Skip to main content Module basis Copy item path Source closed_form_penalty Closed-form scalar building blocks for Riesz, Matérn, and isotropic
hybrid Duchon kernels. AnisoBasisPsiDerivatives Per-axis psi_a derivative package for anisotropic spatial terms. AnisoPenaltyCrossProvider BSplineBasisSpec 1D B-spline basis configuration. BasisBuildResult Standardized basis build result for engine-level composition. BasisOptions Options for basis generation, controlling derivative order. BasisPsiDerivativeBundle BasisPsiDerivativeResult BasisPsiSecondDerivativeResult BasisWorkspace Explicit per-run workspace for reusable basis-construction caches. CanonicalPenaltyBlock CollocationOperatorMatrices Dense Marker type for dense basis matrix output. DuchonBasisSpec Duchon-like basis configuration with explicit low-frequency null-space
control and explicit spectral power. DuchonOperatorPenaltySpec DuchonSplineBasis Duchon radial basis with triple operator regularization and explicit
low-frequency null-space order. The penalty blocks are collocation
operator Gram matrices, not the native spectral Duchon seminorm. ImplicitDesignPsiDerivative Implicit representation of ∂X/∂ψ_d that supports matrix-vector products
without materializing the full (n x p) derivative matrices. KroneckerFactoredBasis Factored tensor-product basis metadata for operator-backed downstream use. MaternBasisSpec Matérn basis configuration. MaternSplineBasis Matérn radial basis and penalties. PenaltyCandidate PenaltyInfo Sparse Marker type for sparse basis matrix output. SpatialBasisPlan Resource-aware plan for a spatial smooth (Duchon / Matérn / TPS). SplineScratch Scratch memory for B-spline evaluation to avoid allocations in tight loops. ThinPlateBasisSpec Thin-plate basis configuration. ThinPlateSplineBasis Thin-plate regression spline basis and penalty (order m=2). BSplineIdentifiability Per-smooth identifiability policy for 1D B-spline bases. BSplineKnotPlacement Internal-knot placement strategy when knots are automatically inferred. BSplineKnotSpec Which knot strategy to use for 1D B-spline bases. BasisError A comprehensive error type for all operations within the basis module. BasisEvalKind BasisFamily Basis-family selector for 1D spline evaluation. BasisMetadata Metadata returned by generic basis builders. CenterCountRequest How plan_spatial_basis CenterStrategy Spatial center selection strategy. CenterStrategyKind DuchonNullspaceOrder Duchon null-space polynomial degree.
0 keeps the constant null space (m=1), 1 keeps
[1, x_1, ..., x_d] (m=2), and Degree(k) keeps all monomials
with total degree <= k. KnotSource Specifies the source of knots for basis generation. MaternIdentifiability Per-smooth identifiability policy for Matérn kernel coefficients. MaternNu Matérn smoothness parameter nu (half-integer variants with closed forms). OperatorPenaltySpec PenaltyDropReason PenaltySource SpatialIdentifiability Per-smooth identifiability policy for spatial (TPS / Duchon) bases. SpatialStorageMode Storage mode recommended by plan_spatial_basis BasisOutput Trait for selecting basis storage format at compile time. BasisOutputFormat Trait for building basis matrices with different storage formats.
