## Contaminated Data ix <- sample(1:n, 10) y[ix] <- y0[ix] + 2 plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) sol <- L2E_TF_lasso(y=y, D=D, lambdaSeq=lambda) plot(x, y, pch=16, cex.lab=1.5, cex.axis=1.5, cex.sub=1.5, col='gray') lines(x, f, lwd=3) lines(x, sol$Beta[,1], col='blue', lwd=3) ## 1st lambda
myGetDkn computes the kth order differencing matrix for use in
trend filtering regression
myGetDkn(k, n)myGetDkn(k, n)
k |
Order of the differencing matrix |
n |
Number of time points |
Returns a Matrix object as the kth order differencing matrix
objective computes the objective of the L2E regression in terms of eta
objective(eta, r, method = "mean")objective(eta, r, method = "mean")
eta |
The current estimate of eta |
r |
Vector of residuals |
method |
Mean or median |
Returns the output of the objective function (scalar)
objective_tau computes the objective of the L2E regression in terms of tau
objective_tau(tau, r, method = "mean")objective_tau(tau, r, method = "mean")
tau |
The current estimate of tau |
r |
Vector of residuals |
method |
Mean or median |
Returns the output of the objective function (scalar)
update_beta_convex updates beta for L2E convex regression using PG
update_beta_convex(y, b, tau, max_iter = 100, tol = 1e-04)update_beta_convex(y, b, tau, max_iter = 100, tol = 1e-04)
y |
Response vector |
b |
Current estimate for beta |
tau |
Current estimate for tau |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_isotonic updates beta for L2E isotonic regression using PG
update_beta_isotonic(y, b, tau, max_iter = 100, tol = 1e-04)update_beta_isotonic(y, b, tau, max_iter = 100, tol = 1e-04)
y |
Response vector |
b |
Current estimate for beta |
tau |
Current estimate for tau |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_MM_convex updates beta for L2E convex regression using MM
update_beta_MM_convex(y, beta, tau, max_iter = 100, tol = 1e-04)update_beta_MM_convex(y, beta, tau, max_iter = 100, tol = 1e-04)
y |
Response |
beta |
Initial vector of regression coefficients |
tau |
Precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_MM_isotonic updates beta for L2E isotonic regression using MM
update_beta_MM_isotonic(y, beta, tau, max_iter = 100, tol = 1e-04)update_beta_MM_isotonic(y, beta, tau, max_iter = 100, tol = 1e-04)
y |
response |
beta |
initial vector of regression coefficients |
tau |
precision estimate |
max_iter |
maximum number of iterations |
tol |
relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_MM_ls updates beta for L2E multivariate regression using MM
update_beta_MM_ls(y, X, beta, tau, max_iter = 100, tol = 1e-04)update_beta_MM_ls(y, X, beta, tau, max_iter = 100, tol = 1e-04)
y |
Response |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Precision estimate |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_MM_sparse updates beta for L2E sparse regression using the distance penalty
update_beta_MM_sparse( y, X, beta, tau, k, rho, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04 )update_beta_MM_sparse( y, X, beta, tau, k, rho, stepsize = 0.9, sigma = 0.5, max_iter = 100, tol = 1e-04 )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
k |
The number of nonzero entries in the estimated coefficients |
rho |
The parameter in the proximal distance algorithm |
stepsize |
The stepsize parameter for the MM algorithm (0, 1) |
sigma |
The halving parameter sigma (0, 1) |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_MM_TF updates beta in L2E trend filtering regression using the distance penalty
update_beta_MM_TF(y, X, beta, tau, D, k, rho, max_iter = 100, tol = 1e-04)update_beta_MM_TF(y, X, beta, tau, D, k, rho, max_iter = 100, tol = 1e-04)
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
D |
The fusion matrix |
k |
The number of nonzero entries in D*beta |
rho |
The parameter in the proximal distance algorithm |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_qr updates beta for L2E multivariate regression via a QR solve
update_beta_qr(y, X, QRF, tau, b, max_iter = 100, tol = 1e-04)update_beta_qr(y, X, QRF, tau, b, max_iter = 100, tol = 1e-04)
y |
Response vector |
X |
Design matrix |
QRF |
QR factorization object for X (obtained via 'QRF=qr(X)') |
tau |
Current estimate for tau |
b |
Current estimate for beta |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_sparse_ncv updates beta for L2E sparse regression using existing penalization methods
update_beta_sparse_ncv( y, X, beta, tau, lambda, penalty, max_iter = 100, tol = 1e-04 )update_beta_sparse_ncv( y, X, beta, tau, lambda, penalty, max_iter = 100, tol = 1e-04 )
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
lambda |
Tuning parameter |
penalty |
Available penalties include lasso, MCP and SCAD. |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_beta_TF_lasso updates beta in L2E trend filtering regression using the Lasso penalty
update_beta_TF_lasso(y, X, beta, tau, D, lambda, max_iter = 100, tol = 1e-04)update_beta_TF_lasso(y, X, beta, tau, D, lambda, max_iter = 100, tol = 1e-04)
y |
Response vector |
X |
Design matrix |
beta |
Initial vector of regression coefficients |
tau |
Initial precision estimate |
D |
The fusion matrix |
lambda |
The tuning parameter |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for beta (vector) and the number of iterations (scalar) the update step utilized
update_eta_bktk updates the precision parameter tau = e^eta for L2E regression using Newton's method
update_eta_bktk(r, eta, max_iter = 100, tol = 1e-10)update_eta_bktk(r, eta, max_iter = 100, tol = 1e-10)
r |
Vector of residual |
eta |
Initial estimate of eta |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for eta (scalar), the number of iterations (scalar) the update step utilized, the eta and objective function solution paths (vectors), and the first and second derivatives calculated via Newton's method (vectors)
update_tau_R updates the precision parameter tau
update_tau_R(r, tau, sd_y, max_iter = 100, tol = 1e-10)update_tau_R(r, tau, sd_y, max_iter = 100, tol = 1e-10)
r |
Residual vector |
tau |
Current estimate for tau |
sd_y |
Standard deviation of y |
max_iter |
Maximum number of iterations |
tol |
Relative tolerance |
Returns a list object containing the new estimate for tau (scalar) and the number of iterations (scalar) the update step utilized