, Wei-Ying Wu [aut],
Toshio Honda [aut], Ching-Kang Ing [aut]
| Title: | Adaptively Weighted Group Lasso for Semiparametric Quantile Regression Models |
|---|---|
| Description: | Implements an adaptively weighted group Lasso procedure for simultaneous variable selection and structure identification in varying coefficient quantile regression models and additive quantile regression models with ultra-high dimensional covariates. The methodology, grounded in a strong sparsity condition, establishes selection consistency under certain weight conditions. To address the challenge of tuning parameter selection in practice, a BIC-type criterion named high-dimensional information criterion (HDIC) is proposed. The Lasso procedure, guided by HDIC-determined tuning parameters, maintains selection consistency. Theoretical findings are strongly supported by simulation studies. (Toshio Honda, Ching-Kang Ing, Wei-Ying Wu, 2019, <DOI:10.3150/18-BEJ1091>). |
| Authors: | Wen-Ting Wang [aut, cre]
, Wei-Ying Wu [aut],
Toshio Honda [aut], Ching-Kang Ing [aut]
|
| Maintainer: | Wen-Ting Wang <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0.0 |
| Built: | 2024-10-13 07:25:37 UTC |
| Source: | CRAN |
Generate a set of orthogonalized B-splines using the Gram-Schmidt algorithm applied to the built-in function splines::bs().
orthogonize_bspline( knots, boundary_knots, degree, predictors = NULL, is_approx = FALSE )orthogonize_bspline( knots, boundary_knots, degree, predictors = NULL, is_approx = FALSE )
knots |
Array. The knots that define the spline. |
boundary_knots |
Array. The breakpoints that define the spline. |
degree |
Integer. The degree of the piecewise polynomial. |
predictors |
Array. The predictor variables with size p. |
is_approx |
Boolean. The default is |
A list containing:
bsplines |
Matrix of orthogonalized B-splines with dimensions |
z |
Predictors used in generation |
# Example: Generate and plot the first 5 orthogonalized B-splines p <- 30 total_knots <- 10 degree <- 3 boundaries <- c(0, 1) x <- seq(from = 0, to = 1, length.out = total_knots) knots <- x[2:(total_knots - 1)] predictors <- runif(p, min = 0, max = 1) bsplines <- orthogonize_bspline(knots, boundaries, degree, predictors) # Plot the first 5 B-splines index <- order(bsplines$z) original_par <- par(no.readonly = TRUE) par(mfrow = c(1, 5)) for (i in 1:5) plot(bsplines$z[index], bsplines$bsplines[index, i], main = i, type = "l") par(original_par)# Example: Generate and plot the first 5 orthogonalized B-splines p <- 30 total_knots <- 10 degree <- 3 boundaries <- c(0, 1) x <- seq(from = 0, to = 1, length.out = total_knots) knots <- x[2:(total_knots - 1)] predictors <- runif(p, min = 0, max = 1) bsplines <- orthogonize_bspline(knots, boundaries, degree, predictors) # Plot the first 5 B-splines index <- order(bsplines$z) original_par <- par(no.readonly = TRUE) par(mfrow = c(1, 5)) for (i in 1:5) plot(bsplines$z[index], bsplines$bsplines[index, i], main = i, type = "l") par(original_par)
qrglasso
Visualize the HDIC BIC results corresponding to hyperparameters obtained from qrglasso.
## S3 method for class 'qrglasso' plot(x, ...)## S3 method for class 'qrglasso' plot(x, ...)
x |
An object of class |
... |
Additional parameters not used directly. |
NULL
set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) # Visualize the BIC results plot(result)set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) # Visualize the BIC results plot(result)
qrglasso
Visualize the predicted coefficient functions selected by BIC.
## S3 method for class 'qrglasso.predict' plot(x, ...)## S3 method for class 'qrglasso.predict' plot(x, ...)
x |
An object of class |
... |
Additional parameters not used directly. |
NULL
set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) # Predict the top-k coefficient functions estimate <- predict(result, top_k = 2) # Display the predicted coefficient functions plot(estimate)set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) # Predict the top-k coefficient functions estimate <- predict(result, top_k = 2) # Display the predicted coefficient functions plot(estimate)
Predict the top-k coefficient functions based on a qrglasso class object.
predict( qrglasso_object, metric_type = "BIC", top_k = 5, degree = 2, boundaries = c(0, 1), is_approx = FALSE )predict( qrglasso_object, metric_type = "BIC", top_k = 5, degree = 2, boundaries = c(0, 1), is_approx = FALSE )
qrglasso_object |
A |
metric_type |
Character. Metric type for gamma selection, e.g., |
top_k |
Integer. The number of top estimated functions to predict. Default is 5. |
degree |
Integer. Degree of the piecewise polynomial. Default is 2. |
boundaries |
Array. Two boundary points for the piecewise polynomial. Default is c(0, 1). |
is_approx |
Logical. If TRUE, the size of covariate indexes will be 1e6; otherwise, 1e4. Default is FALSE. |
A list containing:
coef_functions |
Matrix. The estimated top-k coefficient functions with dimension ( |
z |
Array. Index predictors used in the generation. |
set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) estimate <- predict(result) print(dim(estimate$coef_functions))set.seed(123) n <- 100 p <- 5 L <- 5 Y <- matrix(rnorm(n), n, 1) W <- matrix(rnorm(n * p * (L - 1)), n, p * (L - 1)) # Call qrglasso with default parameters result <- qrglasso(Y = Y, W = W, p = p) estimate <- predict(result) print(dim(estimate$coef_functions))
The function qrglasso performs Adaptively Weighted Group Lasso for semiparametric quantile regression models. It estimates the coefficients of a quantile regression model with adaptively weighted group lasso regularization. The algorithm supports the use of B-spline basis functions to model the relationship between covariates and the response variable. Regularization is applied across different groups of covariates, and an adaptive weighting scheme is employed to enhance variable selection.
