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REVIEW 3 major objections 5 minor 47 references

Shadowing the rotating annulus. Part I: Measuring candidate trajectory shadowing times

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A residual-vector consistency test measures how long high-dimensional model trajectories shadow noisy observations, and the measured times obey a simple logarithmic law.

desk verdict A practical shadowing-time method for high-dimensional models, honestly demonstrated on the rotating annulus, with a load-bearing IID assumption that needs a correlation check. read the letter →

arxiv 1909.04488 v1 pith:ZPFZ7IC7 submitted 2019-09-04 physics.data-an math.DSnlin.CDphysics.ao-phphysics.flu-dyn

Roland M. B. Young , Roman Binter , Falk Niehörster This is my paper · ORCID
classification physics.data-anmath.DSnlin.CDphysics.ao-phphysics.flu-dyn
keywords shadowingtrajectoriesrotatingannulusperfectmodelscenariocandidatetimeorderstatisticschaoticdynamicshigh-dimensionalforecastingMORALS
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to make shadowing—the time a model trajectory stays consistent with a sequence of noisy observations—measurable in high-dimensional models, and to verify that the measurement behaves as physical intuition demands. The authors replace the earlier sequence-based test, which samples the noise distribution and collapses residuals to distances, with a "state" method that checks each residual vector directly: at every verification time, the median and 90th percentile of the scaled residual field must lie inside analytically computed confidence intervals for the same percentiles of the observational noise. Applied to a 24,192-variable rotating-annulus model in the perfect model scenario, the method yields candidate shadowing times that match visual divergence and a simple distance threshold, and that decrease as the initial distance from truth increases. The quantitative payoff is an empirical scaling law, $\tau_S \approx 250(1-\log_{10}\delta)$ seconds, which predicted a 2000 s shadowing time for $\delta = 10^{-7}$ that the holdout experiment confirmed. A sympathetic reader would care because this supplies a benchmark for judging candidate trajectories in weather- and climate-scale models, where shadowing questions have been well posed but rarely answered.

What carries the argument

Load-bearing machinery is the order-statistic identity that makes the test computational: for IID components, the CDF of the $r$-th order statistic of a sample of size $N$ is the incomplete $\beta$ function $I_{P(x)}(r, N-r+1)$, so confidence intervals for the median and 90th percentile of the observational noise can be computed by inverting it instead of drawing $M$ Monte Carlo samples. This identity is what lifts the method from low-dimensional toy systems to $N \approx 24{,}000$. The second piece of machinery is the Šidák bound, $p < 1-(1-R/E)^{1/2n}$, which sets the per-test significance level so that fewer than $R$ of $E$ candidates suffer a false rejection over $n$ verification times; at the candidate counts used here it forces $p \approx 10^{-5}$–$10^{-6}$. The sanity check is a simple threshold on the Euclidean distance $D(t)$, using the fact that the expected distance between truth and observations is $\sigma\sqrt{N}$, so divergence is declared when $D(t) > m\sigma\sqrt{N}$; the paper uses $m=2$ and finds $\tau_D$ insensitive to $m$.

What would settle it

Run the state method on perfect initial conditions ($\delta = 0$) with observational noise that is spatially correlated across grid points instead of independent, and compare the observed fraction of false rejections with the Šidák prediction of Eq. (13); a systematic mismatch would show that the IID assumption is load-bearing and would require recalibrating the significance levels and shadowing times.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that candidate shadowing times in a high-dimensional chaotic model can be obtained by testing, time step by time step, whether the distribution of the residual state vector is consistent with the known observational error distribution, and that the resulting times are reasonable. The procedure uses the perfect model scenario: MORALS, discretized on a staggered grid with $N = 24{,}192$ independent variables, is both system and model; observations are the true state plus independent Gaussian noise scaled by the 99\% natural variability range at each grid point; candidates begin at fixed scaled distances $\delta$ from truth. For each candidate, the shadowing time $\tau_S$ is the longest prefix over which both the median and the 90th percentile of the scaled residual vector fall within confidence intervals obtained from the incomplete $\beta$ function. The measured distributions agree with the time of visual divergence and with the distance-based divergence time $\tau_D$, are largely insensitive to position on the attractor and to the significance level once Type I errors are controlled, and follow $\tau_S \approx 250(1-\log_{10}\delta)$ across eight perturbation sizes; a prediction at $\delta = 10^{-7}$ gave a median shadowing time near 2000 s as expected.

