Additive Nonparametric Terms for rqss Fitting
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, w = rep(1, length(x)))
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.rqssas 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.
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
The various pieces returned are stored in sparse matrix.csr form.
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
For the bivariate case, package tripack (and for plotting also akima) are required (automatically, by the R code).
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