{ "schemaVersion": "1.6", "slug": "canonical-covariance-localisation-failure", "title": "When canonical covariance localisation fails", "shortTitle": "Where you localise matters: canonical coordinates are not always the best basis", "url": "https://evidencepress.org/releases/canonical-covariance-localisation-failure/", "oneLine": "Snyder and Hakim's conjecture that canonical coordinates are the best basis for ensemble covariance localisation fails when error is measured in physical variables, at every ensemble size for a single observation, but holds at leading order when error is measured relative to the forecast uncertainty; in their own test case symmetric whitening removes about half the canonical basis's excess error.", "abstract": "Exact finite-ensemble counterexamples, a sharp leading-order criterion, and the Snyder–Hakim test case. Ensemble Kalman filters estimate forecast uncertainty from a small ensemble and localise the noisy sample covariance, discarding correlations that are probably noise. Snyder and Hakim (2022) conjectured that canonical coordinates (whiten the prior and observation errors, then rotate to the singular vectors of the whitened observation operator) are the best basis in which to localise. For one analysis step with an independent Gaussian ensemble, a known (oracle) prior covariance used to build every basis and deterministic tapers, the answer depends on the loss. Under physical squared error, as in Snyder and Hakim's experiments, a single observation that is not an eigenvector direction of the prior covariance makes the conjecture fail at every ensemble size N ≥ 2: a small rotation followed by complete localisation beats every canonical taper with unit observed diagonal, and with the same deterministic variance rescaling it beats every canonical taper with any observed-variance factor. Two two-dimensional examples are certified in exact rational arithmetic; in one the canonical basis is a strict local maximum of the risk. With several observations, coupling between an observed and an unobserved canonical direction makes canonical complete localisation non-stationary at every N. A uniform large-ensemble expansion gives a leading objective F(U) that ranks bases, and a sharp criterion says exactly when canonical coordinates minimise it; under prior-normalised (Mahalanobis) loss they always do, and for a scalar observation they are a strict local minimum at finite N provided a scalar inequality holds that is supported numerically but unproved. In a reconstruction of Snyder and Hakim's 200-variable test case, canonical coordinates leave about 42–54% of the avoidable excess error, and symmetric (ZCA) whitening does almost as well as the best basis found, equivalent to roughly 1.7–2 times more ensemble members; bases estimated from small pilot samples keep much less of the gain. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-03", "dateModified": "2026-11-03", "version": "0.2.0-candidate", "doi": "10.5281/zenodo.23941881", "doiUrl": "https://doi.org/10.5281/zenodo.23941881", "conceptDoi": "10.5281/zenodo.23926883", "pdfUrl": "https://github.com/ipitchford/canonical-covariance-localisation-failure/releases/download/v0.2.0-candidate/canonical-covariance-localisation-failure-0.2.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23941881", "repoUrl": "https://github.com/ipitchford/canonical-covariance-localisation-failure", "releaseUrl": "https://github.com/ipitchford/canonical-covariance-localisation-failure/releases/tag/v0.2.0-candidate", "markdownUrl": "https://evidencepress.org/releases/canonical-covariance-localisation-failure/index.md", "bibtexUrl": "https://evidencepress.org/releases/canonical-covariance-localisation-failure/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/canonical-covariance-localisation-failure.mp3", "imageUrl": "https://evidencepress.org/assets/og/canonical-covariance-localisation-failure.png?v=447b7cf208", "coverArtUrl": "https://evidencepress.org/assets/art/canonical-covariance-localisation-failure.svg?v=22f93b1727", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/canonical-covariance-localisation-failure.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/canonical-covariance-localisation-failure.