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Taming Uncertainty in a Complex World: The Rise of Uncertainty Quantification -- A Tutorial for Beginners

Nan Chen, Stephen Wiggins, Marios Andreou

TL;DR

This tutorial introduces uncertainty quantification for beginners by using simple, interpretable examples tied to probability densities and information measures. It guides readers from fundamental concepts like Shannon entropy and KL divergence to practical tools such as Bayes data assimilation, Kalman filtering, and Lagrangian data assimilation, highlighting how uncertainty propagates in linear and nonlinear dynamical systems. The work illustrates how posterior uncertainty is reduced with observations, how nonlinear diagnostics can amplify or distort uncertainty, and how stochastic surrogates can be calibrated to reproduce forecast statistics. Together, these insights illuminate the practical steps for incorporating UQ into modeling, data analysis, and diagnostics across geophysical, engineering, and related disciplines.

Abstract

This paper provides a tutorial about uncertainty quantification (UQ) for those who have no background but are interested in learning more in this area. It exploits many very simple examples, which are understandable to undergraduates, to present the ideas of UQ. Topics include characterizing uncertainties using information theory, UQ in linear and nonlinear dynamical systems, UQ via data assimilation, the role of uncertainty in diagnostics, and UQ in advancing efficient modeling. The surprisingly simple examples in each topic explain why and how UQ is essential. Both MATLAB and Python codes are made available for these simple examples.

Taming Uncertainty in a Complex World: The Rise of Uncertainty Quantification -- A Tutorial for Beginners

TL;DR

This tutorial introduces uncertainty quantification for beginners by using simple, interpretable examples tied to probability densities and information measures. It guides readers from fundamental concepts like Shannon entropy and KL divergence to practical tools such as Bayes data assimilation, Kalman filtering, and Lagrangian data assimilation, highlighting how uncertainty propagates in linear and nonlinear dynamical systems. The work illustrates how posterior uncertainty is reduced with observations, how nonlinear diagnostics can amplify or distort uncertainty, and how stochastic surrogates can be calibrated to reproduce forecast statistics. Together, these insights illuminate the practical steps for incorporating UQ into modeling, data analysis, and diagnostics across geophysical, engineering, and related disciplines.

Abstract

This paper provides a tutorial about uncertainty quantification (UQ) for those who have no background but are interested in learning more in this area. It exploits many very simple examples, which are understandable to undergraduates, to present the ideas of UQ. Topics include characterizing uncertainties using information theory, UQ in linear and nonlinear dynamical systems, UQ via data assimilation, the role of uncertainty in diagnostics, and UQ in advancing efficient modeling. The surprisingly simple examples in each topic explain why and how UQ is essential. Both MATLAB and Python codes are made available for these simple examples.
Paper Structure (24 sections, 77 equations, 15 figures, 1 table)

This paper contains 24 sections, 77 equations, 15 figures, 1 table.

Table of Contents

  1. Basic Probabilities
  2. Computing Shannon's Entropy and Relative Entropy
  3. Shannon's entropy
  4. Gaussian distribution.
  5. Gamma distribution.
  6. General PDFs.
  7. Relative entropy
  8. UQ in Dynamical Systems
  9. Uncertainty propagation in the linear damped system
  10. Uncertainty propagation in the chaotic Lorenz 63 model
  11. Uncertainty Reduction via Bayes Formula and Data Assimilation (DA)
  12. Uncertainties in posterior distributions
  13. Posterior distribution as a function of the number of observations $L$.
  14. Lagrangian DA
  15. Forecast Model.
  16. ...and 9 more sections

Figures (15)

  • Figure 1: Quantifying the uncertainty using Shannon's entropy (S.E.) for Gaussian and non-Gaussian distributions. In all panels, the x-axis spans $20$ units.
  • Figure 2: Solutions of the linear system \ref{['Linear_ODE']}. Panel (a): time evolution of $x(t)$ with a deterministic initial condition. Panel (b): time evolution of $x(t)$ with the initial condition given by a Gaussian distribution. The ensemble size is 1000.
  • Figure 3: Solutions of the nonlinear chaotic Lorenz 63 model \ref{['Lorenz63']}. Panel (a): the Lorenz attractor. Panel (b): time evolution of $z(t)$ with a deterministic initial condition. Panel (b): time evolution of $z(t)$ with the initial condition given by a Gaussian distribution. The ensemble size is 1000.
  • Figure 4: Posterior distribution from Bayes formula \ref{['Bayes_formula']} with $\mu_f=1$, $R_f=1$ and $m=1$. Panels (a)--(d): the PDFs with different observational uncertainty $r^o$ and number of observations $L$. To keep the figure concise, the PDFs corresponding to the $L=5$ or $10$ noisy observations are omitted from Panels (c)--(d). Panel (e): the asymptotic behavior of the posterior mean $\mu_a$ and the posterior variance $R_a$ as a function of $L$. Due to the randomness in observations, the shading area shows the variation of the results with 100 sets of independent observations for each fixed $L$.
  • Figure 5: UQ in LaDA. Panel (a): uncertainty reduction in the signal and dispersion parts as a function of $L$. Panels (b)--(e): comparing the true flow field with the recovered ones using 2, 10 and 50 tracers.
  • ...and 10 more figures