In the formula specification of rqss nonparametric terms
are specified with qss. Both univariate and bivariate
specifications are possible, and qualitative constraints may also be specified
for the qss terms.
qss(x, constraint = "N", lambda = 1, ndum = 0, dummies = NULL,
Dorder = 1, w = rep(1, length(x)))
Fidelity component of the design matrix
List of dummy vertices
Penalty component of the design matrix
Constraint component of the design matrix
Constraint component of the rhs
The covariate determining the nonparametric component, if x is a matrix with two columns then the qss function will construct a penalized triogram term.
The smoothing parameter governing the tradeoff between fidelity and the penalty component for this term. Larger lambdas produce smoother fits. In future versions there should be an automatic mechanism for default choice of the lambdas. For now, this is the responsibility of the user.
Optional specification of qualitative constraints
on the fitted univariate qss functions, take the values: "N","I","D","V","C"
"VI","VD","CI","CD" for none, increasing, decreasing, convex,
concave, convex and increasing, etc. And for bivariate qss
components can take the values "N","V","C" for none, convex, and concave.
Note that confidence bands for constrained fits of this sort, while
available from plot.rqss as of yet lack a formal justification.
number of dummy vertices: this is only relevant for qss2 terms. In addition to vertices at the observed (x,y) points ndum dummy vertices are generated -- distributed uniformly over the rectangle given by the Cartesian product of the ranges of x and y -- observations that fall in the convex hull of the observations are retained. So the actual number of dummy vertices used is smaller than ndum. The values of these vertices are returned in the list dummies, so that they can be reused.
Order of the total variation penalty, the default of 1
implies a penalty on the first derivative of the fitted function,
a value of 0 implies total variation of the fitted function
itself will be penalized. Note that only monotonicity constraints,
"I" and "D" are allowed when Dorder = 0, and result in estimates
that are equivalent to a form of isotonic regression when lambda is
sufficiently near zero. Results in this case from the package isotone
may differ slightly when plotted due to multiple solutions so it is prudent
to evaluate the objective function for both solutions.
list of dummy vertices as generated, for example by triogram.fidelity when ndum > 0. Should be a list with x and y components. These points should lie inside the convex hull of the real xy points, but no explicit checking of this assertion is currently done.
weights not yet unimplemented
Roger Koenker
The various pieces returned are stored in sparse matrix.csr form.
See rqss for details on how they are assembled. To preserve the
sparsity of the design matrix the first column of each qss term is dropped.
This differs from the usual convention that would have forced qss terms
to have mean zero. This convention has implications for prediction that need to be
recognized. The penalty components for qss terms are based on total
variation penalization of the first derivative (and gradient, for bivariate x)
as described in the references appearing in the help for rqss.
When Dorder = 0, fitting is like the taut string methods of Davies (2014), except
for the fact that fidelity is quantilesque rather than quadratic,
and that no provision is made for automatic selection of the smoothing
parameter.
For the bivariate case, package interp (and for plotting also interp) are required (automatically, by the R code).
Davies, Laurie (2014) Data Analysis and Approximate Models, CRC Press.
rqss