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Uncertainty-aware automated machine learning toolbox

  • Tanja Dorst

    Tanja Dorst studied Mathematics at Saarland University and received her Master of Science degree in November 2013. After that, she studied Mechanical Engineering at University of Applied Sciences in Saarbrücken and received her Bachelor of Engineering degree in September 2017. Since July 2020 she has been working at Center for Mechatronics and Automation Technology (ZeMA) gGmbH as a scientific researcher. Her research interests include measurement uncertainties in ML for condition monitoring of technical systems.

     ORCID logo EMAIL logo     , Tizian Schneider

    Tizian Schneider studied Microtechnologies and Nanostructures at Saarland University and received his Master of Science degree in January 2016. Since that time, he has been working at the Lab for Measurement Technology (LMT) of Saarland University and at Center for Mechatronics and Automation Technology (ZeMA) gGmbH leading the research group Data Engineering & Smart Sensors. His research interests include ML methods for condition monitoring of technical systems, automatic ML model building and interpretable AI.

         , Sascha Eichstädt

    Dr. Sascha Eichstädt is the leader of the Physikalisch-Technische Bundesanstalt (PTB) department “Metrology for digital transformation”. He received his Diploma in Mathematics in 2008 at the HU Berlin, and his PhD in Theoretical Physics in 2012 at the TU Berlin. From 2008 to 2017 he joined the group “Mathematical modelling and data analysis” at PTB. His main research areas are signal processing and sensor networks.

     ORCID logo     und Andreas Schütze

    Andreas Schütze received his diploma in physics from RWTH Aachen in 1990 and his doctorate in Applied Physics from Justus-Liebig-Universität in Gießen in 1994 with a thesis on microsensors and sensor systems for the detection of reducing and oxidizing gases. From 1994 until 1998 he worked for VDI/VDE-IT, Teltow, Germany, mainly in the fields of microsystems technology. From 1998 until 2000 he was professor for Sensors and Microsystem Technology at the University of Applied Sciences in Krefeld, Germany. Since April 2000 he is professor for Measurement Technology in the Department Systems Engineering at Saarland University, Saarbrücken, Germany and head of the Laboratory for Measurement Technology (LMT). His research interests include smart gas sensor systems as well as data engineering methods for industrial applications.

     ORCID logo
Veröffentlicht/Copyright: 22. September 2022
tm - Technisches Messen
Aus der Zeitschrift tm - Technisches Messen Band 90 Heft 3

Abstract

Measurement data can be considered complete only with an associated measurement uncertainty to express knowledge about the spread of values reasonably attributed to the measurand. Measurement uncertainty also allows to assess the comparability and the reliability of measurement results as well as to evaluate decisions based on the measurement result. Artificial Intelligence (AI) methods and especially Machine Learning (ML) are often based on measurements, but so far, uncertainty is widely neglected in this field. We propose to apply uncertainty propagation in ML to allow estimating the uncertainty of ML results and, furthermore, an optimization of ML methods to minimize this uncertainty. Here, we present an extension of a previously published automated ML toolbox (AMLT), which performs feature extraction, feature selection and classification in an automated way without any expert knowledge. To this end, we propose to apply the principles described in the “Guide to the Expression of Uncertainty in Measurement” (GUM) and its supplements to carry out uncertainty propagation for every step in the AMLT. In previous publications we have presented the uncertainty propagation for some of the feature extraction methods in the AMLT. In this contribution, we add some more elements to this concept by also including statistical moments as a feature extraction method, add uncertainty propagation to the feature selection methods and extend it to also include the classification method, linear discriminant analysis combined with Mahalanobis distance. For these methods, analytical approaches for uncertainty propagation are derived in detail, and the uncertainty propagation for the other feature extraction and selection methods are briefly revisited. Finally, the use the uncertainty-aware AMLT is demonstrated for a data set consisting of uncorrelated measurement data and associated uncertainties.

