Approximate Propagation of both Epistemic and Aleatory Uncertainty through Dynamic Systems

When ignorance due to the lack of knowledge, modeled as epistemic uncertainty using Dempster-Shafer structures on closed intervals, is present in the model parameters, a new uncertainty propagation method is necessary to propagate both aleatory and epistemic uncertainty. The new framework proposed here, combines both epistemic and aleatory uncertainty into a second-order uncertainty representation which is propagated through a dynamic system driven by white noise. First, a finite parametrization is chosen to model the aleatory uncertainty by choosing a representative approximation to the probability density function conditioned on epistemic variables. The epistemic uncertainty is then propagated through the moment evolution equations of the conditional probability density function. This way we are able to model the ignorance when the knowledge about the system is incomplete. The output of the system is a Dempster-Shafer structure on sets of cumulative distributions which can be combined using different rules of combination and eventually transformed into a singleton cumulative distribution function using Smets’ pignistic transformation when decision making is needed.

💡 Research Summary

The paper introduces a comprehensive framework for propagating both epistemic (knowledge‑based) and aleatory (random) uncertainties through stochastic dynamic systems. Recognizing that model parameters often suffer from incomplete knowledge, the authors model epistemic uncertainty using Dempster‑Shafer (DS) structures defined on closed intervals, while aleatory uncertainty is represented by probability density functions (PDFs) of the system states.

The methodology proceeds in two hierarchical layers. In the first layer, the aleatory component is approximated by a finite‑parameter family (the authors adopt a Gaussian approximation) and its parameters are evolved in time via moment evolution equations derived from the Itô lemma applied to the governing stochastic differential equation. Assuming the system nonlinearity can be expressed as a polynomial, the authors obtain closed‑form ordinary differential equations (ODEs) for the first two moments. To truncate the infinite moment hierarchy, a Gaussian closure is employed, thereby retaining only the mean and variance.

In the second layer, epistemic uncertainty is propagated through the same moment ODEs but now the initial conditions and model coefficients are interval‑valued DS focal elements. The authors first discuss Yager’s convolution rule for combining independent DS structures, noting that naïve interval arithmetic can lead to overly conservative bounds due to dependence problems. To overcome this, they embed the interval problem into a stochastic one by assuming uniform distributions over the intervals and then apply Polynomial Chaos Expansion (PCE) with Legendre polynomials (Wiener‑Askey scheme). The stochastic variables (initial moments and parameters) are expanded, and a Galerkin projection yields a deterministic system of ODEs for the PCE coefficients.

A key innovation is the transformation of the resulting polynomial chaos representation into the Bernstein form. Because Bernstein polynomials possess a range‑enclosure property, the minimum and maximum values of each state variable can be computed exactly, providing tight bounds for the DS focal elements at any time. This approach avoids the conservatism of traditional interval arithmetic while preserving the epistemic information encoded in the DS structure.

After time integration, the output of the dynamic system is expressed as a DS structure on sets of cumulative distribution functions (CDFs), i.e., a probability box (p‑box). For decision‑making, the authors employ Smets’ pignistic transformation to convert the DS‑p‑box into a single “pignistic” PDF, enabling the use of expected utility theory. Recognizing that the pignistic transformation discards ignorance, they propose a scalar metric called the Normalized Integral of Degree of Ignorance (NIDI), which quantifies the area between the cumulative belief and plausibility functions. A low NIDI indicates that the pignistic PDF is a reliable surrogate for the original imprecise information.

The paper’s contributions are threefold: (1) a unified mathematical framework that separates and simultaneously propagates epistemic and aleatory uncertainties; (2) a novel combination of moment evolution, PCE, and Bernstein‑form interval propagation that yields tight, non‑conservative bounds for nonlinear dynamics; (3) a complete pipeline from DS‑p‑box representation to decision‑ready pignistic PDFs with an accompanying ignorance metric.

Limitations are acknowledged. The Gaussian closure restricts the method to near‑Gaussian state PDFs; non‑Gaussian behavior would require alternative closures. The accuracy of the PCE depends on the chosen order and on the dimensionality of the uncertain inputs, potentially leading to high computational cost for large systems. The independence assumption underlying the DS convolution may be violated in practice, causing residual conservatism. Future work could explore non‑Gaussian closures, adaptive PCE order selection, and dependence modeling within DS structures.

Overall, the study offers a rigorous yet implementable approach for uncertainty quantification in stochastic dynamic models, bridging the gap between theoretical imprecision handling and practical risk‑based decision making.