Per-cell constrained inverse-precision sub-block for a joint fit

Runs the same per-outer-grid-cell selected-inversion the joint nested-Laplace driver's own fixed-effect retention uses internally (extract_inner_vcov_block_cell(), see the note above .joint_attach_grid_fixed()), for a caller-chosen index set. A consumer package computing its own standard-error correction on a subset of a joint fit's latent coordinates – a different idx than whatever the fit itself retained on $cov_block_per_grid – reruns the extraction through this door instead of reimplementing the sparse selected inversion.

Usage

tulpa_joint_inner_vcov_blocks(
  Q_p_per_grid,
  Q_i_per_grid,
  Q_x_per_grid,
  n_x,
  idx,
  n_dense,
  A_cols_list,
  field_marginal = TRUE,
  n_threads = 1L
)

Arguments

Q_p_per_grid, Q_i_per_grid, Q_x_per_grid

Per-outer-grid-cell CSC storage of the latent precision Q (fit$Q_csc_p_per_grid / _i_per_grid / _x_per_grid), one list element per cell.

n_x

Integer, the latent dimension Q is built over (fit$Q_csc_n).

idx

Integer vector (1-based) of latent coordinates to extract the sub-block for.

n_dense

Integer, the number of leading idx entries that are dense fixed effects rather than field coordinates (see Details).

A_cols_list

A list of integer vectors, one per sum-to-zero constraint group, each the (1-based) latent coordinates that group averages to zero.

field_marginal

Logical; if TRUE (default) and n_dense < length(idx), only the field coordinates' marginal variances are computed (a single Takahashi pass) rather than their full dense sub-block.

n_threads

Integer, grid cells processed concurrently.

Value

A list of length n_grid (one per outer-grid cell), each element either NULL (the cell stored no Q, or its sparse Cholesky failed) or a dense length(idx) x length(idx) matrix: the constrained covariance sub-block for idx, with the sum-to-zero constraints in A_cols_list conditioned out by the same kriging correction the fit's own posterior draws use.

Details

When a field is present (n_dense < length(idx)), the extraction takes a cheap selected-inversion path: the dense (fixed-effect) block and the fixed-effect x field cross terms are exact; the field x field block is either its marginal diagonal (field_marginal = TRUE) or left at zero (FALSE) – never the full field x field covariance.

Examples

# Two outer-grid cells of a latent precision over two fixed effects and a
# ten-node random-walk field (sum-to-zero), each stored as the lower-triangle
# CSC a joint fit keeps per cell.
make_Q <- function(tau) {
  Q <- matrix(0, 12, 12)
  Q[1:2, 1:2] <- diag(2, 2) + 0.1
  Q[3:12, 3:12] <- tau * crossprod(diff(diag(10))) + diag(0.3, 10)
  Q[1, 3:12] <- Q[3:12, 1] <- 0.1
  L <- Matrix::tril(methods::as(methods::as(Q, "generalMatrix"),
                                "CsparseMatrix"))
  list(p = L@p, i = L@i, x = L@x)
}
cells <- list(make_Q(1), make_Q(4))
blocks <- tulpa_joint_inner_vcov_blocks(
  lapply(cells, `[[`, "p"), lapply(cells, `[[`, "i"), lapply(cells, `[[`, "x"),
  n_x = 12L, idx = 1:12, n_dense = 2L, A_cols_list = list(3:12))
blocks[[1]][1:2, 1:2]        # fixed-effect covariance in the first cell
diag(blocks[[2]])[3:12]      # field marginal variances in the second