Load-bearing premise

The load-bearing premise is that the components of the residual vector are independent and identically distributed draws from the observational error distribution; if spatial correlations among grid points or inhomogeneous scaling break that assumption, the analytic confidence intervals and every reported shadowing time would shift systematically.

Editorial extensions

If this is right

  • In the perfect model scenario, candidate shadowing times measured from random-direction perturbations are lower bounds on the true shadowing time; candidates placed on the attractor or its stable manifold can shadow longer, but such initial conditions occur with probability zero under the paper's perturbation scheme.
  • The empirical law $\tau_S \approx 250(1-\log_{10}\delta)$ gives a benchmark for future shadowing searches: once a candidate's distance from truth is known, its expected shadowing time is fixed, and any candidate that beats this expectation is doing genuine work.
  • With an imperfect model or real observations the same measurement should produce upper bounds rather than lower bounds, because model and system attractors are typically disjoint, so model error—not initial-condition uncertainty—will limit shadowing.
  • The method is deliberately inapplicable to low-dimensional systems ($N = O(100)$ or below), since the 90th percentile of a three- or ten-component residual vector carries no statistical meaning; this defines the regime of applicability of the technique.
  • Type I errors accumulate across repeated verification times, so significance levels must be set by the Šidák bound; at practical candidate counts this forces $p \approx 10^{-5}$–$10^{-6}$ rather than conventional 0.05.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the logarithmic scaling survives in other chaotic parameter regimes of the annulus, it offers a cheap predictability diagnostic: the slope of 250 s per decade of initial distance could be compared across flow regimes and model resolutions without running full shadowing searches.
  • A testable extension is to replace the IID noise model with spatially correlated observational error; the incomplete-beta calibration would then be approximate, and the size of the resulting bias in $\tau_S$ would quantify how much the method depends on that assumption.
  • The same per-state percentile consistency test could be applied to operational ensemble forecasts, declaring an ensemble member to have stopped shadowing when its median and 90th-percentile residual leave the noise-model intervals; this would give a state-vector-level version of useful forecast duration.
  • Because the test is applied per time step, small persistent outliers can go undetected, the paper's own caveat in Sect. 2.4.2; combining the state method with a whole-trajectory cumulative test would exploit both sensitivities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a "state" method for measuring candidate trajectory shadowing times in high-dimensional models, building on the sequence method of Smith et al. (2010). At each verification time, the residual vector between model trajectory and observations is tested for consistency with the observational error distribution using the median and 90th percentile of the residual components; the sampling distributions of these order statistics are obtained analytically from the incomplete beta function under an IID assumption, avoiding Monte Carlo sampling. The method is applied to MORALS, a rotating-annulus model with N=24,192 state variables, in the perfect model scenario, with observations generated by adding IID Gaussian noise scaled by each grid point's natural variability. The authors calibrate the significance level using null trajectories (Type I error control), measure shadowing times for candidate trajectories started at eight distances δ from truth, and report an empirical scaling law τS ≈ 250(1−log10δ), which successfully predicts a shadowing time near 2000 s for δ=10^-7. The candidate shadowing times are compared with a simple distance metric τD and with visual divergence of the trajectories. The paper is explicitly a record of work submitted in 2010 and not accepted; the current version is posted to arXiv for citation and follow-up.

