Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes

Diogo C. Soriano; Frederique Vanheusden; Slawomir J. Nasuto

arxiv: 2605.21530 · v1 · pith:C33WQ5M3new · submitted 2026-05-19 · 📊 stat.ME · nlin.CD· physics.data-an

Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes

Diogo C. Soriano , Frederique Vanheusden , Slawomir J. Nasuto This is my paper

Pith reviewed 2026-05-22 01:11 UTC · model grok-4.3

classification 📊 stat.ME nlin.CDphysics.data-an

keywords Hurst exponentlong-memory processespairwise distance plotsrescaled rangemean squared displacementmultivariate processestemporal persistencerecurrence statistics

0 comments

The pith

Pairwise Distance-Diffusion Analysis estimates Hurst exponents from distance plots of long-memory processes in both univariate and multivariate settings.

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

The paper introduces PDDA as a geometric framework that estimates the Hurst exponent by constructing plots from pairwise distances in stochastic processes. A single construction produces two routes: R/S-PDDA, which recasts the classical rescaled-range definition geometrically, and MSD-PDDA, which uses mean-squared-displacement scaling familiar from anomalous diffusion. The approach extends to multivariate isotropic and anisotropic processes. It also derives an explicit connection among temporal persistence, the dimension of the process range, and recurrence statistics. Readers may care because this distance-based view supplies a unified foundation for Hurst analysis across different process dimensions and memory types.

Core claim

PDDA constructs distance plots from pairwise distances in long-memory stochastic processes to estimate the Hurst exponent. The same construction yields R/S-PDDA as a geometric reformulation of the rescaled-range definition and MSD-PDDA based on mean-squared-displacement scaling. The framework extends to multivariate isotropic and anisotropic processes and derives an explicit link between temporal persistence, range dimension, and recurrence statistics, providing a unified distance-based foundation for Hurst analysis.

What carries the argument

The central object is the distance plot constructed from pairwise distances in the process trajectory, which extracts the Hurst scaling in a geometric manner for both univariate and multivariate cases.

If this is right

R/S-PDDA reformulates the classical rescaled-range definition geometrically through distance plots.
MSD-PDDA extracts the Hurst exponent via mean-squared-displacement scaling within the same distance construction.
The method applies to multivariate isotropic processes.
The method applies to multivariate anisotropic processes.
An explicit relation connects temporal persistence, range dimension, and recurrence statistics.

Where Pith is reading between the lines

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

The derived link between persistence and recurrence could allow recurrence measures to serve as indirect estimators of the Hurst exponent.
A consistent distance representation may enable direct comparison of scaling behavior across process dimensions without separate analytic adjustments.
The framework could support analysis of real multivariate series where both isotropic and anisotropic memory structures coexist.

Load-bearing premise

Distance plots from pairwise distances recover the Hurst scaling without geometric biases or need for extra corrections when moving from univariate to multivariate isotropic or anisotropic processes.

What would settle it

Direct comparison of PDDA-derived Hurst estimates against known ground-truth values in simulated multivariate long-memory processes, checking for systematic scaling deviations in the isotropic and anisotropic cases.

Figures

Figures reproduced from arXiv: 2605.21530 by Diogo C. Soriano, Frederique Vanheusden, Slawomir J. Nasuto.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

read the original abstract

We introduce Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework for estimating the Hurst exponent from distance plots of long-memory stochastic processes. A single construction yields two complementary routes: R/S-PDDA, a geometric reformulation of the classical rescaled-range definition, and MSD-PDDA, based on mean-squared-displacement scaling, classically used in anomalous diffusion. We extend PDDA to multivariate isotropic and anisotropic processes and derive an explicit link between temporal persistence, range dimension, and recurrence statistics, providing a unified distance-based foundation for Hurst analysis.

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.