Zusammenfassung

Messdaten können nur dann als vollständig angesehen werden, wenn sie mit einer Messunsicherheit versehen sind, die das Wissen über die Streuung der Werte ausdrückt, die der Messgröße zugeordnet werden kann. Die Messunsicherheit ermöglicht zudem die Beurteilung der Vergleichbarkeit und Zuverlässigkeit von Messergebnissen sowie die Bewertung von Entscheidungen auf der Grundlage von Messergebnissen. Methoden der künstlichen Intelligenz (KI) und insbesondere des maschinellen Lernens (ML) basieren häufig auf Messungen, aber bisher wurde die Unsicherheit in diesem Bereich weitgehend vernachlässigt. Wir schlagen daher in diesem Beitrag vor, die Unsicherheitsfortpflanzung beim ML anzuwenden, um die Unsicherheit von ML-Ergebnissen abzuschätzen und darüber hinaus eine Optimierung von ML-Methoden zur Minimierung dieser Unsicherheit zu ermöglichen. Dazu stellen wir eine Erweiterung einer bereits veröffentlichten automatisierten ML-Toolbox (AMLT) vor, die Merkmalsextraktion, Merkmalsselektion und Klassifikation automatisiert und ohne Expertenwissen durchführt. Die im „Guide to the Expression of Uncertainty in Measurement“ (GUM) und seinen Supplementen beschriebenen Prinzipien werden angewandt, um eine Unsicherheitsfortpflanzung für jeden Schritt in der AMLT durchzuführen. In früheren Veröffentlichungen haben wir bereits die Unsicherheitsfortpflanzung für einige der Merkmalsextraktionsmethoden in der AMLT vorgestellt. In diesem Beitrag fügen wir nun diesem Konzept einige weitere Elemente hinzu, indem wir auch statistische Momente als Merkmalsextraktionsmethode einbeziehen, die Unsicherheitsfortpflanzung zu den Merkmalselektionsmethoden hinzufügen und sie auch auf die Klassifikationsmethode, die lineare Diskriminanzanalyse in Kombination mit der Mahalanobis-Distanz, ausweiten. Für diese Methoden werden analytische Ansätze für die Unsicherheitsfortpflanzung im Detail abgeleitet, und die Unsicherheitsfortpflanzungen für die anderen Merkmalsextraktions- und -selektionsmethoden werden kurz aufgegriffen. Abschließend wird die Anwendung der zuvor vorgestellten Version der AMLT, welche Unsicherheiten berücksichtig, für einen Datensatz, welcher aus unkorrelierten Messdaten und dazugehörigen Unsicherheiten besteht, demonstriert.

Keywords: Measurement uncertainty; uncertainty propagation; statistical moments; linear discriminant analysis; machine learning
Schlagwörter: Messunsicherheit; Unsicherheitsfortpflanzung; statistische Momente; lineare Diskriminanzanalyse; maschinelles Lernen

Funding source: Horizon 2020

Award Identifier / Grant number: 17IND12

Funding source: Bundesministerium für Bildung und Forschung

Award Identifier / Grant number: 16ES0419K

Funding statement: Part of this work has received funding within the project 17IND12 Met4FoF from the EMPIR program co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation program. The basic version of the automated ML toolbox was developed at ZeMA as part of the MoSeS-Pro research project funded by the German Federal Ministry of Education and Research in the call “Sensor-based electronic systems for applications for Industry 4.0 – SElekt I 4.0”, funding code 16ES0419K, within the framework of the German Hightech Strategy.

About the authors

Tanja Dorst

Tanja Dorst studied Mathematics at Saarland University and received her Master of Science degree in November 2013. After that, she studied Mechanical Engineering at University of Applied Sciences in Saarbrücken and received her Bachelor of Engineering degree in September 2017. Since July 2020 she has been working at Center for Mechatronics and Automation Technology (ZeMA) gGmbH as a scientific researcher. Her research interests include measurement uncertainties in ML for condition monitoring of technical systems.

Tizian Schneider

Tizian Schneider studied Microtechnologies and Nanostructures at Saarland University and received his Master of Science degree in January 2016. Since that time, he has been working at the Lab for Measurement Technology (LMT) of Saarland University and at Center for Mechatronics and Automation Technology (ZeMA) gGmbH leading the research group Data Engineering & Smart Sensors. His research interests include ML methods for condition monitoring of technical systems, automatic ML model building and interpretable AI.

