Fitting function for additive quantile regression models with possible univariate
and/or bivariate nonparametric terms estimated by total variation regularization.
See summary.rqss and plot.rqss for further details on inference and
confidence bands.
rqss(formula, tau = 0.5, data = parent.frame(), weights, subset, na.action,
method = "sfn", lambda = NULL, contrasts = NULL, ztol = 1e-5, control, ...)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.
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 univariate or
bivariate. See qss for details on how to
specify these terms.
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 and it is strongly recommended that they be strictly positive, since zero weights are ambiguous.
an optional vector specifying a subset of observations to be used in the fitting. This can be a vector of indices of observations to be included, or a logical vector.
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.omit, which
deletes observations that contain one or more missing values.
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 Cholesky decomposition as described in Koenker and Ng (2003).
Option "sfnc" is used if the user specifies inequality constraints.
Option "sfn" is used if there are no inequality constraints.
Linear inequality constraints on the fitted coefficients are specified
by a matrix R and a vector r, specified inside the qss
terms, representing the constraints in the form \(Rb \ge r\).
The option method = "lasso" allows one to penalize the coefficients
of the covariates that have been entered linearly as in rq.fit.lasso;
when this is specified then there should be an additional lambda
argument specified that determines the amount of shrinkage.
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 be used as coordinate dependent shrinkage factors.
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 full-rank
matrix with as many rows as there are levels in the factor),
or else a function to compute such a matrix given the number of levels.
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
Roger Koenker
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("rq"). 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.
Although modeling with rqss typically imposes smoothing penalties on
the total variation of the first derivative, or gradient, of the fitted functions,
for univariate smoothing, it is also possible to penalize total variation of
the function itself using the option Dorder = 0 inside qss terms.
In such cases, estimated functions are piecewise constant rather than piecewise
linear. See the documentation for qss for further details.
[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.
qss
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
if(requireNamespace("interp")){
if(requireNamespace("interp")){
data(CobarOre)
fCO <- rqss(z ~ qss(cbind(x,y), lambda= .08), data=CobarOre)
plot(fCO)
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