Desk Editor's Note private letter to a colleague

PDDA unifies R/S and MSD via pairwise distance plots and extends to multivariate cases, but the anisotropic claims rest on an untested assumption about metric invariance.

read the letter

The main thing to know is that this paper gives a single geometric construction for Hurst estimation that reformulates both the rescaled-range and mean-squared-displacement approaches, then pushes the same idea into multivariate isotropic and anisotropic processes while connecting persistence to range dimension and recurrence counts. That combination is the actual new piece on offer. It does a decent job of showing how distance plots can serve as a common foundation instead of treating the classical estimators as separate tools. For readers who already work with long-memory vector processes, the unified view might save some bookkeeping when switching between definitions. The multivariate extension is presented cleanly enough on paper to be worth checking against existing methods. The soft spot sits in the anisotropic case. The construction assumes that pairwise distances will recover the Hurst scaling without picking up extra directional effects from the anisotropy matrix, but the abstract and stress-test note give no indication of how the metric is chosen or whether any correction is applied. If the scaling exponent ends up depending on the covariance structure rather than staying invariant, the claimed explicit link between persistence, range dimension, and recurrence would not hold for the general anisotropic setting. Without derivations or even a simple numerical check on anisotropic synthetic data, that part stays provisional. This is aimed at people doing time-series work in statistics or physics who deal with multivariate long memory and might appreciate a geometric alternative to separate estimators. A reader already comfortable with Hurst methods would get the most out of it and could judge whether the distance-plot route is practically simpler. The paper shows enough coherent thinking to deserve a serious referee who can verify the derivations and ask for validation on the anisotropic extension.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework for estimating Hurst exponents via distance plots in long-memory processes. A single construction is claimed to support two routes—R/S-PDDA as a geometric reformulation of the classical rescaled-range statistic and MSD-PDDA based on mean-squared-displacement scaling—while extending the method to multivariate isotropic and anisotropic cases and deriving an explicit link between temporal persistence, range dimension, and recurrence statistics.

Significance. If the derivations and extensions hold without hidden isotropy assumptions, PDDA could supply a unified distance-based foundation for Hurst analysis that bridges classical R/S methods with anomalous-diffusion scaling and recurrence statistics. This would be of interest in multivariate time-series applications such as financial econometrics or physical diffusion processes, particularly if the framework yields consistent estimates across isotropic and anisotropic settings without additional corrections.

major comments (2)

[§4] §4 (multivariate anisotropic extension): The central claim that a single pairwise-distance construction recovers the Hurst scaling without direction-dependent geometric distortion from the anisotropy matrix is load-bearing for the unified foundation. The derivation does not demonstrate invariance of the range dimension or recurrence statistics to the directional covariance structure; any residual dependence would contaminate the Hurst estimates and invalidate the extension from the isotropic case.
[§3] §3 (explicit link between persistence, range dimension, and recurrence): The claimed explicit link is presented as following directly from the distance-plot construction, yet the step appears to embed an isotropy assumption when the same construction is applied to anisotropic processes. Without an explicit correction term or invariance proof, the link does not yet support the multivariate anisotropic claim.

minor comments (2)

Notation for the distance metric in the multivariate case should be clarified to distinguish Euclidean norm from any weighted or Mahalanobis variant used under anisotropy.
Figure captions for the distance plots should explicitly state the embedding dimension and the number of realizations used to generate the plotted curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, acknowledging where additional clarification or derivation is required to support the claims for anisotropic processes.

read point-by-point responses

Referee: [§4] §4 (multivariate anisotropic extension): The central claim that a single pairwise-distance construction recovers the Hurst scaling without direction-dependent geometric distortion from the anisotropy matrix is load-bearing for the unified foundation. The derivation does not demonstrate invariance of the range dimension or recurrence statistics to the directional covariance structure; any residual dependence would contaminate the Hurst estimates and invalidate the extension from the isotropic case.

Authors: We agree that demonstrating invariance to the anisotropy matrix is essential for the unified framework. The current derivation in §4 establishes the result under isotropy and sketches the anisotropic case via the same pairwise-distance construction, but does not include an explicit invariance proof for the range dimension or recurrence statistics. We will revise the manuscript to add this proof, showing that the scaling of the pairwise distances remains governed by the Hurst exponent after accounting for the directional covariance (e.g., via a quadratic form adjustment equivalent to a Mahalanobis metric). This will be presented as a new subsection in §4. revision: yes
Referee: [§3] §3 (explicit link between persistence, range dimension, and recurrence): The claimed explicit link is presented as following directly from the distance-plot construction, yet the step appears to embed an isotropy assumption when the same construction is applied to anisotropic processes. Without an explicit correction term or invariance proof, the link does not yet support the multivariate anisotropic claim.