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematics candidate. The main results have written proofs; Conjecture 5.4 (two observed directions) and Open problem 8.2 (the prior-normalised finite-N inequality) are supported numerically only; the testbed is a numerical reconstruction, not a replication. Internal AI-assisted review in five roles, a separate-model adversarial review, one revision round with extensions, a proof audit and separate-model audit of the new proofs, and a re-review. No human peer review, independent reproduction or formal verification has been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23941881", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "All four scripts were rerun from a fresh copy of the package with Python -I -B and with -O (CPython 3.11.14, NumPy 2.3.5, SciPy 1.17.0, Matplotlib 3.10.8, mpmath 1.4.1); all checks passed, recorded values matched the shipped results apart from timings and version strings, and figures were byte-identical. GitHub CI on Linux reruns all four in both modes. Version 0.2.0-candidate changes no script, result or figure (byte-identical to 0.1.0-candidate), so that replay is reused; two scripts were rerun under -O from the frozen 0.2.0 archive." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Reviewers recomputed several constants and the testbed independently, but this is internal AI-assisted checking, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation; exact rational arithmetic checks only the finite constants used in the proofs." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review; no data-assimilation specialist has reviewed the paper." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Package versions are pinned in requirements.txt and every random step has a fixed seed; not a locked container. No real observations or operational data are used." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematics and numerical verification", "AI-assisted internal review, separate-model adversarial review and audit of new proofs, proof audit, re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [ { "date": "2026-11-04", "scope": "claim", "fixedIn": "0.2.0-candidate", "summary": "Corrected how the paper relates its results to Furrer and Bengtsson (2007). Version 0.1.0-candidate cited that paper only for its subject and said its expansions had not been compared with ours; version 0.2.0-candidate compares them term by term.", "detail": "Section 1.5 now credits their optimal taper (their Eq. 20), which has Flowerdew's form and predates it; their diagonal-prior taper result (Eq. 17); their Jensen gain bias (Eq. 6, Corollaries 2-3); and their gain-bias inflation rule (Eq. 26). It also states that their expansion (Proposition 6) is a different object from Theorem 6.1: covariance error in a fixed eigenbasis, for the untapered estimator, with an unproved remainder and no basis objective. They never optimise over bases, so the main results are not anticipated. No theorem, proof, computation or recorded output changed. Version 0.1.0-candidate remains archived at doi:10.5281/zenodo.23926884." } ], "publicCorrections": [ { "date": "2026-11-04", "scope": "claim", "fixedIn": "0.2.0-candidate", "summary": "Corrected how the paper relates its results to Furrer and Bengtsson (2007). Version 0.1.0-candidate cited that paper only for its subject and said its expansions had not been compared with ours; version 0.2.0-candidate compares them term by term.", "detail": "Section 1.5 now credits their optimal taper (their Eq. 20), which has Flowerdew's form and predates it; their diagonal-prior taper result (Eq. 17); their Jensen gain bias (Eq. 6, Corollaries 2-3); and their gain-bias inflation rule (Eq. 26). It also states that their expansion (Proposition 6) is a different object from Theorem 6.1: covariance error in a fixed eigenbasis, for the untapered estimator, with an unproved remainder and no basis objective. They never optimise over bases, so the main results are not anticipated. No theorem, proof, computation or recorded output changed. Version 0.1.0-candidate remains archived at doi:10.5281/zenodo.23926884." } ], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript of 3 November 2026 (31 pages), its README, verification ledger, agent index, internal review summary and recorded outputs, and the replay of all four scripts from a fresh copy of the package.", "grandfatheredAtSchemaVersion": null, "keywords": [ "ensemble Kalman filter", "covariance localisation", "canonical coordinates", "data assimilation", "whitening", "Snyder–Hakim conjecture", "sampling error", "Schur product taper" ], "keyResults": [ "Physical squared error, scalar observation (Theorem 3.3, Corollary 3.4, Theorem 3.7, proof): at every