🤖 AI Summary
This work addresses the limitations of traditional Gaussian assumptions in accurately representing complex uncertainties, which often lead to information loss and reduced accuracy in multi-stage measurement and control processes. To overcome these challenges, the paper proposes a scalable precision framework based on Gaussian Mixture Models (GMMs), leveraging GMMs as universal approximators of probability density functions. The approach integrates closed-form uncertainty propagation algorithms with memory-efficient computational strategies, thereby transcending the representational constraints of Gaussian methods while maintaining computational tractability. Experimental evaluations in manufacturing and metrology scenarios—such as circular factories—demonstrate that the proposed method significantly enhances the fidelity of uncertainty characterization and propagation, outperforming conventional Gaussian-based techniques.
📝 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.