Significance. If the state method is valid, it provides a computationally cheap way to assign candidate shadowing times in intermediate- and high-dimensional systems, and the empirical scaling law would be a useful benchmark for future shadowing studies in the annulus and similar systems. The paper has genuine strengths: the analytic order-statistic result removes a sampling bottleneck; the Type I error calibration in Fig. 4 is careful and is checked against actual noise realizations; the out-of-sample prediction at δ=10^-7 is a real falsifiable check; and the agreement with an independent distance metric and visual divergence provides external corroboration. The central limitation is that the order-statistic formula in Eq. (3) assumes the N residual components are IID. That assumption is exact for a perfect initial condition, but it is violated for the diverging candidates the method is intended to measure, because residual fields are spatially coherent. Unless this is shown to be quantitatively harmless, the reported shadowing times and the scaling law Eq. (16) may be systematically too short.

major comments (3)
  1. [Section 2.4.1, Eq. (3)] The IID assumption is load-bearing in the derivation of F(r)(x)=IP(x)(r,N−r+1). Under the null hypothesis δ=0, the residual vector e_t∘r^{-1} is IID observational noise by construction, so the Type I calibration in Fig. 4 is meaningful. For a diverging candidate, however, e_t∘r^{-1} develops large-scale, spatially coherent structure (visible in Figs. 5b and 9), so the effective number of independent components is far smaller than N=24,192. The analytic confidence intervals for the median and 90th percentile are then much narrower than the true sampling distributions, causing premature rejection. The shadowing times in Fig. 7 and the scaling law Eq. (16) therefore reflect not only the marginal distribution of the residuals but also their spatial correlation structure. I request a block-bootstrap or spatial-decorrelation estimate of the effective sample size, with the order-statistic test repeated using that effective N, and a comparison of the resulting τS values with those reported.
  2. [Section 4.2, Fig. 7] The insensitivity of τS to the significance level p is presented as evidence that the method is robust, but this does not address the IID concern. For p between 0.1 and 0.001, Fig. 4 shows that Type I errors dominate for a perfect trajectory, so near-constancy of τS in Fig. 7 is expected whenever divergence is fast relative to the critical-value differences in that range. The key check is whether rejection occurs because the residual marginal distribution has moved by more than the correlation-corrected sampling error, rather than simply because the IID critical intervals are too tight. Without that check, the agreement between τS and τD in Sect. 4.3 supports the qualitative claim that the method detects divergence, but not the specific numerical values entering Eq. (16).
  3. [Section 4.1, Eq. (14)] The Šidák correction r=1−(1−p)^{2n} assumes independence of successive verification times and of the two percentile statistics. This assumption is valid for the null candidate, where additive noise is IID across components and over time, but it is not valid for the diverging candidates used to estimate τS, because the test statistics are serially and spatially correlated. The paper should state explicitly that Eq. (14) calibrates the null case only, and should provide a sensitivity analysis (for example, recomputing τS with the correlation-corrected effective N) to show that the conclusions are not an artifact of miscalibrated critical values.
minor comments (5)
  1. [Section 3.4] The shadowing time definition is ambiguous if the test fails at one time and passes at a later time: the text says "the longest time for which the null hypothesis is retained" and also "the final t at which the test is passed, counting upwards from zero." Please clarify the stopping rule used in the implementation.
  2. [Section 2.3] There is a typo in the phrase "use the the Euclidean length" in the description of the Smith et al. (2010) method.
  3. [Section 4.2] The notation "e.10^-2" and "e.10^-1" is nonstandard and is never defined; these presumably mean e·10^-2 and e·10^-1, but the notation should be introduced explicitly.
  4. [Section 4.2, Eq. (16)] The empirical fit τS≈250(1−log10δ) is presented without uncertainty estimates or goodness-of-fit measures; given that Eq. (17) is used for prediction, reporting the fitted slope and its uncertainty would strengthen the claim.
  5. [Figure 9] The caption correctly notes that the order of grid points along the x-axis is unimportant, but the axis is otherwise unlabeled; adding a label such as "grid point index" would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the δ=10⁻⁷ shadowing-time prediction is out-of-sample and independently validated by the τD metric and visual checks.

