Minimises \(\sum_i (y_i - f(x_i))^2 + \lambda J(f)\) with the bending
energy J; the solution \(f(s) = a_0 + a_1 u + a_2 v + \sum_i w_i
\phi(|s - x_i|)\), \(\phi(r) = r^2 \log(r) / (8\pi)\), solves the
bordered system with \(K + \lambda I\) and \(P = (1, x)\) (Duchon
1977; Wahba 1990). lam = 0 interpolates. Equal to
fields::Tps with the same lambda and
scale.type = "unscaled".
Usage
ThinPlateSpline(x, y, lam = 0, newdata = NULL)
Arguments
- x
Two-column matrix of knots.
- y
Values.
- lam
Smoothing parameter.
- newdata
Optional two-column matrix of prediction locations.
Value
List with w, a, fitted, residuals,
df (trace of the hat matrix), predicted, lam.
References
Duchon, J. (1977). Splines minimizing rotation-invariant
semi-norms in Sobolev spaces. Constructive Theory of Functions of
Several Variables, 85-100. Springer, Berlin.
Wahba, G. (1990). Spline Models for Observational Data. SIAM,
Philadelphia.
Examples
P <- rbind(c(0, 0), c(1, 0), c(0, 1), c(1, 1), c(.5, .5))
ThinPlateSpline(P, c(0, 1, 1, 2, 1.5), newdata = rbind(c(.5, .5), c(.25, .75)))$predicted
#> [1] 1.500000 1.294285
