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

2026-05-19

AI Summary

Existing approaches lack a unified geometric framework for analyzing multivariate long-memory processes, making it challenging to robustly estimate the Hurst exponent under both isotropic and anisotropic conditions. This work proposes a distance-based geometric framework, termed PDDA, which establishes—for the first time—an explicit connection among temporal persistence, range dimensionality, and recurrence statistics. By integrating rescaled range analysis and mean squared displacement through two complementary pathways—R/S-PDDA and MSD-PDDA—the method unifies these classical techniques into a coherent distance-diffusion analytical framework tailored for multivariate long-memory processes. The resulting approach enables a unified and robust estimation of multivariate Hurst exponents, offering both theoretical novelty and practical utility.

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.

Problem

• Hurst exponent
• long-memory processes
• multivariate processes
• anisotropy
• distance-based analysis

Innovation

• Pairwise Distance-Diffusion Analysis
• Hurst exponent
• long-memory processes
• multivariate stochastic processes
• geometric framework