pqrBayes

Bayesian Penalized Quantile Regression

The quantile varying coefficient model is robust to data heterogeneity, outliers and heavy-tailed distributions in the response variable. In addition, it can flexibly model dynamic patterns of regression coefficients through nonparametric varying coefficient functions. In this package, we have implemented the Gibbs samplers of the penalized Bayesian quantile varying coefficient model with spike-and-slab priors [Zhou et al.(2023)]doi:10.1016/j.csda.2023.107808 for efficient Bayesian shrinkage estimation, variable selection and statistical inference. In particular, valid Bayesian inferences on sparse quantile varying coefficient functions can be validated on finite samples. The Markov Chain Monte Carlo (MCMC) algorithms of the proposed and alternative models can be efficiently performed by using the package.

How to install

• To install from github, run these two lines of code in R

install.packages("devtools")
devtools::install_github("cenwu/pqrBayes")

• Released versions of pqrBayes are available on CRAN , and can be installed within R via

install.packages("pqrBayes")

Example

Data Generation

Data <- function(n,p,quant){
  sig1 = matrix(0,p,p)
  diag(sig1)=1
  for (i in 1: p)
  {
  for (j in 1: p)
  {
  sig1[i,j]=0.5^abs(i-j)
  }
 }
x = mvrnorm(n,rep(0,p),sig1)
x = cbind(1,x)
error=rt(n,2) -quantile(rt(n,2),probs = quant)
u = runif(n,0.01,0.99)
gamma0 = 2+2*sin(u*2*pi)
gamma2 = -6*u*(1-u)
gamma1 = 2*exp(2*u-1)
gamma3= -4*u^3
y = gamma1*x[,2] + gamma2*x[,3]  + gamma3*x[,4] + gamma0 + error
dat = list(y=y, u=u, x=x, gamma=cbind(gamma0,gamma1,gamma2,gamma3))
return(dat)
}

95% coverage probability for the varying coefficients

n=250; p=100; rep=200;
quant = 0.5; # focus on median for Bayesian inference

CI_RBGLSS = CI_RBGL = CI_BGLSS = CI_BGL= c()

for (h in 1:rep) {
dat = Data(n,p,quant)
y = dat$y
u = dat$u
x = dat$x
g = x[,-1]
kn=2
degree=2
u.grid = (1:n)*0.005
gamma_0_grid = 2+2*sin(2*u.grid*pi)
gamma_1_grid = 2*exp(2*u.grid-1)
gamma_2_grid = -6*u.grid*(1-u.grid)
gamma_3_grid = -4*u.grid^3
coefficient = cbind(gamma_0_grid,gamma_1_grid,gamma_2_grid,gamma_3_grid)

fit = pqrBayes(g, y, u, e=NULL,quant=quant, iterations=10000, kn=2, degree=2, robust = TRUE, sparse=TRUE, hyper=NULL,debugging=FALSE)
posterior=fit$posterior
coverage = coverage(posterior,coefficient,u.grid,kn,degree)

fit1 = pqrBayes(g, y, u, e=NULL,quant=quant, iterations=10000, kn=2, degree=2, robust = TRUE, sparse=FALSE, hyper=NULL,debugging=FALSE)
posterior1=fit1$posterior
coverage1 = coverage(posterior1,coefficient,u.grid,kn,degree)

fit2 = pqrBayes(g, y, u, e=NULL,quant=quant, iterations=10000, kn=2, degree=2, robust = FALSE, sparse=TRUE, hyper=NULL,debugging=FALSE)
posterior2=fit2$posterior
coverage2 = coverage(posterior2,coefficient,u.grid,kn,degree)

fit3 = pqrBayes(g, y, u, e=NULL,quant=quant, iterations=10000, kn=2, degree=2, robust = FALSE, sparse=FALSE, hyper=NULL,debugging=FALSE)
posterior3=fit3$posterior
coverage3 = coverage(posterior3,coefficient,u.grid,kn,degree)

CI_RBGLSS = rbind(CI_RBGLSS,coverage)
CI_RBGL   = rbind(CI_RBGL,coverage1)
CI_BGLSS  = rbind(CI_BGLSS,coverage2)
CI_BGL    = rbind(CI_BGL,coverage3)
cat("iteration = ", h, "\n")

}
# the intercept gamma_0 has not been regularized
cp_RBGLSS =  colMeans(CI_RBGLSS) # 95% coverage probabilities for the varying coeffcients under the default setting
cp_BGLSS  =  colMeans(CI_BGLSS)
cp_RBGL   =  colMeans(CI_RBGL)
cp_BGL    =  colMeans(CI_BGL)

News

pqrBayes 1.0.4 [2025-01-24]

• Added functions to compute the coverage probability of varying coefficients.
• Added examples to obtain the inference results reported in the reference.

pqrBayes 1.0.3 [2024-12-20]

• Fixed the issue of no output in the VCselect() function and added examples.
• Updated the list of pqrBayes output objects.
• Added non-robust methods and corresponding examples.
• Updated the documentation.

Methods

This package provides implementation for methods proposed in

• Zhou, F., Ren, J., Ma, S. and Wu, C. (2023). The Bayesian Regularized Quantile Varying Coefficient Model. {, 107808}