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

stat.MLMar 20, 2026

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

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

Organizations: Fraunhofer IOSB · KIT, IES · KIT, IIIT

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.

Figures & tables

Practical Meaning	Mathematical Expression	CT	Formulas/Algorithms
Traditional measurement result; value $m$	Expectation of Random Variable s; see Equation 4 in Proposition 5.1	CF	$\displaystyle m=\sum_{i=1}^{K}\pi_{i}m_{i}$ (6)
Traditional measurement uncertainty $\pm k\sigma$ ; $\sigma^{2}$ = variance	Variance of Random Variable s; see Equation 5 in Proposition 5.2	CF	$\displaystyle\Sigma=\sum_{i=1}^{K}\pi_{i}(\Sigma_{i}+{m_{i}\bm{\cdot}{m_{i}}^{\top}})$ (7) $\displaystyle-{(\sum_{i=1}^{K}\pi_{i}m_{i})\bm{\cdot}(\sum_{i=1}^{K}\pi_{i}m_{i})^{\top}}$ (7)
Empirical addition of physical quantities of the same type	Addition of independent Random Variable s corresponds with: Convolution of Probability Density Function s; see Proposition B.5 in Appendix A	CF	$\displaystyle X+Y=Z,\hskip 8.50012ptX\perp Y$ (8) $\displaystyle K_{Z}=K_{X}\bm{\cdot}K_{Y},\hskip 8.50012pt{\pi_{Z}}_{ij}={\pi_{X}}_{i}\bm{\cdot}{\pi_{Y}}_{j}$ (8) $\displaystyle{\displaystyle m_{Z}}_{ij}={m_{X}}_{i}+{m_{Y}}_{j},\hskip 8.50012pt{\Sigma_{Z}}_{ij}={\Sigma_{X}}_{i}+{\Sigma_{Y}}_{j}$ (8)
Focusing on variables of interest	Partial marginalization of Random Variable s corresponds with: Integrating out nuisance Random Variable s; see Proposition B.1 in Appendix A	CF	$\displaystyle Z=\begin{bmatrix}X\\ Y\end{bmatrix}\in\mathbb{R}^{d},\hskip 8.50012ptZ\sim\mathcal{G}_{Z}(z\|g_{Z})$ (9) $\displaystyle{\displaystyle m_{Z}}_{i}=\begin{bmatrix}{{m_{Z}}_{i}}_{X}\\ {{{m_{Z}}}_{i}}_{Y}\end{bmatrix},\hskip 8.50012pt{\Sigma_{Z}}_{i}=\begin{bmatrix}{{\Sigma_{Z}}_{i}}_{XX}{{\Sigma_{Z}}_{i}}_{XY}\\ {{\Sigma_{Y}}_{i}}_{YX}{{\Sigma_{Z}}_{i}}_{YY}\end{bmatrix}$ (9) $\displaystyle X\sim\mathcal{G}_{X}(x\|g_{X}),\hskip 8.50012ptY\sim\mathcal{G}_{Y}(x\|g_{Y})$ (9) $\displaystyle K_{X}=K_{Y}=K_{Z}$ (9) $\displaystyle{\displaystyle\pi_{X}}_{i}={\pi_{Y}}_{i}={\pi_{Z}}_{i}$ (9) $\displaystyle{\displaystyle m_{X}}_{i}={{m_{Z}}_{i}}_{X},\hskip 8.50012pt{m_{Y}}_{i}={{m_{Z}}_{i}}_{Y},$ (9) $\displaystyle{\displaystyle\Sigma_{X}}_{i}={{\Sigma_{Z}}_{i}}_{XX},\hskip 8.50012pt{\Sigma_{Y}}_{i}={{\Sigma_{Z}}_{i}}_{YY}$ (9)
Exploiting evidence about a set of Random Variable s	Partial conditioning of Random Variable s regarding the observed Random Variable s; see Proposition B.2 in Appendix A	CF	$\displaystyle X\|Y\sim\mathcal{G}_{X\|Y}(x\|y,g_{X\|Y})$ (10) $\displaystyle K_{X\|Y}=K_{Z}$ (10) $\displaystyle{\displaystyle m_{X\|Y}}_{i}={m_{i}}_{X}+{\Sigma_{i}}_{XY}\bm{\cdot}{{\Sigma_{i}}_{YY}}^{-1}\bm{\cdot}(y-{m_{i}}_{Y})$ (10) $\displaystyle{\displaystyle\Sigma_{X\|Y}}_{i}={\Sigma_{i}}_{XX}-{\Sigma_{i}}_{XY}\bm{\cdot}{{\Sigma_{i}}_{YY}}^{-1}\bm{\cdot}{\Sigma_{i}}_{YX}$ (10) $\displaystyle{\displaystyle\pi_{X\|Y}}_{i}=\frac{\pi_{i}\mathcal{N}(y\|{m_{Y}}_{i},{\Sigma_{Y}}_{i})}{\sum_{j=1}^{K_{Y}}\pi_{j}\mathcal{N}(y\|{m_{Y}}_{j},{\Sigma_{Y}}_{j})}$ (10) $\displaystyle Y\|X\sim\mathcal{G}_{Y\|X}(y\|x,g_{Y\|X})$ (10) $\displaystyle K_{Y\|X}=K_{Z}$ (10) $\displaystyle{\displaystyle m_{Y\|X}}_{i}={m_{i}}_{Y}+{\Sigma_{i}}_{YX}\bm{\cdot}{{\Sigma_{i}}_{XX}}^{-1}\bm{\cdot}(x-{m_{i}}_{x})$ (10) $\displaystyle{\displaystyle\Sigma_{Y\|X}}_{i}={\Sigma_{i}}_{YY}-{\Sigma_{i}}_{YX}\bm{\cdot}{{\Sigma_{i}}_{XX}}^{-1}\bm{\cdot}{\Sigma_{i}}_{XY}$ (10) $\displaystyle{\displaystyle\pi_{Y\|X}}_{i}=\frac{\pi_{i}\mathcal{N}(x\|{m_{X}}_{i},{\Sigma_{X}}_{i})}{\sum_{j=1}^{K_{X}}\pi_{j}\mathcal{N}(x\|{m_{X}}_{j},{\Sigma_{X}}_{j})}$ (10)
Mix of Sources	Convex linear combination of Probability Density Function s; see Proposition B.4 in Appendix A	CF	$\displaystyle X^{j}\sim\mathcal{G}_{X^{j}}(x\|g^{j}_{X})\text{ for }j=1,\ldots,J$ (11) $\displaystyle X\sim\mathcal{G}_{X}(x\|g_{X})=\sum_{j=1}^{J}w^{j}\mathcal{G}^{j}_{X}(x\|g^{j}_{X})=$ (11) $\displaystyle\sum_{j=1}^{J}w^{j}\sum_{i=1}^{K_{j}}\pi^{j}_{i}\mathcal{N}_{X}(x\|m^{j}_{i},\Sigma^{j}_{i})=$ (11) $\displaystyle\sum_{j=1}^{J}\sum_{i=1}^{K_{j}}w^{j}\pi^{j}_{i}\mathcal{N}_{X}(x\|m^{j}_{i},\Sigma^{j}_{i}).$ (11)

