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Propagation of Uncertainty from Random Error

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...

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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...

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Related Experiment Video

Updated: May 30, 2026

Split Point Analysis and Uncertainty Quantification of Thermal-Optical Organic/Elemental Carbon Measurements

Split Point Analysis and Uncertainty Quantification of Thermal-Optical Organic/Elemental Carbon Measurements

Published on: September 7, 2019

Improving uncertainty analysis in kinetic evaluations using iteratively reweighted least squares.

Zhenglei Gao¹, John W Green, Jan Vanderborght

¹Bayer CropScience, Monheim, Germany.

Environmental Toxicology and Chemistry

|July 26, 2011

Summary

Standard nonlinear least squares (NLS) methods make unrealistic assumptions about data variance. An iteratively reweighted least squares (IRLS) algorithm provides more accurate confidence intervals for environmental fate model parameters, especially when error variances differ.

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Area of Science:

Environmental chemistry
Chemical kinetics
Environmental modeling

Background:

Environmental fate processes are modeled using kinetic parameters.
Standard nonlinear least squares (NLS) methods assume equal data variances, which is often unrealistic for degradation and metabolite formation data.
Unequal error variances between parent compounds and metabolites complicate parameter estimation.

Purpose of the Study:

To introduce and evaluate an iteratively reweighted least squares (IRLS) algorithm for estimating kinetic parameters and error variances.
To compare the performance of IRLS with standard NLS and Markov-Chain Monte-Carlo (MCMC) approaches for environmental fate modeling.

Main Methods:

Application of an iteratively reweighted least squares (IRLS) algorithm to obtain maximum likelihood estimates.
Comparison of IRLS with nonlinear least squares (NLS) and Markov-Chain Monte-Carlo (MCMC) methods.
Analysis of simulated data and real-world data from aerobic transformation of chemicals in soil.

Main Results:

IRLS provides reliable parameter estimates and realistic confidence intervals, even with unequal error variances.
NLS yields significantly different confidence intervals when error variances differ substantially between species.
IRLS-derived confidence intervals are consistent with those obtained from MCMC, suggesting greater accuracy.

Conclusions:

IRLS is a more robust method than NLS for estimating confidence intervals of kinetic parameters in environmental fate models.
The assumption of equal error variances in NLS can lead to unrealistic parameter uncertainty estimates.
IRLS offers a superior approach for handling heteroscedasticity in environmental data analysis.