Propagation of uncertainty

Propagation of uncertainty refers to the process of determining the uncertainty of an outcome calculated from multiple measurements, each with their own uncertainty. In many scientific and engineering fields, when measurements are used in calculations to derive other quantities, the uncertainties of those measurements propagate through the calculations and affect the certainty of the resulting output.

Overview

The concept of uncertainty propagation is crucial in many scientific disciplines, including physics, engineering, and chemistry, where it is essential to estimate the uncertainty of derived quantities based on the uncertainties of the measurements that were used in their calculation. The propagation of uncertainty allows researchers and engineers to quantify how measurement errors influence the final result, enabling them to provide more accurate and reliable conclusions.

Mathematical Formulation

The most common approach to propagate uncertainty is through the use of Taylor series expansions of the measurement functions, assuming that the uncertainties are small relative to the measurements themselves. For a function f that depends on multiple variables x, y, ..., the uncertainty σ_f in the function value due to uncertainties σ_x, σ_y, ... in the variables can be approximated as:

σ_f^2 ≈ (∂f/∂x)^2σ_x^2 + (∂f/∂y)^2σ_y^2 + ...

where ∂f/∂x and ∂f/∂y are the partial derivatives of f with respect to x and y, respectively. This formula assumes that the uncertainties in the variables are independent and normally distributed.

Types of Uncertainty

There are two main types of uncertainty in measurements: Type A uncertainty, which is evaluated by statistical methods, and Type B uncertainty, which is evaluated by other means, such as published specifications or personal experience. The propagation of uncertainty must consider both types to provide a comprehensive uncertainty estimate.

Applications

The propagation of uncertainty is applied in various fields for different purposes. In metrology, it is used to ensure the accuracy of measurements and calibrations. In environmental science, it helps in assessing the reliability of pollution measurements and the potential impacts on public health. In finance, uncertainty propagation is used in risk assessment and the valuation of complex financial instruments.

Challenges

One of the main challenges in the propagation of uncertainty is the assumption of linearity and normality. Many systems and models are nonlinear, and the assumption of normal distribution may not always be valid, especially for Type B uncertainties. Advanced methods, such as Monte Carlo simulations, are often used to address these challenges, providing more accurate uncertainty estimates for complex systems.

Conclusion

Understanding and accurately estimating the propagation of uncertainty is essential for making reliable measurements and decisions in science, engineering, and beyond. It provides a quantitative measure of confidence in the results obtained from experimental and calculated data.