ccd_grid: Central composite design (CCD) grid for nested-Laplace...

In tulpa: Templated Unified Library for Posterior Approximation in Bayesian Hierarchical Models

Central composite design (CCD) grid for nested-Laplace integration

Description

Produces a structured set of standardised hyperparameter points z \in \mathbb{R}^k for use as integration nodes in a nested Laplace approximation when k \ge 3. CCD scales much better than the full tensor expand.grid() used by the 1D and 2D backends: a CCD has 1 + 2k + 2^{k - q} points (1 centre, 2k axial, 2^{k - q} factorial), versus m^k for an m-per-axis tensor product.

Point layout (with centre+axial+factorial scaling f_0):

• 1 centre point at the origin;

• 2k axial points at \pm f_0 along each coordinate axis;

• 2^{k - q} factorial points at corners of the hypercube, scaled to lie on a sphere of radius f_0.

For k \le 6 the factorial portion is the full 2^k design. For k \ge 7 a half-fraction (q = 1) using the defining word x_1 \cdots x_k keeps the point count reasonable while preserving Resolution V.

Used by tulpa_nested_laplace() for higher-dimensional hyperparameter blocks. The standardised z-coordinates are mapped to physical hyperparameters \theta via ccd_to_theta().

Usage

ccd_grid(k, f_0 = sqrt(k))

Arguments

k	Number of hyperparameters (length of theta). Must be >= 1.
f_0	Radius of the design sphere (default \sqrt{k}). Larger f_0 spreads points further out; smaller concentrates near the centre. INLA's default scales like \sqrt{k}.

Value

A list with components:

• z: numeric matrix ⁠[n_points x k]⁠ of standardised hyperparameter coordinates.

• n_points: integer; total grid size.

• kind: character vector labelling each point as "center", "axial", or "factorial".

• f_0: the sphere radius used.

See Also

ccd_to_theta() to map z-coordinates to physical theta.

Examples

g <- ccd_grid(3)
g$n_points
table(g$kind)

tulpa documentation built on Oct. 19, 2026, 5:09 p.m.