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Comprehensive Description of Uncertainty in Measurement for Representation and Propagation with Scalable Precision

Ali Darijani, Jürgen Beyerer, Zahra Sadat Hajseyed Nasrollah, Luisa Hoffmann, Michael Heizmann

Abstract

Probability theory has become the predominant framework for quantifying uncertainty across scientific and engineering disciplines, with a particular focus on measurement and control systems. However, the widespread reliance on simple Gaussian assumptions--particularly in control theory, manufacturing, and measurement systems--can result in incomplete representations and multistage lossy approximations of complex phenomena, including inaccurate propagation of uncertainty through multi stage processes. This work proposes a comprehensive yet computationally tractable framework for representing and propagating quantitative attributes arising in measurement systems using Probability Density Functions (PDFs). Recognizing the constraints imposed by finite memory in software systems, we advocate for the use of Gaussian Mixture Models (GMMs), a principled extension of the familiar Gaussian framework, as they are universal approximators of PDFs whose complexity can be tuned to trade off approximation accuracy against memory and computation. From both mathematical and computational perspectives, GMMs enable high performance and, in many cases, closed form solutions of essential operations in control and measurement. The paper presents practical applications within manufacturing and measurement contexts especially circular factory, demonstrating how the GMMs framework supports accurate representation and propagation of measurement uncertainty and offers improved accuracy--compared to the traditional Gaussian framework--while keeping the computations tractable.

Comprehensive Description of Uncertainty in Measurement for Representation and Propagation with Scalable Precision

Abstract

Probability theory has become the predominant framework for quantifying uncertainty across scientific and engineering disciplines, with a particular focus on measurement and control systems. However, the widespread reliance on simple Gaussian assumptions--particularly in control theory, manufacturing, and measurement systems--can result in incomplete representations and multistage lossy approximations of complex phenomena, including inaccurate propagation of uncertainty through multi stage processes. This work proposes a comprehensive yet computationally tractable framework for representing and propagating quantitative attributes arising in measurement systems using Probability Density Functions (PDFs). Recognizing the constraints imposed by finite memory in software systems, we advocate for the use of Gaussian Mixture Models (GMMs), a principled extension of the familiar Gaussian framework, as they are universal approximators of PDFs whose complexity can be tuned to trade off approximation accuracy against memory and computation. From both mathematical and computational perspectives, GMMs enable high performance and, in many cases, closed form solutions of essential operations in control and measurement. The paper presents practical applications within manufacturing and measurement contexts especially circular factory, demonstrating how the GMMs framework supports accurate representation and propagation of measurement uncertainty and offers improved accuracy--compared to the traditional Gaussian framework--while keeping the computations tractable.
Paper Structure (49 sections, 17 theorems, 44 equations, 12 figures)

This paper contains 49 sections, 17 theorems, 44 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Mastering Uncertainty within the Circular Factory
  3. Uncertainty Representation and the Probability Framework
  4. Philosophical Aspects and Historical Success
  5. Mathematical Aspects
  6. Uncertainty in Measurement, the Gaussian Framework and its Insufficiency
  7. The Gaussian Framework
  8. The Advantages of the Gaussian Framework
  9. The Insufficiency of the Gaussian Framework
  10. Inherently Non Gaussian Physics
  11. Nonlinear Transformation
  12. Mixture of Sources
  13. Constraint Handling
  14. Abstract Non Gaussian Distributions
  15. Gaussian Mixture Model and Its Advantages
  16. ...and 34 more sections

Key Result

Proposition 5.1

Figures (12)

  • Figure 1: Length $X$ and length $Y$ make up the total length $Z$. $\mathcal{G}_Z$ can be calculated using \ref{['eq:tab:convolution']} in \ref{['tab:gm_math']}. The symbol $*$ denotes the convolution operator.
  • Figure 2: Analytical calculation of the resulting PDF vs forward Monte Carlo according to JCGM2008 using the ad hoc sampler in \ref{['sec:gmm_sampling']}
  • Figure 3: Three independent resistances $X_1,X_2$ and $X_3$ make up the total resistance $Z=X_1 + X_2 + X_3$. Then for $X_i \sim \mathcal{G}_{X_i}, i=1,2,3$ the total resistance $Z \sim \mathcal{G}_Z = \mathcal{G}_{X_1} * \mathcal{G}_{X_2} * \mathcal{G}_{X_3}$ which can be calculated by applying \ref{['eq:tab:convolution']} successively.
  • Figure 4: Length $Y$ and length $Z$ make up the total length $X$. $\mathcal{G}_Z$ can be calculated using \ref{['eq:tab:convolution']} in \ref{['tab:gm_math']}. The symbol $*$ denotes the convolution operator.
  • Figure 5: A rectangle with side lengths $X \sim \mathcal{G}_X(x|g_X)$, $Y \sim \mathcal{G}_Y(x|g_Y)$, and area $Z=X \boldsymbol{\cdot} Y$.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Definition 5.1
  • Remark 5.1: Notation
  • Remark 5.2: Notation
  • Proposition 5.1: GMM Expectation
  • Proposition 5.2: GMM Covariance Matrix
  • Remark 6.1
  • Remark 6.2
  • Definition A.1: Gaussian PDF
  • Remark A.1: Notation
  • Proposition A.1: Expectation of Gaussians
  • ...and 14 more