[Paper Review] The Lower Bound Error for polynomial NARMAX using an Arbitrary Number of Natural Interval Extensions

Priscila F. S. Guedes, Márcia L. C. Peixoto+3 more|arXiv (Cornell University)|Nov 21, 2017

Chaos control and synchronization6 references3 citations

TL;DR

This paper proposes a novel method to estimate the lower bound error in polynomial NARMAX models by comparing k pseudo-orbits derived from k distinct natural interval extensions, leveraging recursive function propagation and Lyapunov exponent validation. The approach improves error estimation accuracy and demonstrates strong agreement with literature values for both the sine map (Lyapunov exponent ≈1.15 bits/s) and Duffing-Ueda oscillator (≈0.1202), confirming its reliability for chaotic system simulations.

ABSTRACT

The polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is a model that represents the dynamics of physical systems. This polynomial contains information from the past of the inputs and outputs of the process, that is, it is a recursive model. In digital computers this generates the propagation of the rounding error. Our procedure is based on the estimation of the maximum value of the lower bound error considering an arbitrary number of pseudo-orbits produced from different natural interval extensions, and a posterior Lyapunov exponent calculation. We applied successfully our technique for two identified models of the systems: sine map and Duffing-Ueda oscillator

Motivation & Objective

• To address the lack of robust error propagation analysis in polynomial NARMAX simulations, particularly regarding rounding errors in digital computation.
• To extend prior work that used only two pseudo-orbits for lower bound error estimation to an arbitrary number of interval extensions.
• To enhance the reliability of numerical simulations of nonlinear dynamic systems by quantifying the minimum possible error growth.
• To validate the proposed method using well-known chaotic systems—sine map and Duffing-Ueda oscillator—through comparison with established literature values.
• To demonstrate that the maximum distance among k pseudo-orbits provides a tighter and more accurate lower bound error estimate than pairwise comparisons.

Proposed method

• The method generates k pseudo-orbits from k different natural interval extensions of the same polynomial NARMAX function, preserving mathematical equivalence but differing in arithmetic operation order.
• Each pseudo-orbit is computed using interval arithmetic, with initial conditions set to X₀ = 0.1 for both test systems.
• The maximum pairwise distance among all k pseudo-orbits at each time step is computed, forming the basis for the lower bound error estimate.
• A theorem is proven stating that the maximum distance among k pseudo-orbits exceeds the lower bound error derived from any two, thereby tightening the error estimate.
• The Lyapunov exponent is calculated for the maximum lower bound error trajectory using the method from [6], enabling comparison with literature values.
• The approach is applied to two benchmark systems: the sine map and the Duffing-Ueda oscillator, with results validated against known dynamical parameters.

Experimental results

Research questions

• RQ1 Can the use of k pseudo-orbits from k distinct natural interval extensions yield a more accurate and tighter lower bound error estimate than the traditional two-pseudo-orbit method in polynomial NARMAX models?
• RQ2 How does the maximum distance among k pseudo-orbits relate to the true lower bound error in recursive numerical simulations?
• RQ3 Does the proposed method preserve the dynamical characteristics of chaotic systems, such as the Lyapunov exponent, under interval arithmetic error propagation?
• RQ4 To what extent does increasing the number of interval extensions improve the reliability of error estimation in polynomial NARMAX models?
• RQ5 Can the method be successfully applied to well-known chaotic systems like the sine map and Duffing-Ueda oscillator with results consistent with established literature?

Key findings

• For the sine map, the Lyapunov exponent calculated using the proposed method was approximately 1.15 bits/s, closely matching the literature value of 1.15 bits/s.
• For the Duffing-Ueda oscillator, the computed Lyapunov exponent was 0.1202, which is in strong agreement with the literature value of 0.115.
• The maximum distance among k pseudo-orbits was shown to be a valid and tighter lower bound error estimate than pairwise comparisons, as proven by the theoretical extension of the lower bound error theorem.
• The method successfully captured the chaotic dynamics of both systems despite numerical error propagation, confirming its robustness.
• The use of multiple interval extensions improved error estimation accuracy without increasing computational complexity unduly, as the method scales efficiently with k.
• The results confirm that the proposed k-pseudo-orbit approach enhances the reliability of free-run simulations of polynomial NARMAX models in chaotic regimes.