Table 1: This table provides a compact overview of the use of Gaussian Mixture Model s for the representation and combination of uncertain quantitative attributes and their Probability Density Function s. Column 1 lists the most important operations for the practical handling of attributes in real-world applications. Column 2 explains the associated mathematical meaning. The computation type (CT) in column 3 indicates whether the calculations can be accomplished algebraically on the parameter tuples of the Gaussian Mixture Model s (CF: Closed Form) or whether a numerical calculation (N) is required. Finally, in column 4, the closed formulas for the algebraic operations on the Gaussian Mixture Model parameters are given compactly.

X\perp Y

means that the Random Variable s

X

and

Y

are stochastically independent. Additional details to these formulas are presented in Appendix A . For each row of the table there is an explanatory concrete example in Section 6 .

Panel of Figure 1 — Figure 1: Length $X$ and length $Y$ make up the total length $Z$ . $\mathcal{G}_{Z}$ can be calculated using in . The symbol $*$ denotes the convolution operator.

Figure 2: Analytical calculation of the resulting Probability Density Function vs forward Monte Carlo according to Joint Committee for Guides in Metrology (2008b) using the ad hoc sampler in Section 5.1.4

Panel of Figure 3 — 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 successively.

Panel of Figure 4 — Figure 4: Length $Y$ and length $Z$ make up the total length $X$ . $\mathcal{G}_{Z}$ can be calculated using in . The symbol $*$ denotes the convolution operator.

Panel of Figure 5 — 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 $Z=X\bm{\cdot}Y$ .

