Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2109.14881

Extracting stochastic dynamical systems with $\alpha$-stable L\'evy noise from data

Yang Li·Yubin Lu·Shengyuan Xu·Jinqiao Duan
View original source ↗Read source document ↗Export metadata ↓

Abstract

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) L\'evy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with $\alpha$-stable L\'evy noise from short burst data based on the properties of $\alpha$-stable distributions. More specifically, we first estimate the L\'evy jump measure and noise intensity via computing mean and variance of the amplitude of the increment of the sample paths. Then we approximate the drift coefficient by combining nonlocal Kramers-Moyal formulas with normalizing flows. Numerical experiments on one- and two-dimensional prototypical examples illustrate the accuracy and effectiveness of our method. This approach will become an effective scientific tool in discovering stochastic governing laws of complex phenomena and understanding dynamical behaviors under non-Gaussian fluctuations.

Explore related subjects

stat.MLcs.LGmath.PRphysics.data-an

Keep this discovery

BibTeXRIS

Yang Li, Yubin Lu, Shengyuan Xu, Jinqiao Duan. 2021-09-30. Extracting stochastic dynamical systems with $\alpha$-stable L\'evy noise from data. https://arxiv.org/abs/2109.14881

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs

Feature-based explanations quantify features' influence on model predictions, but are primarily designed for scalar outputs. In many applications, however, outputs are functional or multivariate, such as time-dependent trajectories in demand forecasting. Consequently, existing approaches typically explain each output location independently, ignoring dependencies across the output components. We address this limitation by developing a unified framework for feature-based explanations of time-dependent outputs. Specifically, we generalize functional decomposition to Hilbert-valued prediction functions and extend an existing feature-based explanation framework to this setting. Our framework introduces kernel-based output representations that enable time-dependency-aware explanations at multiple levels of temporal granularity, including time-specific, time-resolved, and time-aggregated, while providing a unified view in which existing methods arise as special cases. We validate our framework on synthetic and real-world data, including intraday financial market volatility prediction and energy demand forecasting.

stat.ML↗
arXiv2026

Risk-Averse Decision Making with Multi-Level Reliability Guarantees

Many applications in engineering, including wireless broadcasting, require designs that provide performance certificates at different target outage levels. This paper studies the problem of maximizing the weighted average of such certificates in the presence of uncertainty about the true system state. The problem is shown to be equivalent to an optimization over nested prediction sets, connecting to the literature on conformal prediction and extending prior art on single-level risk-averse decision making. Furthermore, we derive a dual formulation that decouples optimization across input values. Numerical experiments on a diversity-based wireless transmission system illustrate the cost of enforcing multi-level certificates with a single shared policy and trace the Pareto trade-off between multiple reliability levels.

stat.ML↗
arXiv2026

A distribution-free certification framework for trustworthy crash-severity prediction

Crash-severity models inform screening, dispatch and site prioritization, yet are deployed without a finite-sample statement of what one prediction means. Off-the-shelf guarantees fail here, because the features that make crash severity distinctive defeat them: the KABCO outcome is ordinal, the recorded label is a field assessment agreeing with medical severity about half the time, erring in a structured way, and deployment crosses jurisdictions and years calibration never saw. We develop a certification layer that wraps any severity model unmodified, with distribution-free guarantees using this structure: contiguous ordinal sets that read as "B or worse"; per-class validity for any pre-declared partition, with an oracle efficiency characterization; transfer of coverage to unobserved true severity through a declared reporting band, with a worst-case sharpness result; a one-sided certificate under deployment shift; and severity-weighted risk control. The guarantees compose with an attributable slack budget. The same analysis bounds what certification can achieve. A certified set's informativeness is governed by a functional of the true law that no base model can evade and that cannot be lower-bounded distribution-free; given a declared misreporting channel identified from record-linkage data, a nonvacuous lower bound on that floor becomes computable. On 5.2 million Texas records across seven base models spanning four decades, the layer attaches identical validity and certifies, on the vulnerable road users, a model-independent floor on set width that no base model beats, separating it from a remainder that stays bounded but distribution-free unidentifiable. The framework is released as an open-source package with theorem-level tests.

stat.ML↗