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arxiv: 2605.22692 · v1 · pith:2QVHGPTGnew · submitted 2026-05-21 · 🧮 math.DS

Mechanisms and Pathways of Extreme Events in Partially-Observed Stochastic Dynamical Systems

Charlotte Moser , Nan Chen , Marios Andreou This is my paper

Pith reviewed 2026-05-22 03:33 UTC · model grok-4.3

classification 🧮 math.DS
keywords extreme eventspartially observed systemsdata assimilationstochastic dynamicshidden precursorslatent pathwaysconditional Gaussian models
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The pith

A framework integrates data assimilation with trajectory and statistical diagnostics to trace how hidden states drive observed extreme events in stochastic systems.

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

The paper develops a mathematical framework for studying the mechanisms and pathways of extreme events in partially-observed stochastic dynamical systems that include hidden variables. It combines data assimilation with information-theoretic and trajectory-based tools to infer latent precursor dynamics from observations, quantify their uncertainty, and track how their influence reaches the observed extremes. A sympathetic reader would care because most prior work on extremes examines only directly observable variables, yet many high-impact events are shaped by unseen processes whose timing and pathways remain opaque. Conditional Gaussian models supply a setting where closed-form diagnostics can be derived analytically, and the same logic extends numerically to broader nonlinear cases. The method is demonstrated on three examples that reveal distinct classes of hidden drivers and multiple routes to the same class of extreme.

Core claim

By comparing filtering and smoothing distributions along individual trajectories, the framework detects the onset of hidden precursors and measures their temporal influence; by constructing event-conditioned distributions over the hidden state, it isolates sensitive triggering directions, latent pathways, and distinct mechanism classes via clustering. Conditional Gaussian models permit closed-form expressions for these quantities, while numerical methods generalize the approach to nonlinear systems.

What carries the argument

The joint use of filtering-versus-smoothing discrepancies for trajectory-wise precursor detection together with event-conditioned hidden-state distributions for pathway identification and clustering.

If this is right

  • Hidden damping dynamics can be shown to precede observed bursts through systematic differences between filter and smoother estimates.
  • Damping-induced, forcing-driven, and mixed pathways to extremes become separable by examining the conditional hidden-state distributions.
  • Distinct blocking and unblocking mechanisms appear as separate clusters when the framework is applied to a nonlinear topographic-flow model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same diagnostics could be applied to real-world data streams such as climate or financial time series to generate earlier alerts based on inferred hidden states rather than waiting for the observed extreme to begin.
  • Clustering the hidden-state pathways offers a way to classify extreme events into mechanistically distinct families even when only partial observations are available.
  • The numerical extension step implies that the framework can be tested on fully nonlinear models by replacing analytic expressions with particle or ensemble approximations.

Load-bearing premise

The approach assumes conditional Gaussian models yield tractable closed-form diagnostics that remain reliable when extended numerically to more general nonlinear dynamics.

What would settle it

In a controlled simulation of a partially observed stochastic system whose hidden variables are known, the inferred precursor onset times and pathway clusters fail to align with the true hidden dynamics or do not improve extreme-event timing predictions over methods that use only the observed variables.

Figures

Figures reproduced from arXiv: 2605.22692 by Charlotte Moser, Marios Andreou, Nan Chen.

Figure 1.1
Figure 1.1. Figure 1.1: Schematic overview of the proposed framework for diagnosing hidden mechanisms and [PITH_FULL_IMAGE:figures/full_fig_p004_1_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Simulation and hidden-state mechanisms of extreme events in system ( [PITH_FULL_IMAGE:figures/full_fig_p019_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Mechanisms of representative individual extreme events in system ( [PITH_FULL_IMAGE:figures/full_fig_p020_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Simulation of system (4.3) and cluster diagnostics for the detected extreme events. Panel (a): True trajectory of u, where colored markers indicate event peaks assigned to different clusters; dashed horizontal lines denote the positive and negative extreme-event thresholds. Panel (b): Hidden stochastic damping γ(t). The dashed horizontal line marks the anti-damping thresh￾old above which the net linear d… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Cluster-wise representative pathways for the extreme events in system ( [PITH_FULL_IMAGE:figures/full_fig_p022_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Basic statistical properties of the topographic flow model ( [PITH_FULL_IMAGE:figures/full_fig_p025_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Mechanism-based clustering of blocking and unblocking events in the topographic [PITH_FULL_IMAGE:figures/full_fig_p026_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Hidden-state distributions conditioned on each event cluster (Panels (a), (b), (c) for [PITH_FULL_IMAGE:figures/full_fig_p027_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Representative trajectories, hidden-energy evolution, and flow patterns for the three [PITH_FULL_IMAGE:figures/full_fig_p028_4_8.png] view at source ↗
read the original abstract

