qreg_spline: Quantile regression with splines.

In BSquare: Bayesian Simultaneous Quantile Regression

Description

Quantile regression using spline basis functions for the quantile process.

Usage

qreg_spline(X, Y = NULL, Y_low = NULL, Y_high = NULL, 
status = NULL, knots_inter = c(0.1, 0.5, 0.9), 
Pareto = TRUE, 
varying_effect = NULL,  
tau = seq(0.05, 0.95, 0.05), 
burn = 10000, iters = 50000,  
q_low = 0.01, q_high = 0.99, 
sig_a = .1, sig_b = .1, 
mu_var = 10^2, cbf_var = 10^3, 
tail_mean = -1.2, tail_var = .4,
cbf_eps = 0.5, theta_eps = 0.5, 
tuning_theta = 1, tuning_tail = rep(1, 2), 
cred_low = 0.025, cred_high = 0.975, 
seed = 1, verbose = TRUE)

Arguments

X	Matrix of predictors with the first column consisting of all ones and all other values between -1 and 1.
Y	A vector of responses.
Y_low,Y_high	Vectors of endpoints for interval-censored values.
status	Censoring status taking values 0 if uncensored 1 if left-censored on the interval (-Inf,Y) 2 if right-censored on the interval (Y,Inf) 3 if censored on the interval (Y_low,Y_high).
knots_inter	An ordered sequence of numbers in (0,1) specifying knot locations.
Pareto	An indicator for whether Pareto or exponential tails are fit, corresponding to heavy and light tailed densities respectively. Distributions that decay slowly in the tail (e.g. t-distribution) are better
varying_effect	If varying_effect = j, then only the covariates in the first j columns of X have different effects on different quantile levels.
tau	Vector of quantile levels for output.
burn	Number of MCMC samples to discard as burn-in.
iters	Number of MCMC samples to generate after the burn-in.
q_low	The quantile level below which the lower parametric tail is fit.
q_high	The quantile level above which the upper parametric tail is fit.
sig_a	The shape hyperparameter for the prior precision of the basis functions.
sig_b	The scale hyperparameter for the prior precision of the basis functions.
mu_var	The prior variance of the prior mean of the basis functions.
cbf_var	The prior variance of the constant basis functions.
tail_mean	The prior mean for the log of the shape parameters for Pareto tails. Only used if Pareto = TRUE.
tail_var	The prior variance for the log of the shape parameters for Pareto tails. Only used if Pareto = TRUE.
cbf_eps	A parameter in [.1,.5] indicating the degree of stochastic thinning for the location basis functions. Lower is stronger.
theta_eps	A parameter in [.1,.5] indicating the degree of stochastic thinning for the other basis functions. Lower is stronger.
tuning_theta	The initial candidate variance for the parameters updated by random walk Metropolis Hastings.
tuning_tail	The initial candidate variance for the tail parameters.
cred_low	The lower limit of the posterior credible intervals.
cred_high	The upper limit of the posterior credible intervals.
seed	MCMC seed.
verbose	An indicator for outputting real-time MCMC updates.

Details

See http://www4.stat.ncsu.edu/~reich/QR/ for more detailed descriptions and examples.

Value

q	A (iters x N_tau x P) array of posterior quantile effects
q_lower	An (N_tau x P) array of lower limits of credible sets for posterior quantile effects
q_upper	An (N_tau x P) array of upper limit of credible sets for posterior quantile effects
q_mean	An (N_tau x P) array of posterior means for posterior quantile effects
theta	An (iters x M x P) array of posterior parameters
tuning_parms	An (M x P) array of candidate precisions
acc_theta	An (M x P) array of acceptance ratios of theta for the keepers
post_mu	An (N_tau x M x P) array of posterior mean hyperparameters
post_sigma2	An (N_tau x M x P) array of posterior precision hyperparameters
post_rho	An (N_tau x M x P) array of posterior correlation hyperparameters
post_xi_low	An (iters x 1) array of posterior lower tail shape parameters
post_xi_high	An (iters x 1) array of posterior upper tail shape parameters
tau	A vector specifying the quantile levels of interest (includes the thresholds)
MCMC_time	The MCMC run time.
LPML	The log pseudo marginal likelihood.
iters	The number of MCMC iterations kept.
burn	The number of MCMC iterations burned.

Note

The example is used to illustrate the method. In practice MCMC chains should be longer. Code for modeling multiple quantile functions correlated across space or time or for running on graphics processing units is available from the authors on request.

References

Smith LB, Fuentes M, Herring AH, Reich BJ (2013) Bayesian dependent quantile regression processes for birth outcomes. Submitted.

Reich BJ, Fuentes M, Dunson DB (2011) Bayesian spatial quantile regression. JASA, 106, 6-20.

See Also

qr_plot

Examples

data(airquality)
ozone=airquality[,1]
solar=airquality[,2]

#Remove missing observations
missing=is.na(ozone) | is.na(solar)
ozone=ozone[!missing]
solar=solar[!missing]

#Create design matrix.  First column must be all ones, others must be between -1 and 1
solar_std = 1.8 * (solar - min(solar))/(max(solar)-min(solar)) - 0.9
X = cbind(1,solar_std)

tau=seq(0.05,0.95,0.05)
#use longer chains in practice
fit<-qreg_spline(X,Y = ozone,iters = 1000, burn = 1000, knots_inter = c(.5))

qr_plot(fit,index=2, main = "Solar Effect")

Example output

Loading required package: quadprog
Loading required package: quantreg
Loading required package: SparseM

Attaching package: 'SparseM'

The following object is masked from 'package:base':

    backsolve

Loading required package: VGAM
Loading required package: stats4
Loading required package: splines
burn, burn, burn... 
Burn-in Finished. 
Keepers 0 % done.
[1] "Compiling results..."

BSquare documentation built on May 1, 2019, 8:21 p.m.