sqr: Spline Quantile Regression by Formula

In qfa: Quantile-Frequency Analysis (QFA) of Time Series and Spline Quantile Regression (SQR)

Spline Quantile Regression by Formula

Description

This function computes the spline quantile regression solution SQR, SQR1, or SAR3 on a given set of quantile levels from a regression formula with or without user-supplied smoothing parameter. SQR represents the regression coefficients as cubic spline functions of the quantile level and employs the L1-norm of their second derivative as roughness penalty. SQR1 represents the regression coefficients as linear spline functions and employs the total variation of their first derivatives as roughness penalty. SQR3 also represents the regression coefficients as cubic spline functions but employs the L2-norm of their second derivatives as roughness penalty. SQR and SQR1 are solved as linear program (LP) using sqr.fit() and sqr1.fit(), respectively. SQR3 is solved as quadratic program (QP) using sqr3.fit().

Usage

sqr(
  formula,
  tau = seq(0.1, 0.9, 0.2),
  tau0 = tau,
  spar = NULL,
  w = rep(1, length(tau0)),
  mthreads = FALSE,
  criterion = c("AIC", "BIC", "GIC"),
  method = c("sqr", "sqr1", "sqr3"),
  type = c("dual", "primal"),
  ztol = NULL,
  solver = NULL,
  interval = NULL,
  npar = c(1, 2),
  all.knots = FALSE,
  control = list(),
  data,
  subset,
  na.action,
  model = TRUE
)

Arguments

formula	formula object, with the response on the left of a ~ operator, and the terms, separated by + operators, on the right.
tau	sequence of quantile levels for evaluation
tau0	sequence of quantile levels for fitting (min(tau0) <= tau <= max(tau0); default = tau)
spar	smoothing parameter, selected automatically by criterion if spar = NULL (default)
w	weight sequence in penalty (default = rep(1,length(tau0)))
mthreads	if FALSE (default), set RhpcBLASctl::blas_set_num_threads(1)
criterion	criterion for smoothing parameter selection ("AIC", "BIC", or "GIC")
method	'sqr' (default), 'sqr1', or 'sqr3'
type	type of QP formulation for SQR3: 'dual' (default) or 'primal'
ztol	zero-tolerance parameter to determine the model complexity (default = NULL: set internally to 1e-5 for SQR and SQR1 or 1e-4 for SQR3)
solver	'fnb' or 'sfn' for SQR and SQR1; 'piqp' or 'osqp' for SQR3 (default = NULL: set internally to 'fnb' for SQR and SQR1 or 'piqp' for SQR3)
interval	interval for spar optimization (default: c(-1.5,1.5) for SQR and SQR1 or c(1.0,2.5) for SQR3)
npar	experimental parameter (default = 1)
all.knots	TRUE or FALSE (default), same as in stats::smooth.spline()
control	list of control parameters for QP solvers 'piqp' and 'osqp'
data	data.frame object containing the observations
subset	an optional vector specifying a subset of observations to be used
na.action	a function to filter missing data (see rq() in the 'quantreg' package)
model	if TRUE then the model frame is returned (needed for calling summary subsequently)

Value

object of sqr.fit(), sqr1.fit(), or sqr3.fit()

Examples

library(quantreg)
data(engel)
engel$income <- (engel$income - mean(engel$income))/1000
tau <- seq(0.1,0.9,0.05)
fit <- rq(foodexp ~ income,tau=tau,data=engel)
fit.sqr1 <- sqr(foodexp ~ income,tau=tau,spar=0.5,method="sqr1",data=engel)
fit.sqr3 <- sqr(foodexp ~ income,tau=tau,spar=0.5,method="sqr3",data=engel)
par(mfrow=c(1,1),pty="m",lab=c(10,10,2),mar=c(4,4,2,1)+0.1,las=1)
plot(tau,fit$coef[2,],xlab="Quantile Level",ylab="Coeff1")
lines(tau,fit.sqr1$coef[2,])
lines(tau,fit.sqr3$coef[2,],col=2)

qfa documentation built on March 30, 2026, 5:07 p.m.