Koopman Reduced Order Modeling with Confidence Bounds

Equations (17)–(19) in Section 2.3.1 of the paper follow immediately from the modal/innovation decomposition and the fact that PM is the orthogonal projector onto M; see the setup and decomposition (5)–(11) and Algorithm 1 definitions (12)–(16) for D^t_ξ f(x), r(t), ρ(t)=PM r(t), and η(t)=r(t)−ρ(t) . The paper’s own justification under (17)–(19) is the same: apply PM, use η ⟂ M to get PMη=0, then take expectations and variances . The candidate solution reproduces this argument carefully, adding only minor clarifications (existence of first/second moments and that adding a deterministic constant does not change variance).