qrglasso( Y, W, p, omega = NULL, tau = 0.5, qn = 1, lambda = NULL, maxit = 1000, thr = 1e-04 )qrglasso( Y, W, p, omega = NULL, tau = 0.5, qn = 1, lambda = NULL, maxit = 1000, thr = 1e-04 )
Y |
A |
W |
A |
p |
A numeric indicating the number of covariates. |
omega |
A |
tau |
A numeric quantile of interest. Default value is 0.5. |
qn |
A numeric bound parameter for HDIC. Default value is 1. |
lambda |
A sequence of tuning parameters. Default value is NULL. |
maxit |
The maximum number of iterations. Default value is 1000. |
thr |
Threshold for convergence. Default value is |
A list with the following components:
gamma |
A target estimate. |
xi |
An auxiliary estimate in the ADMM algorithm. |
phi |
An auxiliary estimate in the ADMM algorithm. |
BIC |
A sequence of BIC values with respect to different lambdas. |
lambda |
A sequence of tuning parameters used in the algorithm. |
L |
The number of groups. |
omega |
A |
Wen-Ting Wang
Toshio Honda, Ching-Kang Ing, Wei-Ying Wu (2019). Adaptively weighted group Lasso for semiparametric quantile regression models. Bernoulli 225 4B.
# Example: One true non-linear covariate function # Define the function g1 g1 <- function(x) { (3 * sin(2 * pi * x) / (2 - sin(2 * pi * x))) - 0.4641016 } # Set parameters n <- 100 p <- 50 err_sd <- 0.1 ** 2 tau <- 0.7 # Generate synthetic data set.seed(1234) x <- matrix(runif(n * p, min = 0, max = 1), n, p) error_tau <- rnorm(n, sd = err_sd) - qnorm(tau, sd = err_sd) y <- g1(x[, 1]) + error_tau y <- y - mean(y) # B-spline parameters total_knots <- 5 degree <- 2 boundaries <- c(0, 1) xx <- seq(from = 0, to = 1, length.out = total_knots) knots <- xx[2:(total_knots - 1)] # Create B-spline matrix W L <- total_knots + degree - 1 bspline_results <- lapply(1:n, function(i) orthogonize_bspline(knots, boundaries, degree, x[i, ])) W <- matrix( t(sapply(bspline_results, function(result) sqrt(L) * result$bsplines[, -1])), ncol = p * (L - 1), byrow = TRUE ) # Perform quantile regression with group Lasso n_lambda <- 10 max_lambda <- 10 lambda <- c(0, exp(seq(log(max_lambda / 1e4), log(max_lambda), length = (n_lambda - 1)))) result <- qrglasso(as.matrix(y), W, p) # BIC Results plot(result) # Prediction estimate = predict(result, top_k = 1) plot(estimate)# Example: One true non-linear covariate function # Define the function g1 g1 <- function(x) { (3 * sin(2 * pi * x) / (2 - sin(2 * pi * x))) - 0.4641016 } # Set parameters n <- 100 p <- 50 err_sd <- 0.1 ** 2 tau <- 0.7 # Generate synthetic data set.seed(1234) x <- matrix(runif(n * p, min = 0, max = 1), n, p) error_tau <- rnorm(n, sd = err_sd) - qnorm(tau, sd = err_sd) y <- g1(x[, 1]) + error_tau y <- y - mean(y) # B-spline parameters total_knots <- 5 degree <- 2 boundaries <- c(0, 1) xx <- seq(from = 0, to = 1, length.out = total_knots) knots <- xx[2:(total_knots - 1)] # Create B-spline matrix W L <- total_knots + degree - 1 bspline_results <- lapply(1:n, function(i) orthogonize_bspline(knots, boundaries, degree, x[i, ])) W <- matrix( t(sapply(bspline_results, function(result) sqrt(L) * result$bsplines[, -1])), ncol = p * (L - 1), byrow = TRUE ) # Perform quantile regression with group Lasso n_lambda <- 10 max_lambda <- 10 lambda <- c(0, exp(seq(log(max_lambda / 1e4), log(max_lambda), length = (n_lambda - 1)))) result <- qrglasso(as.matrix(y), W, p) # BIC Results plot(result) # Prediction estimate = predict(result, top_k = 1) plot(estimate)