Constructs a set of basis vectors C_0 and K_1 used to
constrain distributed lag coefficients, β,
using splines. The basis vectors depend on the radii that define
ring-shaped areas around participant locations.
Typical usage relies on calling basis application functions, like
cr (e.g. in
formulas); users should not often have to interact with
radii that define ring-shaped areas around participant locations
a function to define the type of basis. The default is to compute a cubic radial basis based on pairwise cubed absolute differences among the radii. See Details
other parameters passed to
Alternative distance functions,
.fun, may be specified, and
error checking on the user's choice of
.fun is deliberately
missing. Proper candidates for
.fun should return an
(L \times L) matrix, where L is the same as
elements of this matrix are typically non-negative.
In addition, new distance function definitions should follow the idiom:
function(x, y, ...)
if (missing(y)) y <- x
The default value of
.fun computes cubic radial distance,
which amounts to
abs(outer(x, y, "-"))^3; the computed vectors are
then transformed following Rupert, Wand, and Carroll (2003), such that
the spline can be fitted (and penalized) as a mixed-model.
An object of class
Once a basis function (δ()) and radii (r) are chosen, define the matrix, C_1[i, j] = δ(r_i, r_j), and let,
C_0 = [1, r]
C_1 = Q * R
M_1 = Q[-(1:2)]
K_1 = C_1 * M_1 * (M_1' * C_1 * M_1)^-0.5
where A[-j] denotes a matrix A with column(s) j removed. Then the (scaled) distributed lag effects are β = C_0 * α + K_1 * b, where b_l ~ N(0, σ^2_b), for l = 1, ..., L - 2.
Rupert D, Wand MP, & Carroll RJ (2003) Semiparametric Regression. New York: Cambridge University Press.
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