full rationale

The derivation chain is self-contained. The state method's null distribution comes from the classical order-statistics result (David 1981; NIST 2010), not from the paper's own fitted values, and the significance level is calibrated against the analytic Type I error expression Eq. (13) with null runs in Fig. 4. The empirical scaling law Eq. (16) is fitted to shadowing times from eight perturbation sets, and the claimed test at δ=10⁻⁷ uses a newly generated candidate set not included in that fit; the agreement with the predicted ~2000 s is therefore an out-of-sample check rather than a recirculation of inputs. The paper's sanity checks (τD metric, visual inspection, repeated attractor positions) are external to the state method and do not presuppose its output. Citation of Smith et al. (2010) is a starting point for the sequence method and is explicitly modified, with no uniqueness claim or load-bearing self-citation imported. The possible violation of the IID assumption by spatially correlated residual fields (Sect. 2.4.1) is a statistical robustness concern, not a circularity, because the paper states the assumption explicitly and does not hide a fitted parameter as a prediction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claimed contribution rests primarily on the IID assumption for residual vectors, which is likely violated by spatial correlations, and on several hand-chosen parameters including σ, m, significance levels, and the natural-variability percentile range. The empirical scaling law is fit to the data. No new physical entities are introduced.

free parameters (6)
  • Observational noise scale σ = 0.1
    Fixed fraction of natural variability r used to generate observational noise (Sect. 3.2); chosen ad hoc and held constant throughout.
  • Distance metric threshold m = 2
    Hand-chosen threshold in the τD definition (Sect. 3.4); the authors state it is based on test data and that τD is not very sensitive to m, but it remains a free choice.
  • Scaling law intercept = 250 s
    Coefficient in Eq. 16, fitted to the eight candidate sets to give τS ≈ 250(1 − log10 δ).
  • Scaling law slope magnitude = 250 s per decade
    Coefficient in Eq. 16, equivalent to the intercept in the logarithmic fit; fitted to the eight candidate sets.
  • Significance levels p = 1e-5 for 64-candidate sets, 1e-6 for 256-candidate set
    Chosen via Eq. 14 with R < 1 expected Type I error; a conservative but arbitrary cap on false rejections.
  • Natural variability percentile range = 0.5% to 99.5%
    Choice for defining r in Eq. 5, which scales all distances and noise; affects the entire scaled-space analysis.
assumptions (6)
  • domain assumption The components of the residual vector e_t are IID samples from a common distribution.
    Invoked in Sect. 2.4.1 to apply the order-statistic distribution from David (1981); never empirically validated for MORALS state vectors, which are spatially correlated and have heterogeneous natural variability.
  • domain assumption Observational errors are Gaussian, independent, and identically distributed in scaled space.
    Eq. 6 defines ~e ~ σ r N(0,1); this is the noise model underlying all hypothesis tests.
  • domain assumption In the perfect model scenario, the numerical model output at the given timestep is the true system, so ε-shadowing issues are avoided.
    Sect. 2.1; this is the central modeling assumption that allows the system and model to be identified.
  • domain assumption The 1500 s spin-up and 1500 s sample adequately characterize the natural variability range r.
    Sect. 3.2, Eq. 5; the 99% range is estimated from a single 1500 s time series, with no convergence check.
  • domain assumption The flow is chaotic with a positive largest Lyapunov exponent.
    Sect. 3.1; the Lyapunov exponent 0.0061±0.0025 bits/s is estimated from a 100000 s time series using the procedure of Young and Read (2008).
  • standard math Standard order-statistics formulas (David 1981) and the inverse incomplete beta function (NIST 2010) provide exact sampling distributions.
    Used in Sect. 2.4.1 to avoid Monte Carlo sampling; they are unproved background results accepted as correct.

