Shrinkage Thin Plate Regression Spline.
Usage
Equivalent to mgcv’s bs="ts" basis. Ordinary penalized smooths (such as TPRS) always leave a low-dimensional null space — typically the constant and linear terms — completely unpenalized, so even a very large smoothing parameter lambda cannot remove the term from the model entirely: the fit can be flattened to a straight line, but not to exactly zero. ShrinkageTPRS fixes this by adding a second penalty that acts specifically on the null space, using the double-penalty construction of Marra & Wood (2011). With two independently-chosen smoothing parameters, the fitting machinery can drive both the wiggly part and the null-space part of the smooth to zero simultaneously, so the whole term can be shrunk out of the model. This makes ShrinkageTPRS a good choice over plain TPRS whenever a term’s inclusion is itself uncertain and you want automatic variable selection via GCV or REML rather than a separate hypothesis test.
The basis functions are identical to TPRS (bs="tp"); only the penalty structure differs.
ShrinkageTPRS reuses the exact basis construction of TPRS: the first M columns span the polynomial null space (degree <= m - 1 monomials) and the remaining k - M columns are the truncated eigenbasis of the projected thin-plate kernel. What changes is the penalty. Instead of a single penalty matrix, penalty_matrices() returns two matrices:
\mathbf{S}_{\text{wiggle}} = \operatorname{diag}(0, \ldots, 0, \lambda_1, \ldots, \lambda_{k-M}), \qquad \mathbf{S}_{\text{null}} = \begin{pmatrix} \mathbf{I}_M & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix},
where S_wiggle is exactly the ordinary TPRS penalty (zero on the null-space block) and S_null is a projection matrix that is the identity on the null-space block and zero elsewhere. During fitting, each matrix is scaled by its own smoothing parameter, lambda_wiggle and lambda_null, and the two contributions are added together:
\lambda_{\text{wiggle}} \, \boldsymbol{\beta}^\top \mathbf{S}_{\text{wiggle}} \boldsymbol{\beta} + \lambda_{\text{null}} \, \boldsymbol{\beta}^\top \mathbf{S}_{\text{null}} \boldsymbol{\beta}.
Because S_wiggle + S_null is strictly positive definite (it has no zero eigenvalues once both penalties act together), the combined penalty null space is empty, which is why null_space_dimension() returns 0 for this basis — none of the k basis functions is exempt from penalization once both smoothing parameters are positive. If lambda_null is estimated to be very large during fitting, the null-space coefficients are effectively zeroed and the term is excluded from the model, which is the mechanism behind automatic term selection.
| Name | Description |
|---|---|
| null_space_dimension() |
Return 0: the combined double penalty leaves no unpenalized null space.
|
| penalty_matrices() | Return the two penalty matrices used for shrinkage. |
Return 0: the combined double penalty leaves no unpenalized null space.
Usage
int0 for ShrinkageTPRS, since S_wiggle + S_null is positive definite.
Return the two penalty matrices used for shrinkage.
Usage
list[NDArray][S_wiggle, S_null], each of shape (k, k). S_wiggle is the ordinary TPRS wiggliness penalty (zero on the null-space block). S_null is a projection matrix that is the identity on the M null-space columns and zero elsewhere. Each matrix is intended to be paired with its own independently-estimated smoothing parameter during fitting.