Observed-system matvec: (S K S^T + diag(noise)) v

Takes an observed-length vector and applies the GP covariance plus a per-observation noise (nugget), all without building any big matrices.

Usage

Amv(v, obs_idx, N, space_mat, time_mat, noise_var)

Arguments

v

Numeric vector of length \(m\) (observed entries).

obs_idx

Integer indices of observed entries in the full vector.

N

Total length of the full vector.

space_mat

Spatial kernel matrix.

time_mat

Temporal kernel matrix.

noise_var

Scalar or length-\(m\) numeric nugget on the observed scale.

Value

A numeric vector of length \(m\), equal to \((S K S^\top + D)v\).

Details

Technically: for \(K = \mathrm{space}\,\otimes\,\mathrm{time}\), returns $$S\,K\,S^{\mathsf T}\,v \;+\; \operatorname{diag}(\sigma^2)\,v,$$ i.e., the observed block of the GP plus a diagonal nugget. Implemented matrix-free as kron_mv(with_nas(v, obs_idx, N), space_mat, time_mat)[obs_idx] + noise_var * v, where \(S^{\mathsf T}\) “scatters’’ into the full vector and \(\sigma^2\) denotes the per-observation noise.