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Cite this review

Pith. "Pith review of Shadowing the rotating annulus. Part I: Measuring candidate trajectory shadowing times." pith.science (2026). https://pith.science/paper/ZPFZ7IC7

@misc{pith2026190904488,
  author       = {Pith},
  title        = {Pith review of: Shadowing the rotating annulus. Part I: Measuring candidate trajectory shadowing times},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPFZ7IC7}},
  note         = {Machine review of arXiv:1909.04488}
}
read the original abstract

An intuitively necessary requirement of models used to provide forecasts of a system's future is the existence of shadowing trajectories that are consistent with past observations of the system: given a system-model pair, do model trajectories exist that stay reasonably close to a sequence of observations of the system? Techniques for finding such trajectories are well-understood in low-dimensional systems, but there is significant interest in their application to high-dimensional weather and climate models. We build on work by Smith et al. [2010, Phys. Lett. A, 374, 2618-2623] and develop a method for measuring the time that individual "candidate" trajectories of high-dimensional models shadow observations, using a model of the thermally-driven rotating annulus in the perfect model scenario. Models of the annulus are intermediate in complexity between low-dimensional systems and global atmospheric models. We demonstrate our method by measuring shadowing times against artificially-generated observations for candidate trajectories beginning a fixed distance from truth in one of the annulus' chaotic flow regimes. The distribution of candidate shadowing times we calculated using our method corresponds closely to (1) the range of times over which the trajectories visually diverge from the observations and (2) the divergence time using a simple metric based on the distance between model trajectory and observations. An empirical relationship between the expected candidate shadowing times and the initial distance from truth confirms that the method behaves reasonably as parameters are varied.

Figures

Figures reproduced from arXiv: 1909.04488 by the authors.

Figure 1
Figure 1. Schematic of the thermally-driven rotating annulus experiment. The inner cylinder is at radius a = 2.5 cm, the outer cylinder is at radius b = 8.0 cm, and the cylinders have a depth of d = 14.0 cm. The fluid between the cylinders is a mixture of glycerol and water, 17%/83% by volume. The cylinders are maintained at constant temperatures Ta = 18 ◦C and Tb = 22 ◦C respectively, giving a constant temperature difference… view at source ↗
Figure 2
Figure 2. In that figure we draw 500 samples of size 1000 from a known noise distribution [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. Schematic of the general order statistics approach to calculating a candidate’s shadowing time. The top panel shows a histogram (grey) of the values generated from a sample of size 1000 from N(0, 1). The black histograms show, for 500 such samples, the distributions of the 50th and 90th percentiles of each sample. These percentile distributions are shown in the middle (50th) and lower (90th) panels. On the lower pan… view at source ↗
Figures from the paper (7 more)
Figure 3
Figure 3. Figure 3: Temperature output from a rotating annulus simulation using the model described in the text with the setup in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 4
Figure 4. Figure 4: Consequences of the Type I error property of the “state” method for low significance levels. For each significance level (indicated by its p-value) the solid line shows the expected fraction of candidates (from Eq. 13) that fail to shadow because of a Type I error as a…
Figure 5
Figure 5. Figure 5: Example annulus candidate trajectory showing the temperature time series and residual error at a single grid point: R = 4.81 cm, θ = 6.19 rad, z = 5.61 cm. This particular candidate trajectory begins a distance δ = 0.1 from the truth in the scaled space (Eq. 10). 12 […
Figure 6
Figure 6. Figure 6: Distance between 256 candidate trajectories and one realisation of the observational noise (using Eq. 15) for candidates begun a distance δ = 0.1 from truth in the scaled state space introduced in Eq. (7). The vertical scale shows the Euclidean distance between candida…
Figure 7
Figure 7. Figure 7: Distributions of candidate shadowing times measured using the “state” method for the 256 candidate trajectories in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Distributions of candidate shadowing times for eight initial distances from truth δ, using significance level p = 10−5 for all but δ = 0.1, which uses p = 10−6 . The boxes are constructed in the same way as in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Contour plot of the unsigned distance |e[t] ◦ r −1 | between candidate trajectory and observations for each grid point in a candidate with δ = 0.1 (same run as [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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    " write newline "" before.all 'output.state := FUNCTION if.digit duplicate "0" = swap duplicate "1" = swap duplicate "2" = swap duplicate "3" = swap duplicate "4" = swap duplicate "5" = swap duplicate "6" = swap duplicate "7" = swap duplicate "8" = swap "9" = or or or or or or...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.