Extreme events occur across the natural, engineering, and socioeconomic sciences, where rare but high-impact episodes can lead to disproportionate consequences that pose major challenges for prediction and risk management. Existing studies have mainly focused on the statistics, sampling, forecasting, and attribution of extremes from observable variables. In this paper, we develop a mathematical framework for studying the mechanisms and pathways of extreme events in partially-observed stochastic dynamical systems with hidden variables. By integrating data assimilation with information-theoretic and trajectory-based diagnostics, we infer latent precursor dynamics from observations, quantify their uncertainty, and determine how their influence propagates toward observed extreme events. Conditional Gaussian models provide a tractable analytical setting for deriving closed-form diagnostics, while the framework extends through numerical methods. The analysis proceeds from two complementary perspectives. From a trajectory-wise viewpoint, we compare filtering and smoothing distributions to identify the onset of hidden precursors and quantify temporal influence. From a statistical viewpoint, we construct event-conditioned hidden-state distributions to identify sensitive triggering directions, latent pathways, and multiple classes of extreme-event mechanisms through clustering. Three numerical examples illustrate the methodology. In an intermittent stochastic model, hidden damping dynamics emerge before observed bursts, where discrepancies between the filter and smoother provide an onset diagnostic. In a stochastic model with damping and forcing, separate damping-induced, forcing-driven, and mixed pathways to extremes are identified. In a nonlinear topographic-flow model, distinct mechanisms and pathways for blocking and unblocking patterns associated with observed extreme events are revealed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework integrating data assimilation with information-theoretic and trajectory-based diagnostics to infer latent precursor dynamics, quantify uncertainty, and trace pathways to extreme events in partially-observed stochastic dynamical systems. Closed-form diagnostics are derived under conditional Gaussian assumptions, with numerical extensions proposed for nonlinear cases. Analysis proceeds via trajectory-wise filter-smoother comparisons for onset detection and statistical event-conditioned hidden-state distributions with clustering for pathway identification. The approach is illustrated in three numerical examples: an intermittent stochastic model showing hidden damping precursors, a damping-forcing model revealing separate and mixed pathways, and a nonlinear topographic-flow model distinguishing blocking/unblocking mechanisms.

Significance. If the central claims hold under quantitative scrutiny, the work provides a systematic method to uncover hidden mechanisms behind extremes that goes beyond observable-variable statistics, with potential impact on prediction and risk assessment in climate, fluid dynamics, and related fields. The conditional Gaussian setting for closed-form expressions is a clear strength that grounds the numerical extensions.

major comments (2)
  1. [Numerical examples] Numerical examples section: the three examples are described as qualitative illustrations only, with no reported error bars, convergence checks against reference solvers, ablation on assimilation parameters, or stability tests for the identified precursors and pathway classes under numerical approximations. This is load-bearing for the claim that the framework extends reliably through numerical methods to general nonlinear systems, as the skeptic note highlights.
  2. [Statistical viewpoint] Statistical viewpoint (event-conditioned distributions and clustering): there is no sensitivity analysis to post-hoc choices such as the number of clusters, clustering algorithm, or onset-detection thresholds, which could alter the reported distinct mechanisms and multiple pathway classes in the damping-forcing and topographic-flow examples.
minor comments (2)
  1. [Abstract] Abstract: the description of the three models could include one additional sentence on their key features (e.g., intermittency, damping/force terms, topography) to improve accessibility.
  2. [Methodology] Notation: filtering and smoothing distributions are central but their precise definitions and the discrepancy metric used for onset diagnostics would benefit from an explicit equation or box in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and have revised the manuscript accordingly to improve its robustness and clarity.

read point-by-point responses

  1. Referee: [Numerical examples] Numerical examples section: the three examples are described as qualitative illustrations only, with no reported error bars, convergence checks against reference solvers, ablation on assimilation parameters, or stability tests for the identified precursors and pathway classes under numerical approximations. This is load-bearing for the claim that the framework extends reliably through numerical methods to general nonlinear systems, as the skeptic note highlights.

    Authors: We agree that additional quantitative assessments would strengthen the presentation of the numerical examples. In the revised manuscript, we have included error bars computed from ensemble runs, performed convergence tests by varying the ensemble size in the particle filter for the nonlinear topographic-flow model, and added a short ablation study on the data assimilation window length. These results confirm the stability of the identified precursors and pathways. We note that the conditional Gaussian cases provide exact closed-form expressions, serving as the analytical backbone, while the numerical examples illustrate the extension to nonlinear settings. revision: yes

  2. Referee: [Statistical viewpoint] Statistical viewpoint (event-conditioned distributions and clustering): there is no sensitivity analysis to post-hoc choices such as the number of clusters, clustering algorithm, or onset-detection thresholds, which could alter the reported distinct mechanisms and multiple pathway classes in the damping-forcing and topographic-flow examples.

    Authors: We acknowledge the importance of assessing sensitivity to these choices. We have added a new subsection in the revised version that examines the effect of varying the number of clusters from 2 to 5 and compares k-means with Gaussian mixture models. The core pathway classes remain consistent across these choices. Additionally, we have tested different onset-detection thresholds and shown that the precursor identification is robust within a reasonable range. These analyses are now reported in the supplementary material and referenced in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper develops a methodological framework that derives closed-form diagnostics explicitly under the conditional Gaussian assumption and states that it extends via numerical methods to nonlinear cases, with three examples serving as qualitative illustrations rather than quantitative predictions. No equations or steps in the abstract or described methodology reduce a claimed result to a fitted parameter or self-citation by construction; the diagnostics are defined from filtering/smoothing distributions computed under the model, but this is the intended use of data assimilation on a known system rather than a tautological renaming or self-referential fit. The central claims remain independent of the inputs once the model and observations are given, with no load-bearing self-citation chains or ansatz smuggling identified.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Because only the abstract is available, the ledger cannot be populated with specific free parameters, axioms, or invented entities from the full derivations. The central claim rests on the unstated assumption that the chosen diagnostics remain informative once numerical approximations replace closed-form expressions.

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