Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes
Abstract
We introduce Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework that connects the Hurst exponent to the scaling of pairwise distances in long-memory stochastic processes. From a single distance-plot representation, PDDA yields three complementary routes to persistence: R/S-PDDA, based on the growth of geometric extrema; MSD-PDDA, based on the scaling of the second moment of lagged distances; and recurrence-volume scaling, in which the decay of close returns with increasing temporal separation reflects the expansion of the lagged displacement cloud. The framework extends naturally to multivariate isotropic and anisotropic processes, where the local spatial dimension specifies the available degrees of freedom while the Hurst exponents govern their temporal expansion. These results establish the distance plot as a common geometric representation of persistence, providing a unified distance-based foundation for Hurst analysis.
Related
- Fractional Brownian motion with multivariate time: persistence exponents
- A coupled geometric surface and surface reaction-diffusion framework for modelling nuclear mechanics during confined cell migration
- Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents
- Finite Mixtures of Generalized Estimating Equations for Clustering Multivariate Correlated Outcomes
- A multivariate index for estimating seasonal shifts from temperature curves
BibTeX
@misc{soriano2026pairwise,
title = {{Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes}},
author = {Diogo C. Soriano and Frederique Vanheusden and Slawomir J. Nasuto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.21530},
eprint = {2605.21530},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21530},
}
RIS
TY - UNPB
TI - Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes
AU - Diogo C. Soriano
AU - Frederique Vanheusden
AU - Slawomir J. Nasuto
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2605.21530
UR - https://arxiv.org/abs/2605.21530
AB - We introduce Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework that connects the Hurst exponent to the scaling of pairwise distances in long-memory stochastic processes. From a single distance-plot representation, PDDA yields three complementary routes to persistence: R/S-PDDA, based on the growth of geometric extrema; MSD-PDDA, based on the scaling of the second moment of lagged distances; and recurrence-volume scaling, in which the decay of close returns with increasing temporal separation reflects the expansion of the lagged displacement cloud. The framework extends naturally to multivariate isotropic and anisotropic processes, where the local spatial dimension specifies the available degrees of freedom while the Hurst exponents govern their temporal expansion. These results establish the distance plot as a common geometric representation of persistence, providing a unified distance-based foundation for Hurst analysis.
ER -