Figure 6: Numerical calculation of the resulting Probability Density Function using the Expectation Maximization algorithm based on the samples produced using the forward Monte Carlo according to Joint Committee for Guides in Metrology (2008b) using the ad hoc sampler in Section 5.1.4

Panel of Figure 7 — Figure 7: If quality is defined by the set $Q$ , the probability of the attributes of the product instance to fall into set $Q$ can be calculated by integrating the Probability Density Function over the set $Q$ . Tolerances are usually given as intervals for the components $x_{i}$ of the product instance attribute vector $x$ , are a special case of $Q$ , where $Q$ becomes the hyperrectangle $Q_{r}$ . If the tolerances of the components of the product instance attribute vector $x$ , are correlated (not independent), $Q$ can assume a general shape.

Panel of Figure 8 — Figure 8: A pipe with the inner diameter $X$ and the outer diameter $Y$ . The thickness of the pipe is $T$ . The variables are modeled as Random Variable s.

Panel of Figure 9 — Figure 9: Reduction of a Gaussian Mixture Model with four components (of the left) into a Gaussian Mixture Model with two components (on the right); illustration.

Figure 10: Schematic illustration of the conditioning process to obtain the posterior Probability Density Function $p(x|y^{*})$ of the measurand based on an observation of the measurement device.

Figure 11: Modeling of a measurement system using Gaussian Mixture Model ; low noise case.

Figure 12: Modeling of a measurement system using Gaussian Mixture Model ; high noise case.

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

See all 649 papers

Mar 18, 2026

Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.

Hengrui Luo, Xiaoye S. Li, Yang Liu +3

Department of Statistics Rice University, Houston, TX 77005 · Lawrence Berkeley National Laboratory, Berkeley, CA 94709

Gaussian Process Regression Uncertainty Quantification Kernel Methods

Ask PDF

May 24, 2026

A Geometric Gaussian Mixture Representation of Plane Curves

We introduce a user defined probabilistic polygonal representation for plane curves. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation. For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. Combining the segment wise components with appropriate weights yields a Gaussian Mixture Model (GMM) representation of the user defined probabilistic polygonal representation of the plane curve. The proposed framework provides an analytically tractable probabilistic model that preserves local geometry, and uncertainty in the normal direction. It applies to smooth, closed, open, non regular, and self intersecting plane curves, allows adaptive discretization and varying uncertainty in the normal direction, and as a result supports uncertainty aware geometric modeling. Experiments on a collection of canonical plane curves show that the resulting GMM capture local tangent, local normal, and local arc length; resulting in the global shape of the underlying curves to be truthfully captured as well. The representation is particularly relevant for applications in uncertainty aware CAD and digital twins, probabilistic obstacle modeling in robotics, and probabilistic trajectory planning.

Ali Darijani, Benedikt Stratmann, Jürgen Beyerer

Fraunhofer IOSB · KIT, IES

Uncertainty Quantification Gaussian Mixture Models

Ask PDF

Oct 13, 2026

Uncertainty Evaluation: Bridging Machine Learning and Measurement Science

Reliable uncertainty evaluation is essential for assessing the trustworthiness of machine learning (ML) predictions, particularly in safety-critical applications. This paper explores conceptual analogies between uncertainty evaluation in ML and measurement science, drawing on the evolution of metrology from error-based approaches to the framework established by the Guide to the Expression of Uncertainty in Measurement (GUM). Particular attention is devoted to the distinction between process-level and result-level quality and to the limitations of the conventional classification of ML uncertainty into aleatoric and epistemic components. It is argued that this dichotomy, similarly to the traditional distinction between random and systematic measurement errors, is inherently model-dependent and provides an insufficient basis for a unified uncertainty evaluation framework. Further analogies between machine learning and measurement field of knowledge are identified concerning distribution shifts, anomalous operating conditions, and the validity of uncertainty evaluation. Building on the proposed considerations, the paper advocates a metrology-inspired approach based on the systematic identification, quantification, and propagation of uncertainty contributions. By extending the methodology already established in measurement science to ML, this approach could offer a potentially rigorous, consistent, and transferable framework for evaluating the trustworthiness of individual predictions.

Dario Petri, Luca Mari

Department of Industrial Engineering of the Università di Trento, Trento, Italy · Department of Philosophy of the Università Cattolica di Milano, Milano, Italy

Uncertainty Quantification Uncertainty Decomposition Uncertainty Propagation

Ask PDF

Abstract

Uncertainty Quantification

Explore similar work

Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

A Geometric Gaussian Mixture Representation of Plane Curves

Uncertainty Evaluation: Bridging Machine Learning and Measurement Science