Title: Adaptively Weighted Group Lasso for Semiparametric Quantile Regression Models
Version: 1.0.0
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>).
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
ByteCompile: true
BugReports: https://github.com/egpivo/QuantRegGLasso/issues
Depends: R (≥ 3.4.0)
Imports: Rcpp (≥ 1.0.12), ggplot2
LinkingTo: Rcpp, RcppArmadillo
Suggests: knitr, rmarkdown, testthat (≥ 2.1.0)
SystemRequirements: GNU make
Encoding: UTF-8
RoxygenNote: 7.2.3
URL: https://github.com/egpivo/SpatPCA
Config/testthat/edition: 3
NeedsCompilation: yes
Packaged: 2024-01-15 21:57:00 UTC; joseph
Author: Wen-Ting Wang ORCID iD [aut, cre], Wei-Ying Wu [aut], Toshio Honda [aut], Ching-Kang Ing ORCID iD [aut]
Maintainer: Wen-Ting Wang <egpivo@gmail.com>
Repository: CRAN
Date/Publication: 2024-01-16 17:20:02 UTC

Internal function: Quantile Regression with Adaptively Group Lasso without Omega

Description

Internal function: Quantile regression with adaptively group Lasso without Omega.

Usage

awgl(Y, W, lambda, tau, L, qn, zeta, zetaincre, maxit, tol)

Arguments

Y

Data matrix (n \times 1).

W

B-splines with covariates matrix with p \times L columns and n rows.

lambda

A sequence of tuning parameters.

tau

A quantile of interest.

L

The number of groups.

qn

A bound parameter for HDIC.

zeta

A step parameter.

zetaincre

An increment of each step.

maxit

The maximum number of iterations.

tol

A tolerance rate.

Value

A list of selected parameters.

Internal function: Quantile Regression with Adaptively Group Lasso with Omega

Description

Internal function: Quantile regression with adaptively group Lasso with Omega.

Usage

awgl_omega(Y, W, omega, lambda, tau, qn, zeta, zetaincre, maxit, tol)

Arguments

Y

Data matrix (n \times 1).

W

B-splines with covariates matrix with p \times L columns and n rows.

omega

Weights for group lasso.

lambda

A sequence of tuning parameters.

tau

A quantile of interest.

qn

A bound parameter for HDIC.

zeta

A step parameter.

zetaincre

An increment of each step.

maxit

The maximum number of iterations.

tol

A tolerance rate.

Value

A list of selected parameters.

Internal Function: Validate Parameters for Prediction with a qrglasso Object

Description

Internal function to validate parameters for predicting with a qrglasso class object.

Usage

check_predict_parameters(
  qrglasso_object,
  metric_type,
  top_k,
  degree,
  boundaries
)

Arguments

qrglasso_object

A qrglasso class object.

metric_type

Character. Metric type for gamma selection, e.g., BIC, BIC-log. Default is BIC.

top_k

Integer. Top K estimated functions.

degree

Integer. Degree of the piecewise polynomial.

boundaries

Array. Two boundary points.

Value

NULL.

Orthogonalized B-splines

Description

Generate a set of orthogonalized B-splines using the Gram-Schmidt algorithm applied to the built-in function splines::bs().

Usage

orthogonize_bspline(
  knots,
  boundary_knots,
  degree,
  predictors = NULL,
  is_approx = FALSE
)

Arguments

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 FALSE.

Value

A list containing:

bsplines

Matrix of orthogonalized B-splines with dimensions (p, \text{length}(knots) + \text{degree} + 1)

z

Predictors used in generation

Examples

# 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)

Display BIC Results from qrglasso

Description

Visualize the HDIC BIC results corresponding to hyperparameters obtained from qrglasso.

Usage

## S3 method for class 'qrglasso'
plot(x, ...)

Arguments

x

An object of class qrglasso for the plot method.

...

Additional parameters not used directly.

Value

NULL

See Also

qrglasso

Examples

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)

Display Predicted Coefficient Functions from qrglasso

Description

Visualize the predicted coefficient functions selected by BIC.

Usage

## S3 method for class 'qrglasso.predict'
plot(x, ...)

Arguments

x

An object of class qrglasso.predict for the plot method.

...

Additional parameters not used directly.

Value

NULL

See Also

qrglasso

Examples

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)

Internal Function: Plot BIC Results w.r.t. lambda

Description

Internal function to plot BIC results with respect to lambda using ggplot2.

Usage

plot_bic_result(data, variate)

Arguments

data

Dataframe. A dataframe containing columns lambda, bic.

variate

Character. A character representing the title.

Value

A ggplot object.

Internal Function: Plot Coefficient Function

Description

Internal function to plot coefficient functions using ggplot2.

Usage

plot_coefficient_function(data, variate)

Arguments

data

Dataframe. A dataframe containing columns z, coefficient.

variate

Character. A character representing the title.

Value

A ggplot object.

Internal Function: Plot Sequentially

Description

Internal function to plot ggplot2 objects sequentially.

Usage

plot_sequentially(objs)

Arguments

objs

List. Valid ggplot2 objects to be plotted sequentially.

Value

NULL.

Predict Top-k Coefficient Functions

Description

Predict the top-k coefficient functions based on a qrglasso class object.

Usage

predict(
  qrglasso_object,
  metric_type = "BIC",
  top_k = 5,
  degree = 2,
  boundaries = c(0, 1),
  is_approx = FALSE
)

Arguments

qrglasso_object

A qrglasso class object.

metric_type

Character. Metric type for gamma selection, e.g., BIC, BIC-log. Default is BIC.

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.

Value

A list containing:

coef_functions

Matrix. The estimated top-k coefficient functions with dimension (m \times k), where m is the size of z.

z

Array. Index predictors used in the generation.

See Also

qrglasso

Examples

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))

Adaptively Weighted Group Lasso

Description

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.

Usage

qrglasso(
  Y,
  W,
  p,
  omega = NULL,
  tau = 0.5,
  qn = 1,
  lambda = NULL,
  maxit = 1000,
  thr = 1e-04
)

Arguments

Y

A n \times 1 data matrix where n is the sample size.

W

A n \times (p \times L) B-spline matrix where L is the number of groups and p is the number of covariates.

p

A numeric indicating the number of covariates.

omega

A p \times 1 weight matrix. Default value is NULL.

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 10^{-4}.

Value

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 p \times 1 weight matrix used in the algorithm.

Author(s)

Wen-Ting Wang

References

Toshio Honda, Ching-Kang Ing, Wei-Ying Wu (2019). Adaptively weighted group Lasso for semiparametric quantile regression models. Bernoulli 225 4B.

Examples

# 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)