Modal Analysis of Spatiotemporal Data via Multivariate Gaussian Process Regression

The paper derives (i) the correlation C(τ)=∑k σk^2 cos(ωk τ)(W̃2k−1W̃2k−1^T+W̃2kW̃2k^T) from a real 2×2 rotation per conjugate pair with i.i.d. Gaussian modal coordinates, (ii) an LMC-with-linear-kernel construction whose stochastic expectation recovers C(τ), (iii) a spectral/SPOD connection via S(ω)=Γ Δ Γ^H with delta masses at ±ωj implying that MVGPR/LMC columns and SPOD/Koopman modes span the same subspaces, (iv) a projection/back-projection recipe P^+W̃ to recover physical modes, and (v) a mode “rank/weight” surrogate σj^2||wj||^2. These are explicitly shown in the text and equations (C(τ) in Eq. 27; KLMC expectation in Eq. 30; spectral factorization and subspace equivalence in Eqs. 40–45; algorithmic Step 5: P^+W̃; weight formula in Eq. 48) . The candidate solution proves the same statements, but with a slightly different route: it “whitens” the linear kernel coordinates to get E[k_lin,j(g0,g0)]=1 before recovering C(τ), and uses a real-valued B_k argument for the SPOD subspace. The paper’s proof uses a complex spectral density decomposition and a full-rank change of basis between Γ and the SPOD eigenbasis. Apart from these stylistic differences, the logical content agrees. Minor clarifications the paper could add: (a) conditions under which cross-correlation within conjugate pairs yields block structure without affecting subspace conclusions, (b) explicit identifiability assumptions for distinct ωk, and (c) a brief remark justifying the joint-density step in Eq. 30. Overall, both are correct.