The returned matrix is a low-rank approximation of the original P-spline
basis (unless decomposition = "asIs"), that is projected into the
complement of the nullspace of the associated penalty (unless
centerBase = FALSE), i.e. for the default second order difference
penalty, the resulting basis cannot reproduce linear or constant functions
and parameterizes the "wiggly" part of the influence of x only. This
means that it very rarely makes sense to run a model with sm(x)
without also using lin(x) or u(x). The projection
improves the separability between the linear and smooth parts of the
influence of x and centers the resulting function estimates s.t
\(\sum_i f(x_i) = 0\).
sm(
x,
K = min(length(unique(x)), 20),
spline.degree = 3,
diff.ord = 2,
rankZ = 0.999,
centerBase = T,
centerx = x,
decomposition = c("ortho", "MM", "asIs"),
tol = 1e-10
)covariate
number of basis functions in the original basis (defaults to 20)
defaults to 3 for cubic B-plines
order of the difference penalty, defaults to 2 for penalizing deviations from linearity
how many eigenvectors to retain from the eigen decomposition: either a number > 3 or the proportion of the sum of eigenvalues the retained eigenvectors must represent at least. Defaults to .999.
project the basis of the penalized part into the complement of the column space of the basis of the unpenalized part? defaults to TRUE
vector of x-values used for centering (defaults to x)
use a truncated spectral decomposition of the implied
prior covariance of \(f(x)\) for a low rank representation with
orthogonal basis functions and i.i.d. coefficients ("ortho"), or use
the mixed model reparameterization for non-orthogonal basis functions and
i.i.d. coefficients ("MM") or use basis functions as they are with
i.i.d. coefficients ("asIs"). Defaults to "ortho".
count eigenvalues smaller than this as zero
a basis for a smooth function in x
Kneib, T. (2006). Mixed model based inference in structured additive regression. Dr. Hut. https://edoc.ub.uni-muenchen.de/archive/00005011/