ensemble size N ≥ 2, if the observation direction is not an eigenvector of the prior covariance, a small rotation followed by complete localisation beats every canonical taper with unit observed diagonal; with the same deterministic rescaling it also beats every canonical taper with any observed-variance factor.", "Exact certificates (Section 4, Appendix A): an explicit rational rotation lowers the risk by at least 1/20416536 at N = 3, and in an aligned example the canonical basis is a strict local maximum of the risk (bounds −56/773 and −508/773).", "Several observations (Proposition 5.1, Corollaries 5.2–5.3, proof): compared with rotated complete localisations, canonical complete localisation is not stationary at any N when the loss couples an observed and an unobserved canonical direction, and is stationary when [W, C] = 0; the converse is Conjecture 5.4.", "Large ensembles (Theorems 6.1 and 7.1, Corollary 7.6, proof): the leading basis-dependent risk term is 2F(U)/(N−1) with F(U) = Σ(u_iᵀBu_i)(u_iᵀAu_i); canonical coordinates minimise F if and only if A and B commute with oppositely ordered eigenvalues; otherwise they are beaten for all large N. Under prior-normalised loss the criterion always holds.", "Snyder–Hakim test case (Section 10, numerical reconstruction): for N = 8 to 64 the best basis found removes 42–54% and symmetric (ZCA) whitening 41–54% of the canonical basis’s excess error over the Kalman filter (N = 8: 1.137 versus 1.063 times the Kalman-filter error), roughly equivalent to 1.7–2 times more ensemble members; canonical is the best found under prior-normalised loss; pilot-estimated bases keep much less of the gain." ], "reviews": [], "evidencePackage": "Written proofs, not machine-verified. exact_certificates.py checks the finite constants of the proofs in exact rational arithmetic (37 checks, including 4 negative controls); numerical_checks.py (82 checks, deterministic quadrature), theory_checks.py (296 checks, including a seeded Monte Carlo of the actual ensemble procedure with 4 × 10^6 ensembles and negative controls) and sh_testbed.py (39 checks, the reconstructed Snyder–Hakim test case, 4,000 replicates per ensemble size) supply floating-point consistency checks and illustrations. All four scripts were rerun from a fresh copy with Python -I -B and with -O; all passed, every recorded value matched the shipped results apart from timings and version strings, and the figures were byte-identical. GitHub CI on Linux reruns all four in both modes.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Snyder and Hakim (2022). An Optimal Linear Transformation for Data Assimilation. Journal of Advances in Modeling Earth Systems. Source of the canonical-coordinates conjecture tested here and of the reconstructed 200-variable test case.", "url": "https://doi.org/10.1029/2021MS002937", "doi": "10.1029/2021MS002937" }, { "citation": "Flowerdew (2015). Towards a theory of optimal localisation. Tellus A. Optimal localisation in fixed coordinates; Lemma 3.1 is its zero-correlation case.", "url": "https://doi.org/10.3402/tellusa.v67.25257", "doi": "10.3402/tellusa.v67.25257" }, { "citation": "Furrer and Bengtsson (2007). Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. Journal of Multivariate Analysis. Earlier optimal taper, Jensen gain bias, inflation rule and covariance-error expansion in a fixed eigenbasis; compared term by term in Section 1.5.", "url": "https://doi.org/10.1016/j.jmva.2006.08.003", "doi": "10.1016/j.jmva.2006.08.003" }, { "citation": "Atkinson, Flowerdew, Hughes and Roulstone (2024). Comparison of Traditional and Hybrid Forms of Optimal Localisation for Mitigation of Sampling Error in Ensemble Kalman Filters. Tellus A. Optimal-localisation comparisons in fixed coordinates.", "url": "https://doi.org/10.16993/tellusa.35", "doi": "10.16993/tellusa.35" }, { "citation": "Ménétrier, Montmerle, Michel and Berre (2015). Linear Filtering of Sample Covariances for Ensemble-Based Data Assimilation. Part I: Optimality Criteria and Application to Variance Filtering and Covariance Localization. Monthly Weather Review. Localisation chosen by an explicit error criterion in given variables.", "url": "https://doi.org/10.1175/MWR-D-14-00157.1", "doi": "10.1175/MWR-D-14-00157.1" }, { "citation": "Morzfeld and Hodyss (2023). A Theory for Why Even Simple Covariance Localization Is So Useful in Ensemble Data Assimilation. Monthly Weather Review. Theory of localisation in a fixed