This is an implementation detail for the unified create_basis function. analyze_penalty_block analyze_penalty_block_with_op apply_linear_extension_from_first_derivative Applies first-order linear extension outside a knot-domain interval to a basis matrix
that was evaluated at clamped coordinates. apply_sum_to_zero_constraint Applies a sum-to-zero constraint to a basis matrix for model identifiability. apply_sum_to_zero_constraint_sparse applyweighted_orthogonality_constraint Reparameterizes a basis matrix so its columns are orthogonal (with optional weights)
to a supplied constraint matrix. assert_no_dense_derivative_materialization assert_spatial_centers_below_biobank_cap auto_spatial_center_strategy build_bspline_basis_1d Generic 1D B-spline builder returning design + penalty list. build_duchon_basis Generic Duchon builder returning design + penalty list. build_duchon_basis_log_kappa_aniso_derivatives Build per-axis ψ_a derivatives for anisotropic Duchon terms, including
both design-matrix and penalty derivatives. build_duchon_basis_log_kappa_derivative build_duchon_basis_log_kappa_derivatives build_duchon_basis_log_kappa_derivativeswithworkspace build_duchon_basis_log_kappa_derivativewithworkspace build_duchon_basis_log_kappasecond_derivative build_duchon_basis_log_kappasecond_derivativewithworkspace build_duchon_basiswithworkspace build_duchon_collocation_operator_matrices build_duchon_collocation_operator_matriceswithworkspace build_matern_basis Generic Matérn builder returning design + penalty list. build_matern_basis_log_kappa_aniso_derivatives Build per-axis ψ_a derivatives for anisotropic Matérn terms, including
both design-matrix and penalty derivatives. build_matern_basis_log_kappa_derivative build_matern_basis_log_kappa_derivatives build_matern_basis_log_kappa_derivativeswithworkspace build_matern_basis_log_kappa_derivativewithworkspace build_matern_basis_log_kappasecond_derivative build_matern_basis_log_kappasecond_derivativewithworkspace build_matern_basiswithworkspace build_matern_collocation_operator_matrices build_thin_plate_basis Generic thin-plate builder returning design + penalty list. build_thin_plate_basis_log_kappa_derivative build_thin_plate_basis_log_kappa_derivatives build_thin_plate_basis_log_kappa_derivativeswithworkspace build_thin_plate_basis_log_kappa_derivativewithworkspace build_thin_plate_basis_log_kappasecond_derivative build_thin_plate_basis_log_kappasecond_derivativewithworkspace build_thin_plate_basiswithworkspace center_strategy_is_auto center_strategy_kind center_strategy_num_centers center_strategy_with_num_centers closed_form_aniso_psi_derivatives_in_total_basis Closed-form anisotropic penalty S_q and its raw-η derivatives — full
d×d Hessian materialized — in the final (post-transform) basis space.
Returns (S_q, S_q_eta_a per axis, S_q_eta_a_a per axis, S_q_eta_a_b for (a, b) with a < b). All derivatives are with respect to raw η components
directly per math team Letter A §9 — no centering, no apply_raw_psi_scaling. closed_form_anisotropic_pair_block closed_form_anisotropic_pair_block_pure Pure-Duchon (κ=0) variant of closed_form_anisotropic_pair_block closed_form_matern_pair_block Closed-form pair-block penalty for the pure Matérn basis at operator order
q ∈ {0, 1, 2}. closed_form_operator_penalty_in_total_basis closed_form_operator_penalty_in_total_basis_pure Pure-Duchon (κ=0 / length_scale = None) counterpart of
closed_form_operator_penalty_in_total_basisclosed_form_anisotropic_pair_block_pureradial_derivatives_of_isotropic_duchon is defined. Self-pair (R=0)
regularization is handled inside the pair-block routine. closed_form_psi_derivatives_in_total_basis Closed-form penalty value S_q and its log-κ derivatives S_q_psi,
S_q_psi_psi in the final (post-transform) basis space, computed via
pair_block_radial_with_j_second_derivatives from the closed_form_penalty
module. Bundle’s d_kappa and d2_kappa are κ-derivatives; chain rule
converts to log-κ: ∂/∂ψ = κ·∂/∂κ,
∂²/∂ψ² = κ²·∂²/∂κ² + κ·∂/∂κ. closed_form_thin_plate_pair_block Closed-form pair-block penalty for the thin-plate spline (TPS) basis. compute_geometric_constraint_transform Compute the constraint transform Z using Greville abscissae (geometric constraints). compute_greville_abscissae Compute Greville abscissae for a B-spline basis. create_basis Unified B-spline basis generation with configurable storage, knot source, and options. create_difference_penalty_matrix Creates a penalty matrix S for a B-spline basis from a difference matrix D.
