Additive Quantile Regression Smoothing

Fitting function for additive quantile regression models with possible univariate and/or bivariate nonparametric terms estimated by total variation regularization.

robust, regression, smooth
rqss(formula, tau = 0.5, data = parent.frame(), weights, na.action,
	method = "sfn", lambda = NULL, contrasts = NULL, ztol = 1e-5, control, ...)
a formula object, with the response on the left of a `~' operator, and terms, separated by `+' operators, on the right. The terms may include qss terms that represent additive nonparametric components. These terms can be univariat
the quantile to be estimated, this must be a number between 0 and 1,
a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument.
vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative
a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with na.fail) is to create an error if any missing values are found. A possible alternative is na.o
the algorithmic method used to compute the fit. There are currently two options. Both are implementations of the Frisch--Newton interior point method described in detail in Portnoy and Koenker(1997). Both are implemented using sparse Chol
can be either a scalar, in which case all the slope coefficients are assigned this value, or alternatively, the user can specify a vector of length equal to the number of linear covariates plus one (for the intercept) and these values will b
a list giving contrasts for some or all of the factors default = NULL appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any ful
A zero tolerance parameter used to determine the number of zero residuals in the fitted object which in turn determines the effective dimensionality of the fit.
control argument for the fitting routines (see sfn.control
Other arguments passed to fitting routines

Total variation regularization for univariate and bivariate nonparametric quantile smoothing is described in Koenker, Ng and Portnoy (1994) and Koenker and Mizera(2003) respectively. The additive model extension of this approach depends crucially on the sparse linear algebra implementation for R described in Koenker and Ng (2003). There are extractor methods logLik and AIC that is relevant to lambda selection. A more detailed description of some recent developments of these methods is available from within the package with vignette("rqss"). Since this function uses sparse versions of the interior point algorithm it may also prove to be useful for fitting linear models without qss terms when the design has a sparse structure, as for example when there is a complicated factor structure.

If the MatrixModels and Matrix packages are both loadable then the linear in parameters portion of the design matrix is made in sparse matrix form, this is helpful in large applications with many factor variables for which dense formation of the design matrix would take too much space.


  • The function returns a fitted object representing the estimated model specified in the formula. See rqss.object for further details on this object, and references to methods to look at it.


[1] Koenker, R. and S. Portnoy (1997) The Gaussian Hare and the Laplacean Tortoise: Computability of Squared-error vs Absolute Error Estimators, (with discussion). Statistical Science 12, 279--300.

[2] Koenker, R., P. Ng and S. Portnoy, (1994) Quantile Smoothing Splines; Biometrika 81, 673--680.

[3] Koenker, R. and I. Mizera, (2003) Penalized Triograms: Total Variation Regularization for Bivariate Smoothing; JRSS(B) 66, 145--163.

[4] Koenker, R. and P. Ng (2003) SparseM: A Sparse Linear Algebra Package for R, J. Stat. Software.

See Also


  • rqss
  • rqss.fit
  • [.terms
  • untangle.specials
n <- 200
x <- sort(rchisq(n,4))
z <- x + rnorm(n)
y <- log(x)+ .1*(log(x))^2 + log(x)*rnorm(n)/4 + z
plot(x, y-z)
f.N  <- rqss(y ~ qss(x, constraint= "N") + z)
f.I  <- rqss(y ~ qss(x, constraint= "I") + z)
f.CI <- rqss(y ~ qss(x, constraint= "CI") + z)

lines(x[-1], f.N $coef[1] + f.N $coef[-(1:2)])
lines(x[-1], f.I $coef[1] + f.I $coef[-(1:2)], col="blue")
lines(x[-1], f.CI$coef[1] + f.CI$coef[-(1:2)], col="red")

## A bivariate example
fCO <- rqss(z ~ qss(cbind(x,y), lambda= .08), data=CobarOre)
Documentation reproduced from package quantreg, version 4.77, License: GPL (>= 2)

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