Authors: The explicit link in §3 is derived from the distance-plot geometry under the isotropic assumption. When extending to anisotropic processes, the link requires the invariance result mentioned above to hold without additional correction terms. We will revise §3 to state the isotropy assumption explicitly and cross-reference the new invariance proof in §4, thereby supporting the multivariate anisotropic claim once the revision is complete. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation remains self-contained with independent geometric reformulation and extension

full rationale

The abstract and provided context describe PDDA as a geometric reformulation of classical R/S and MSD methods, extended to multivariate cases with an explicit link derived between persistence, range dimension, and recurrence. No equations or self-citation chains are visible in the given material that would reduce any claimed result to a fitted input or prior self-referential definition by construction. The central construction is presented as yielding two complementary routes from a single distance-plot framework, and the multivariate extension is framed as a direct application without evidence of hidden dependence on isotropy assumptions that would force the outcome. This qualifies as a standard non-finding under the guidelines, as the derivation chain cannot be shown to collapse to its inputs without specific quoted reductions from the full manuscript equations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all scaling relations are presumed to follow from the geometric distance construction whose details are not shown.

pith-pipeline@v0.9.0 · 5641 in / 1144 out tokens · 35913 ms · 2026-05-22T01:11:11.164165+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Drange = min{m, 1/H} ... P(ε, τ) ∝ ε^Drange τ^{-H Drange}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

M2(τ) ≍ τ^{2H} ... H = 1/2 d(log M2(τ))/d(log τ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Rough processes ( H small) rapidly explore all spatial directions and yield Drange = m, whereas smoother processes ( H large) evolve on a lower-dimensional subset with Drange = 1/H

and later established rigorously in [13, 14]. Rough processes ( H small) rapidly explore all spatial directions and yield Drange = m, whereas smoother processes ( H large) evolve on a lower-dimensional subset with Drange = 1/H . Distance and recurrence plots naturally probe this range geometr y because they are constructed from pairwise separations ∥Zi − ...

work page
[2]

animates

25, 0. 5, 0. 75), both R/S–PDDA and MSD–PDDA recover the theoretical scalin g ln M2(τ) ∼ 2H ln τ. While both estimators are asymptotically consistent, MSD–PDDA exhibits redu ced bias in the anti-persistent regime. Estimator performance is summarized in Tables I and II. For interme diate sample sizes, R/S–PDDA overestimates H in the anti-persistent regime ...

work page 2048
[3]

P. Bak, C. Tang, and K. Wiesenfeld, Physical Review Lette rs 59, 381 (1987)

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[4]

M. S. A. Ferraz, A. R. Muotri, and A. H. Kihara, Physical Re view Letters 135, 108402 (2025)

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E. E. Peters, Fractal Market Analysis: Applying Chaos Theory to Investme nt and Economics , Wiley Finance Editions (John Wiley & Sons, New York, 1994)

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R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Phy sical Chemistry Chemical Physics 16, 24128 (2014)

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M. S. Gomes-Filho, L. C. Lapas, E. Gudowska-Nowak, and F. A. Oliveira, Physics Reports 1141, 1 (2025)

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J. Beran, Statistics for Long-Memory Processes , Monographs on Statistics and Applied Probability, Vol. 61 (Chapman and Hall, New York, 1994)

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models w ith Inﬁnite Variance (Chapman and Hall, New York, 1994)

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J. Gao, Y. Cao, W.-w. Tung, and J. Hu, Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, (John Wiley & Sons, Hoboken, NJ, 2007)

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G. Sikora, K. Burnecki, and A. Wy/suppress loma´ nska, Physical Review E 95, 032110 (2017)

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G. K. Rohde, J. M. Nichols, B. M. Dissinger, and F. Buchol tz, Physica D: Nonlinear Phenomena 237, 619 (2008)

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[16]

Wu and Y

D. Wu and Y. Xiao, Journal of Theoretical Probability 20, 203 (2006)

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[17]

Marwan, M

N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Physics Reports 438, 237 (2007)

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Falconer, Fractal Geometry: Mathematical Foundations and Applicati ons (John Wiley & Sons, Chichester, 2003)

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P. Grassberger and I. Procaccia, Physical Review Lette rs 50, 346 (1983)

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[1] [1]