basis.", "url": "https://doi.org/10.1175/MWR-D-22-0255.1", "doi": "10.1175/MWR-D-22-0255.1" }, { "citation": "Hodyss and Morzfeld (2023). How Sampling Errors in Covariance Estimates Cause Bias in the Kalman Gain and Impact Ensemble Data Assimilation. Monthly Weather Review. Gain bias from sampling error, the mechanism behind the risk expansion.", "url": "https://doi.org/10.1175/MWR-D-23-0029.1", "doi": "10.1175/MWR-D-23-0029.1" }, { "citation": "Sacher and Bartello (2008). Sampling Errors in Ensemble Kalman Filtering. Part I: Theory. Monthly Weather Review. Sampling-error expansions related to Theorem 6.1.", "url": "https://doi.org/10.1175/2007MWR2323.1", "doi": "10.1175/2007MWR2323.1" }, { "citation": "Anderson (2012). Localization and Sampling Error Correction in Ensemble Kalman Filter Data Assimilation. Monthly Weather Review. Sample-adaptive correction, outside the deterministic-taper class studied.", "url": "https://doi.org/10.1175/MWR-D-11-00013.1", "doi": "10.1175/MWR-D-11-00013.1" }, { "citation": "Buehner and Charron (2007). Spectral and spatial localization of background-error correlations for data assimilation. Quarterly Journal of the Royal Meteorological Society. Localisation in transformed (spectral) variables.", "url": "https://doi.org/10.1002/qj.50", "doi": "10.1002/qj.50" }, { "citation": "Kessy, Lewin and Strimmer (2018). Optimal Whitening and Decorrelation. The American Statistician. Symmetric (ZCA) and other whitening transforms compared in the testbed.", "url": "https://doi.org/10.1080/00031305.2016.1277159", "doi": "10.1080/00031305.2016.1277159" }, { "citation": "Bourin (1999). Some inequalities for norms on matrices and operators. Linear Algebra and its Applications. Classical trace inequality used in Theorem 7.1, not claimed.", "url": "https://doi.org/10.1016/S0024-3795(99)00024-5", "doi": "10.1016/S0024-3795(99)00024-5" }, { "citation": "Fujii (2005). A trace inequality arising from quantum information theory. Linear Algebra and its Applications. Classical trace inequality used in Theorem 7.1, not claimed.", "url": "https://doi.org/10.1016/j.laa.2004.11.009", "doi": "10.1016/j.laa.2004.11.009" }, { "citation": "Houtekamer and Zhang (2016). Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation. Monthly Weather Review. Background on ensemble Kalman filters and localisation.", "url": "https://doi.org/10.1175/MWR-D-15-0440.1", "doi": "10.1175/MWR-D-15-0440.1" }, { "citation": "Al-Ghattas and Sanz-Alonso (2024). Non-asymptotic analysis of ensemble Kalman updates: effective dimension and localization. Information and Inference. Non-asymptotic theory of localised ensemble updates.", "url": "https://doi.org/10.1093/imaiai/iaad043", "doi": "10.1093/imaiai/iaad043" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:canonical-covariance-localisation-failure", "attemptIds": [ "ep-attempt:canonical-covariance-localisation-failure-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "decision-rule", "description": "Whether canonical coordinates are the best basis in which to apply complete covariance localisation in an ensemble Kalman analysis: yes at leading order under prior-normalised loss; under physical loss only when A and B commute with oppositely ordered eigenvalues (Theorem 7.1), and never at finite N for a scalar observation that is not a prior eigenvector direction.", "scope": "One analysis step, independent Gaussian ensemble, oracle prior covariance used to build every basis, deterministic tapers, quadratic loss. Cycling, estimated covariances, adaptive localisation and operational data are not covered." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Section 1.2 restates Snyder and Hakim's definition, localisation convention and loss with section references; Table 2 of the paper lists the notation (P = LLᵀ, W, G, C, B, A, F(U)).", "remainingRisks": [ "The bounded literature check is not exhaustive priority certification.", "The testbed reconstruction may differ from the original configuration." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a data-assimilation theorist check Theorems 3.3, 3.7, 6.1 and 7.1, and compare the testbed reconstruction with the original Snyder–Hakim configuration.", "claimCeiling": "Written proofs of the finite-N counterexamples, the expansion and the sharp criterion under the stated one-step oracle setting; exact certificates for the stated constants; Conjecture 5.4 and Open problem 8.2 numerical only; the testbed is a numerical reconstruction. 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