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arxiv: 2601.06009 · v1 · pith:PBAIPJR6new · submitted 2026-01-09 · 📊 stat.ML · cs.LG· eess.SP· math.PR· stat.AP

Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem

Sunia Tanweer , Firas A. Khasawneh This is my paper

Pith reviewed 2026-05-21 15:32 UTC · model grok-4.3

classification 📊 stat.ML cs.LGeess.SPmath.PRstat.AP
keywords stochastic processesexcursion theoremquadratic variationdiffusion detectionsemimartingalesnonparametric testdiscrete signals
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The pith

A scaling law for excursion counts in semimartingales distinguishes stochastic diffusions from deterministic signals.

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

This paper presents a nonparametric method to detect whether a discrete time series comes from a stochastic diffusion or a deterministic system. It does so by counting the number of excursions larger than a small threshold ε and comparing that count to what is expected from the process's quadratic variation. For any continuous semimartingale the counts follow a universal scaling with the quadratic variation, including cases with nonlinear volatility. Deterministic signals violate this scaling. The test uses the deviation from the expected log-log slope to classify the signal without assuming a specific model.

Core claim

The paper establishes that the number of excursions N_ε of magnitude at least ε in a continuous semimartingale is related to its quadratic variation [X]_T by a universal scaling law that holds for all Ito diffusions with finite quadratic variation. This law fails for deterministic systems, allowing construction of a ratio K(ε) = empirical excursions over theoretical prediction whose log-log slope deviation from the ε^{-2} behavior classifies the dynamics as diffusion-like or not.

What carries the argument

The classical excursion and crossing theorems for continuous semimartingales that provide the theoretical expectation for N_ε based on the quadratic variation [X]_T, which is estimated from the discrete observations to serve as the benchmark.

If this is right

  • The method provides a theoretically grounded alternative to entropy-based or recurrence-based tests for stochasticity.
  • It applies to general Ito diffusions without requiring knowledge of the drift or diffusion coefficient.
  • The test works on a single discrete time series and remains nonparametric.
  • Classification is achieved by summarizing the ratio K(ε) via its log-log slope deviation from the expected scaling.

Where Pith is reading between the lines

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

  • Such a test could be applied to financial tick data or physiological signals to detect underlying stochastic components.
  • Extensions might include adapting the threshold selection or handling irregularly sampled data.
  • Further work could explore the test's power against specific alternatives like fractional Brownian motion or jump processes.

Load-bearing premise

The quadratic variation estimated from the discrete observations accurately reflects the true quadratic variation without the estimation introducing bias that could mimic the stochastic excursion scaling.

What would settle it

Direct computation on a deterministic periodic signal showing that the observed excursion counts do not match the scaling predicted by its estimated quadratic variation, or on a known diffusion where they do match.

Figures

Figures reproduced from arXiv: 2601.06009 by Firas A. Khasawneh, Sunia Tanweer.

Figure 1
Figure 1. Figure 1: Histograms of the fitted log-log slope of the excursion count [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Heatmaps of accuracy for OU process, across various [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heatmaps of accuracy for CIR process, across various [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmaps of accuracy for simple harmonic motion, across various [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Heatmaps of accuracy for Henon map, across various [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Heatmaps of accuracy for logistic map, across various [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Heatmaps of accuracy for LCG map, across various [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Heatmaps of accuracy for Chen and Lu systems, across various [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Heatmaps of accuracy for Duffing and Rayleigh-Duffing systems, across various [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Slope distribution and heatmaps of accuracy for stochastic Duffing system, across various [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: BTC-USD and SPY financial time series retrieved from yfinance on December 25, 2025. [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.

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

3 major / 2 minor

Summary. The paper develops a nonparametric test to distinguish continuous semimartingales (including general Itô diffusions with state-dependent volatility) from deterministic signals in a single discrete time series. It invokes classical excursion theorems to predict that the number of excursions N_ε of size at least ε scales as ε^{-2} times the quadratic variation [X]_T; the ratio K(ε) = N_ε^emp / N_ε^theory is then summarized by its log-log slope to classify the series as diffusion-like or not. The method is demonstrated on canonical stochastic processes, periodic/chaotic maps, additive-noise systems, and the stochastic Duffing oscillator, and is claimed to be model-free and to rely only on the small-scale structure of continuous semimartingales.