Sascha Eichstädt

Dr. Sascha Eichstädt is the leader of the Physikalisch-Technische Bundesanstalt (PTB) department “Metrology for digital transformation”. He received his Diploma in Mathematics in 2008 at the HU Berlin, and his PhD in Theoretical Physics in 2012 at the TU Berlin. From 2008 to 2017 he joined the group “Mathematical modelling and data analysis” at PTB. His main research areas are signal processing and sensor networks.

Andreas Schütze

Andreas Schütze received his diploma in physics from RWTH Aachen in 1990 and his doctorate in Applied Physics from Justus-Liebig-Universität in Gießen in 1994 with a thesis on microsensors and sensor systems for the detection of reducing and oxidizing gases. From 1994 until 1998 he worked for VDI/VDE-IT, Teltow, Germany, mainly in the fields of microsystems technology. From 1998 until 2000 he was professor for Sensors and Microsystem Technology at the University of Applied Sciences in Krefeld, Germany. Since April 2000 he is professor for Measurement Technology in the Department Systems Engineering at Saarland University, Saarbrücken, Germany and head of the Laboratory for Measurement Technology (LMT). His research interests include smart gas sensor systems as well as data engineering methods for industrial applications.

Appendix A Derivations of the sensitivity coefficients and the covariance matrix for statistical moments

A.1 Standard deviation

β p , j = ∂ σ p ∂ d j = 1 2 · ( 1 N p − 1 ∑ i = a p e p ( d i − d p ‾ ) 2 ) − 1 2 · 2 N p − 1 · ∑ i = a p e p ( ( d i − d p ‾ ) · ∂ ∂ d j ( d i − d p ‾ ) ) = 1 2 · σ p − 1 · 2 N p − 1 · ( ( d j − d p ‾ ) · ( 1 − 1 N p ) + ∑ i = a p , i ≠ j e p ( d i − d p ‾ ) · ( − 1 N p ) ) = 1 2 · σ p − 1 · 2 N p − 1 · ( ( d j − d p ‾ ) − 1 N p ∑ i = a p e p ( d i − d p ‾ ) ) = 1 2 · σ p − 1 · 2 N p − 1 · ( d j − d p ‾ − 1 N p ( ∑ i = a p e p d i − N p d p ‾ ) ) = 1 2 · σ p − 1 · 2 N p − 1 · ( d j − d p ‾ − 1 N p ∑ i = a p e p d i + N p N p d p ‾ ) = 1 2 · σ p − 1 · 2 N p − 1 · ( d j − d p ‾ − d p ‾ + d p ‾ ) = 1 2 · σ p · 2 N p − 1 · ( d j − d p ‾ ) = d j − d p ‾ ( N p − 1 ) · σ p

A.2 Skewness

∂ v p denom ∂ d j = 3 N p · ∑ i = a p e p ( ( d i − d p ‾ ) 2 · ∂ ∂ d j ( d i − d p ‾ ) ) = 3 N p · ( ( d j − d p ‾ ) 2 · ( 1 − 1 N p ) + ∑ i = a p , i ≠ j e p ( d i − d p ‾ ) 2 · ( − 1 N p ) ) = 3 N p · ( ( d j − d p ‾ ) 2 − 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 )

∂ v p nom ∂ d j = 3 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 2 · 2 N p · ∑ i = a p e p ( ( d i − d p ‾ ) 1 · ∂ ∂ d j ( d i − d p ‾ ) ) = 3 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 2 · 2 N p · ( ( d j − d p ‾ ) · ( 1 − 1 N p ) + ∑ i = a p , i ≠ j e p ( d i − d p ‾ ) · ( − 1 N p ) ) = 3 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 2 · 2 N p · ( ( d j − d p ‾ ) − 1 N p ∑ i = a p e p ( d i − d p ‾ ) ) = 3 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 2 · 2 N p · ( d j − d p ‾ − 1 N p ∑ i = a p e p d i + N p N p · d p ‾ ) = 3 N p · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 2 · ( d j − d p ‾ )