The penalty is of the form S = D' * D, penalizing the squared order-th
differences of the spline coefficients. This is the core of P-splines. create_duchon_spline_basis Creates a Duchon radial basis whose kernel is derived from
P(w) = ||w||^(2p) * (kappa^2 + ||w||^2)^s
using: create_duchon_spline_basiswithworkspace create_ispline_derivative_dense Compute the k-th derivative of an I-spline basis as a dense matrix. create_matern_spline_basiswithworkspace Creates a Matérn spline basis from data and centers. create_thin_plate_spline_basis Creates a thin-plate regression spline basis from data and knot locations. create_thin_plate_spline_basis_with_knot_count High-level TPS constructor: selects knots from data, then builds basis+penalty. create_thin_plate_spline_basis_with_knot_count_andworkspace create_thin_plate_spline_basiswithworkspace default_num_centers Adaptive default center count for spatial smooths (TPS, Duchon, Matérn). default_spatial_center_strategy duchon_max_active_operator_derivative_order Returns the maximum derivative order required by the active operator
penalties: 2 if stiffness is Active, else 1 if tension is Active, else 0.
Mass-only (or no active operator) penalties only require kernel validity
(2(p+s) > d), tension requires D1 collocation (2(p+s) > d+1), and
stiffness requires D2 collocation (2(p+s) > d+2). duchon_nullspace_dimension evaluate_bspline_basis_scalar Evaluates B-spline basis functions at a single scalar point x into a provided buffer. evaluate_bspline_derivative_scalar Evaluates B-spline basis derivatives at a single scalar point x into a provided buffer. evaluate_bspline_derivative_scalar_into Zero-allocation version: pass pre-allocated buffers for lower_basis and scratch. evaluate_bspline_fourth_derivative_scalar Evaluates B-spline fourth derivatives at a single scalar point x into a provided buffer. evaluate_bspline_fourth_derivative_scalar_into Zero-allocation version for fourth derivatives: pass pre-allocated buffers. evaluate_bsplinesecond_derivative_scalar Evaluates B-spline second derivatives at a single scalar point x into a provided buffer. evaluate_bsplinesecond_derivative_scalar_into Zero-allocation version for second derivatives: pass pre-allocated buffers. evaluate_bsplinethird_derivative_scalar Evaluates B-spline third derivatives at a single scalar point x into a provided buffer. evaluate_bsplinethird_derivative_scalar_into Zero-allocation version for third derivatives: pass pre-allocated buffers. evaluate_ispline_scalar evaluate_ispline_scalarwith_scratch Evaluates I-spline basis functions at a scalar point x into a provided buffer. evaluate_mspline_scalar Evaluates M-spline basis functions at a scalar point x into a provided buffer. filter_active_penalty_candidates filter_active_penalty_candidates_with_ops Same filtering pass as filter_active_penalty_candidatesPenaltyCandidate.op was populated. The ops vector is parallel to the
penalties vector (same length, same order). Each entry is Some(op)
when the candidate carried an operator-form handle bit-equivalent to its
dense matrix, and None for ordinary dense penalties. Consumers in the
PIRLS/REML pipeline that want operator-form matvec for PCG-against-
implicit-H call this and route through the Some entries. initial_aniso_contrasts Compute initial anisotropy contrasts η_a from knot center geometry. knot_cloud_axis_scales Compute per-axis standard deviations of knot center coordinates. minimum_duchon_power_for_operator_penalties operator_penalty_candidates_closed_form operator_penalty_candidates_closed_form_pure Pure-Duchon (κ=0 / length_scale = None) counterpart of
operator_penalty_candidates_closed_form penalty_greville_abscissae_for_knots Selects Greville abscissae for difference-penalty scaling when knot geometry is non-uniform. plan_spatial_basis Build a resource-aware plan for a spatial smooth basis. resolve_duchon_orders Resolve a fully admissible Duchon (nullspace_order, power) pair. select_centers_by_strategy select_thin_plate_knots Deterministically selects thin-plate knots via farthest-point sampling. should_use_implicit_operators_with_policy Determine whether implicit operators should be used based on problem size
and the supplied [ResourcePolicy]. should_use_sparse_basis Returns true if the B-spline basis should be built in sparse form based on density. thin_plate_polynomial_basis_dimension