Significance. If the central scaling relation and its robustness to quadratic-variation estimation can be established, the approach supplies a theoretically grounded, nonparametric alternative to entropy- or recurrence-based heuristics for detecting diffusive behavior. The explicit use of excursion theorems for general semimartingales and the provision of a concrete classification statistic constitute a clear strength.

major comments (3)
  1. [§3] §3 (Method): the theoretical benchmark N_ε^theory is constructed from the realized-volatility estimator ∑(ΔX_i)^2 for [X]_T. For finite sampling grids and state-dependent volatility σ(X), this estimator carries an O(1) relative error that directly rescales N_ε^theory and therefore alters both the level and the log-log slope of K(ε). No error analysis or mesh-refinement study is supplied to show that the classification threshold remains stable under this bias.
  2. [§4.2] §4.2 (Stochastic Duffing example): the reported separation between the stochastic and deterministic cases relies on the same finite-grid quadratic-variation estimate for both the empirical and theoretical counts. Because the estimation step is performed on the identical series, any systematic bias that mimics the ε^{-2} scaling cannot be ruled out; a controlled experiment with known mesh size or with an independent high-frequency proxy for [X]_T is needed.
  3. [Abstract / §2] Abstract and §2: the statement that the ε^{-2} law 'holds universally for all continuous semimartingales with finite quadratic variation' is asserted without a self-contained derivation or reference to the precise excursion theorem (e.g., the exact constant relating N_ε to [X]_T). The absence of this formula makes it impossible to verify that the data-driven implementation matches the continuous-theory benchmark.
minor comments (2)
  1. [Abstract] Abstract: 'theoretically-certfied' is a typographical error.
  2. [§3] Notation: the symbol K(ε) is introduced without an explicit equation number; subsequent references to its log-log slope would be clearer if the definition were numbered.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. The comments highlight important aspects regarding the finite-sample behavior of the quadratic variation estimator and the need for clearer theoretical grounding. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Method): the theoretical benchmark N_ε^theory is constructed from the realized-volatility estimator ∑(ΔX_i)^2 for [X]_T. For finite sampling grids and state-dependent volatility σ(X), this estimator carries an O(1) relative error that directly rescales N_ε^theory and therefore alters both the level and the log-log slope of K(ε). No error analysis or mesh-refinement study is supplied to show that the classification threshold remains stable under this bias.

    Authors: We appreciate this observation. The realized quadratic variation estimator does indeed converge to the true quadratic variation only in the limit of vanishing mesh size, and for finite grids with state-dependent volatility there can be a relative bias. However, our classification relies primarily on the scaling behavior (the log-log slope of K(ε)) rather than the absolute level, which may mitigate the impact of a constant rescaling factor. To rigorously address the concern, we will add an error analysis section and perform mesh-refinement studies in the revised version to confirm that the slope-based threshold remains stable. revision: yes

  2. Referee: [§4.2] §4.2 (Stochastic Duffing example): the reported separation between the stochastic and deterministic cases relies on the same finite-grid quadratic-variation estimate for both the empirical and theoretical counts. Because the estimation step is performed on the identical series, any systematic bias that mimics the ε^{-2} scaling cannot be ruled out; a controlled experiment with known mesh size or with an independent high-frequency proxy for [X]_T is needed.

    Authors: This is a valid point. Using the same series for both empirical counts and the quadratic variation estimate introduces potential circularity in the presence of bias. We will include additional controlled experiments in the revision, such as using a known finer mesh for the quadratic variation proxy or simulating with varying sampling frequencies to demonstrate that the separation persists and is not an artifact of the estimation procedure. revision: yes

  3. Referee: [Abstract / §2] Abstract and §2: the statement that the ε^{-2} law 'holds universally for all continuous semimartingales with finite quadratic variation' is asserted without a self-contained derivation or reference to the precise excursion theorem (e.g., the exact constant relating N_ε to [X]_T). The absence of this formula makes it impossible to verify that the data-driven implementation matches the continuous-theory benchmark.

    Authors: We acknowledge that the manuscript would benefit from a more explicit reference to the underlying excursion theorem. The scaling N_ε ~ [X]_T / ε^2 follows from the occupation time formula and excursion theory for semimartingales. We will add a brief derivation sketch and the precise reference in §2 of the revised manuscript to make the connection transparent and allow verification of the implementation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation relies on external classical theorem

full rationale

The paper grounds its scaling law in classical excursion theorems for continuous semimartingales, which are independent of the present work. The ratio K(ε) is formed by comparing observed excursion counts to a benchmark computed from the standard realized-volatility estimator of quadratic variation; this estimator is an external input drawn from the data rather than a fitted parameter whose value is forced to reproduce the test statistic. The resulting log-log slope is a falsifiable comparison that can (and does, per the abstract) fail for deterministic signals, so the classification outcome is not equivalent to the inputs by construction. No self-citations, ansatzes, or uniqueness claims appear as load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the classical excursion theorem for continuous semimartingales and the assumption that quadratic variation can be recovered from discrete samples; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Excursion-count scaling law holds for all continuous semimartingales with finite quadratic variation
    Invoked as the universal property that deterministic signals violate.

pith-pipeline@v0.9.0 · 5777 in / 1252 out tokens · 54387 ms · 2026-05-21T15:32:07.276692+00:00 · methodology

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Reference graph

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