The infinite-dimensional penalised problem has a finite-dimensional
minimiser: a radial basis expansion in r^2 log r at the data points plus a
linear null-space term, so it is solved exactly by linear algebra. The
penalty annihilates linear functions, so as lambda grows the fit tends to
the least-squares PLANE, not to a constant.
Usage
morie_esl_thin_plate_spline(X, y, lambda_ = 1, newdata = NULL)
Arguments
- X
Locations, n by 2.
- y
Observed values.
- lambda_
Smoothing parameter, non-negative.
- newdata
Locations to predict at; defaults to X.
Value
List with fitted, delta, beta, residuals, edf, rss, gcv.
Details
Solved on an orthonormal basis of null(A') rather than through the
bordered saddle-point system: that matrix has condition number of order
lambda, and a relative-cutoff least-squares solve truncates the O(1) linear
block outright, so the fit drifts instead of converging.
References
Duchon, J. (1977). Splines minimizing rotation-invariant
semi-norms in Sobolev spaces. Springer.
Hastie, T., et al. (2009). ESL (2nd ed.), Sec 5.7. Springer.
Examples
set.seed(1)
P <- matrix(runif(50, -1, 1), ncol = 2)
z <- sin(2 * P[, 1]) + P[, 2]^2
max(abs(morie_esl_thin_plate_spline(P, z, lambda_ = 0)$residuals)) < 1e-6
#> [1] TRUE