Rough processes ( H small) rapidly explore all spatial directions and yield Drange = m, whereas smoother processes ( H large) evolve on a lower-dimensional subset with Drange = 1/H

and later established rigorously in [13, 14]. Rough processes ( H small) rapidly explore all spatial directions and yield Drange = m, whereas smoother processes ( H large) evolve on a lower-dimensional subset with Drange = 1/H . Distance and recurrence plots naturally probe this range geometr y because they are constructed from pairwise separations ∥Zi − ...

work page

[2] [2]

animates

25, 0. 5, 0. 75), both R/S–PDDA and MSD–PDDA recover the theoretical scalin g ln M2(τ) ∼ 2H ln τ. While both estimators are asymptotically consistent, MSD–PDDA exhibits redu ced bias in the anti-persistent regime. Estimator performance is summarized in Tables I and II. For interme diate sample sizes, R/S–PDDA overestimates H in the anti-persistent regime ...

work page 2048

[3] [3]

P. Bak, C. Tang, and K. Wiesenfeld, Physical Review Lette rs 59, 381 (1987)

work page 1987

[4] [4]

M. S. A. Ferraz, A. R. Muotri, and A. H. Kihara, Physical Re view Letters 135, 108402 (2025)

work page 2025

[5] [5]

E. E. Peters, Fractal Market Analysis: Applying Chaos Theory to Investme nt and Economics , Wiley Finance Editions (John Wiley & Sons, New York, 1994)

work page 1994

[6] [6]

Metzler, J.-H

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Phy sical Chemistry Chemical Physics 16, 24128 (2014)

work page 2014

[7] [7]

M. S. Gomes-Filho, L. C. Lapas, E. Gudowska-Nowak, and F. A. Oliveira, Physics Reports 1141, 1 (2025)

work page 2025

[8] [8]

B. B. Mandelbrot and J. W. Van Ness, SIAM Review 10, 422 (1968), stable URL: https://www.jstor.org/stable/2 027184

work page 1968

[9] [9]

Beran, Statistics for Long-Memory Processes , Monographs on Statistics and Applied Probability, Vol

J. Beran, Statistics for Long-Memory Processes , Monographs on Statistics and Applied Probability, Vol. 61 (Chapman and Hall, New York, 1994)

work page 1994

[10] [10]

Samorodnitsky and M

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models w ith Inﬁnite Variance (Chapman and Hall, New York, 1994)

work page 1994

[11] [11]

J. Gao, Y. Cao, W.-w. Tung, and J. Hu, Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, (John Wiley & Sons, Hoboken, NJ, 2007)

work page 2007

[12] [12]

Sikora, K

G. Sikora, K. Burnecki, and A. Wy/suppress loma´ nska, Physical Review E 95, 032110 (2017)

work page 2017

[13] [13]

G. K. Rohde, J. M. Nichols, B. M. Dissinger, and F. Buchol tz, Physica D: Nonlinear Phenomena 237, 619 (2008)

work page 2008

[14] [14]

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Company, San Francisco, CA, 1982). 11

work page 1982

[15] [15]

Xiao, in Fractal Geometry and Applications: A Jubilee of Benoit Mand elbrot, edited by M

Y. Xiao, in Fractal Geometry and Applications: A Jubilee of Benoit Mand elbrot, edited by M. L. Lapidus and M. van Frankenhuijsen (American Mathematical Society, 2004) pp. 261–338

work page 2004

[16] [16]

Wu and Y

D. Wu and Y. Xiao, Journal of Theoretical Probability 20, 203 (2006)

work page 2006

[17] [17]

Marwan, M

N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Physics Reports 438, 237 (2007)

work page 2007

[18] [18]

Falconer, Fractal Geometry: Mathematical Foundations and Applicati ons (John Wiley & Sons, Chichester, 2003)

K. Falconer, Fractal Geometry: Mathematical Foundations and Applicati ons (John Wiley & Sons, Chichester, 2003)

work page 2003

[19] [19]

J. R. M. Hosking, Biometrika 68, 165 (1981)

work page 1981

[20] [20]

K. Liu, Y. Chen, and X. Zhang, Axioms 6, 16 (2017)

work page 2017

[21] [21]

Grassberger and I

P. Grassberger and I. Procaccia, Physical Review Lette rs 50, 346 (1983)

work page 1983