A.3 Kurtosis

∂ w p denom ∂ d j = 4 N p · ∑ i = a p e p ( ( d i − d p ‾ ) 3 · ∂ ∂ d j ( d i − d p ‾ ) ) = 4 N p · ( ( d j − d p ‾ ) 3 · ( 1 − 1 N p ) + ∑ i = a p , i ≠ j e p ( d i − d p ‾ ) 3 · ( − 1 N p ) ) = 4 N p · ( ( d j − d p ‾ ) 3 − 1 N p ∑ i = a p e p ( d i − d p ‾ ) 3 )

∂ w p nom ∂ d j = 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) 1 · 2 N p · ∑ i = a p e p ( ( d i − d p ‾ ) 1 · ∂ ∂ d j ( d i − d p ‾ ) ) = 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) · 2 N p · ( ( d j − d p ‾ ) · ( 1 − 1 N p ) + ∑ i = a p , i ≠ j e p ( d i − d p ‾ ) · ( − 1 N p ) ) = 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) · 2 N p · ( ( d j − d p ‾ ) − 1 N p ∑ i = a p e p ( d i − d p ‾ ) ) = 2 · ( 1 N p ∑ i = a p e p ( d i − d p ‾ ) 2 ) · 2 N p · ( d j − d p ‾ − 1 N p ∑ i = a p e p d i + N p N p · d p ‾ ) = 4 N p 2 · ( ∑ i = a p e p ( d i − d p ‾ ) 2 ) · ( d j − d p ‾ )

A.4 Covariance matrix

U q = J α , β , γ , δ q · U c · ( J α , β , γ , δ q ) ⊤ = A B Γ Δ · U c · A ⊤ , B ⊤ , Γ ⊤ , Δ ⊤ = A B Γ Δ · U c A ⊤ , U c B ⊤ , U c Γ ⊤ , U c Δ ⊤ = A U c A ⊤ A U c B ⊤ A U c Γ ⊤ A U c Δ ⊤ B U c A ⊤ B U c B ⊤ B U c Γ ⊤ B U c Δ ⊤ Γ U c A ⊤ Γ U c B ⊤ Γ U c Γ ⊤ Γ U c Δ ⊤ Δ U c A ⊤ Δ U c B ⊤ Δ U c Γ ⊤ Δ U c Δ ⊤ = A U c A ⊤ A U c B ⊤ A U c Γ ⊤ A U c Δ ⊤ ( A U c B ⊤ ) ⊤ B U c B ⊤ B U c Γ ⊤ B U c Δ ⊤ ( A U c Γ ⊤ ) ⊤ ( B U c Γ ⊤ ) ⊤ Γ U c Γ ⊤ Γ U c Δ ⊤ ( A U c Δ ⊤ ) ⊤ ( B U c Δ ⊤ ) ⊤ ( Γ U c Δ ⊤ ) ⊤ Δ U c Δ ⊤

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Received: 2022-03-29
Accepted: 2022-09-13
Published Online: 2022-09-22
Published in Print: 2023-03-28

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. Measurement systems and sensors with cognitive features II
  4. Research Articles
  5. Uncertainty-aware automated machine learning toolbox
  6. Explainable AI for sensor-based sorting systems
  7. Review Article
  8. Perspectives on AI-driven systems for multiple sensor data fusion
  9. Research Articles
  10. A collection of machine learning assisted distributed fiber optic sensors for infrastructure monitoring
  11. Automated end-of-line quality assurance with visual inspection and convolutional neural networks
  12. Review Article
  13. Automotive mass production of camera systems: Linking image quality to AI performance
https://doi.org/10.1515/teme-2022-0042
Schlagwörter für diesen Artikel
Measurement uncertainty; uncertainty propagation; statistical moments; linear discriminant analysis; machine learning

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. Measurement systems and sensors with cognitive features II
  4. Research Articles
  5. Uncertainty-aware automated machine learning toolbox
  6. Explainable AI for sensor-based sorting systems
  7. Review Article
  8. Perspectives on AI-driven systems for multiple sensor data fusion
  9. Research Articles
  10. A collection of machine learning assisted distributed fiber optic sensors for infrastructure monitoring
  11. Automated end-of-line quality assurance with visual inspection and convolutional neural networks
  12. Review Article
  13. Automotive mass production of camera systems: Linking image quality to AI performance
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