Description Usage Arguments Details Value Decomposition References See Also

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 `dlm`

model
formulas); users should not often have to interact with `basis`

directly.

1 |

`x` |
radii that define ring-shaped areas around participant locations |

`center` |
if |

`scale` |
if |

`.fun` |
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 `length(x)`

;
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 `LagBasis`

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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