Source
Chorin and Lu (2015), PNAS 112(32):9804–9809, LIT-1304, read in full in NOTE-1028. The closure is the paper’s Eqs. 5–7 and 10. The results are its Tables 2–3 and Figs. 1–2.
What was actually shown
The system. dx_k/dt = x_{k−1}(x_{k+1} − x_{k−2}) − x_k + F + z_k, with z_k = (h_x/J)Σ_j y_{j,k}, and εdy_{j,k}/dt = y_{j+1,k}(y_{j−1,k} − y_{j+2,k}) − y_{j,k} + h_y x_k, cyclic indices, at ε = 0.5, K = 18, J = 20, F = 10, h_x = −1, h_y = 1. At ε = 0.5 the 360 eliminated variables are not fast. Data: the full system integrated by RK4 at step 0.001 and observed at spacing δ = 0.01 or δ = 0.05, N = 5 × 10⁵ observations each.
The closure. x^{n+1} = x^n + δR_δ(x^n) + δz^{n+1}, where R_δ is one RK4 step of the resolved equations with the unresolved term dropped, and z^{n+1} = Φ^{n+1} + ξ^{n+1} with ξ iid Gaussian. Each component’s Φ uses only its own past and that of its own x_k (Eq. 10). The orders fitted:
- δ = 0.01: (p, r, s, q) = (1, 2, 0, 1), d_x = 1, so Φ^n = μ + a₁z^{n−1} + b x^{n−1} + b′ x^{n−2} + d₁ξ^{n−1}, with a₁ = 0.9782, b = −0.1271, b′ = 0.1132, d₁ = 0.9997, μ = 0.0115, σ² = 0.0004.
- δ = 0.05: (p, r, s, q) = (1, 1, 1, 0), (d_x, d_R) = (3, 1), so Φ^n = μ + a₁z^{n−1} + cubic in x^{n−1} + c R_δ(x^{n−1}), with a₁ = 0.8879, cubic coefficients −0.0712, −0.0002, 0.0002, c = −0.0084, μ = 0.0556, σ² = 0.0284.
The parameters are maximum-likelihood (least-squares) estimates given the structure; the structure was chosen “by trial and error” so that the long-run mean and autocorrelation fit, the estimates converge with more data, and they respect the system’s symmetry.
The results, which could have come out otherwise.
| δ = 0.01 mean | sd | KS D | δ = 0.05 mean | sd | KS D | |
|---|---|---|---|---|---|---|
| Lorenz 96 | 2.4506 | 3.5230 | 2.3978 | 3.5222 | ||
| POLYAR | 2.5335 | 3.3807 | 0.0183 | 2.6031 | 2.8564 | 0.0747 |
| NARMAX | 2.4113 | 3.5270 | 0.0055 | 2.4293 | 3.5402 | 0.0049 |
In Fig. 1, at δ = 0.05, POLYAR’s pdf of x₁ is visibly bimodal where the true one is not, and its autocorrelation and x₁–x₂ cross-correlation overshoot in amplitude and drift in phase over ten time units. NARMAX’s curves lie on the true ones. At δ = 0.01 both closures are close. In Fig. 2, over 10,000 ten-unit trajectories and ensembles of 1, 5 and 20, at δ = 0.05 NARMAX has the lower RMSE and the higher anomaly correlation at every lead time and ensemble size; with 20 members the anomaly correlation falls below 0.6 at about 4 time units against about 2 for POLYAR. At δ = 0.01 the gain is small and not uniform: NARMAX is better up to a lead time of about 2.5, but beyond about 2.7 POLYAR’s 5- and 20-member RMSE curves lie slightly below NARMAX’s (about 14.3 against 14.7 at lead 5), though NARMAX keeps the higher anomaly correlation at equal ensemble size. The caption’s “NARMAX has smaller RMSE … than the POLYAR” is therefore not true at every lead time for δ = 0.01 (read from the figure at 250 dpi).
The memory used is short in the variables kept: the closure looks back one step in z, one or two in x, and one in ξ, that is 0.01 to 0.05 time units. But the autoregressive and moving-average terms make its implied dependence on the past of x non-zero at every lag, decaying geometrically.
What this does not say
- Not that a finite discrete memory suffices in general, nor in this system with a stated error. There is no bound, no study of how the error moves with the orders, and no out-of-sample rule for choosing them. The statistics used to choose the structure are among those reported.
- Not that the closure approximates the Mori–Zwanzig memory. The paper draws the link by analogy only (NOTE-1028). The fitted Φ is whatever predicts z best within the chosen family; it also absorbs the RK4 scheme’s own error at step δ, which the paper intends.
- Not a property of the system independent of the sampling. The fitted structure differs between δ = 0.01 and δ = 0.05, and the coefficients are tied to the integrator. What has finite memory is the sampled, discretized process, not the flow.
- Not a comparison with the strongest alternatives. The baseline is Wilks’s POLYAR, whose noise has no memory of x. Conditional Markov chain closures are excluded because they need exact observations of z.
- Not more than x₁ and its neighbour. The figures show one component and one cross-correlation; translation symmetry makes this representative of single-site statistics, not of spatial patterns.
Connections
THEORY-760 and THEORY-747 are the continuous-time analogue from turbulence: a finite set of memory coordinates whose length is fitted and whose error is not bounded. This account is the discrete, data-driven version of the same shape, in a chaotic system without a scale gap. The record’s linear check in NOTE-1028 shows the mechanism by which a few lags can be exact: a partially observed linear Gaussian VAR(1) is an ARMA process whose order matches the rank of its MZ memory kernels (CLAIM-389, NOTE-989), so a memory non-zero at every lag is carried exactly by finitely many coordinates. THEORY-773 is the opposite case: where the eliminated variables are fast and contracting, no memory is needed at all. Here ε = 0.5, so that theorem’s hypothesis fails and memory helps.