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Related Concept Videos

Uncertainty: Overview00:59

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
Propagation of Uncertainty from Systematic Error01:10

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular...
Uncertainty in Measurement: Accuracy and Precision03:37

Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or the accepted value. Precise values agree with each other; accurate values agree with a true value.
Uncertainty: Confidence Intervals00:54

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor 't,' or...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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Related Experiment Video

Updated: May 9, 2026

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
11:51

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Published on: February 22, 2018

Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems.

Themistoklis P Sapsis1, Andrew J Majda

  • 1Department of Mathematics and Climate, Atmospheric and Oceanic Sciences, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. sapsis@mit.edu

Proceedings of the National Academy of Sciences of the United States of America
|August 7, 2013
PubMed
Summary

A new framework for predictive statistical modeling and uncertainty quantification in turbulent systems is introduced. This reduced-order, modified quasilinear Gaussian (ROMQG) method accurately captures complex dynamics in geophysical and engineering turbulence.

Keywords:
dynamical systems with many instabilitiesnonlinear response and sensitivityreduced-order modified quasilinear Gaussian closure

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Area of Science:

  • Physics
  • Fluid Dynamics
  • Computational Science

Background:

  • Turbulent dynamical systems are prevalent in geophysics and engineering.
  • Predictive modeling and uncertainty quantification in these systems are challenging due to their complexity.

Purpose of the Study:

  • To develop a novel framework for low-order predictive statistical modeling and uncertainty quantification in turbulent dynamical systems.
  • To create reduced-order, modified quasilinear Gaussian (ROMQG) algorithms applicable to systems with significant linear instability and energy-conserving nonlinear interactions.

Main Methods:

  • Constructing a low-order, nonlinear dynamical system for mean and covariance statistics in a reduced subspace.
  • Optimally incorporating indirect effects of non-Gaussian third-order statistics via a systematic calibration stage using unperturbed equilibrium statistics.
  • Assessing performance on the 40-mode Lorenz 96 model and large-scale baroclinic ocean turbulence models.

Main Results:

  • The ROMQG algorithm with a single mode accurately captures transient responses in the Lorenz 96 model.
  • For ocean turbulence models, ROMQG with a fraction of the total modes (0.2%) effectively captures nonlinear responses in energy, heat flux, and their spectra.

Conclusions:

  • The ROMQG framework provides an efficient and accurate method for statistical modeling and uncertainty quantification in turbulent systems.
  • This approach demonstrates significant potential for applications in geophysical and engineering turbulence, offering substantial computational savings.