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Likelihood-based Sensor Calibration using Affine Transformation

Rüdiger Machhamer, Lejla Begic Fazlic, Eray Guven, David Junk, Gunes Karabulut Kurt, Stefan Naumann, Stephan Didas, Klaus-Uwe Gollmer, Ralph Bergmann, Ingo J. Timm, Guido Dartmann

TL;DR

The work addresses calibrating measurements across identically designed sensors by estimating an affine transformation, enhanced by expert-provided correspondences. It develops three estimation strategies—augmented Gleser–Watson MLE, multivariate regression, and a hybrid approach—demonstrating that denoising via eigenvectors and combining methods improves accuracy. Through Monte Carlo simulations and a real 8-sensor case, it shows that the hybrid method often delivers the best or comparable accuracy with lower computation, and it articulates a pathway toward distributed, expert-guided learning in IoT sensor networks. The approach offers a practical, scalable framework for software-based sensor calibration and drift compensation.

Abstract

An important task in the field of sensor technology is the efficient implementation of adaptation procedures of measurements from one sensor to another sensor of identical design. One idea is to use the estimation of an affine transformation between different systems, which can be improved by the knowledge of experts. This paper presents an improved solution from Glacier Research that was published back in 1973. The results demonstrate the adaptability of this solution for various applications, including software calibration of sensors, implementation of expert-based adaptation, and paving the way for future advancements such as distributed learning methods. One idea here is to use the knowledge of experts for estimating an affine transformation between different systems. We evaluate our research with simulations and also with real measured data of a multi-sensor board with 8 identical sensors. Both data set and evaluation script are provided for download. The results show an improvement for both the simulation and the experiments with real data.

Likelihood-based Sensor Calibration using Affine Transformation

TL;DR

The work addresses calibrating measurements across identically designed sensors by estimating an affine transformation, enhanced by expert-provided correspondences. It develops three estimation strategies—augmented Gleser–Watson MLE, multivariate regression, and a hybrid approach—demonstrating that denoising via eigenvectors and combining methods improves accuracy. Through Monte Carlo simulations and a real 8-sensor case, it shows that the hybrid method often delivers the best or comparable accuracy with lower computation, and it articulates a pathway toward distributed, expert-guided learning in IoT sensor networks. The approach offers a practical, scalable framework for software-based sensor calibration and drift compensation.

Abstract

An important task in the field of sensor technology is the efficient implementation of adaptation procedures of measurements from one sensor to another sensor of identical design. One idea is to use the estimation of an affine transformation between different systems, which can be improved by the knowledge of experts. This paper presents an improved solution from Glacier Research that was published back in 1973. The results demonstrate the adaptability of this solution for various applications, including software calibration of sensors, implementation of expert-based adaptation, and paving the way for future advancements such as distributed learning methods. One idea here is to use the knowledge of experts for estimating an affine transformation between different systems. We evaluate our research with simulations and also with real measured data of a multi-sensor board with 8 identical sensors. Both data set and evaluation script are provided for download. The results show an improvement for both the simulation and the experiments with real data.
Paper Structure (13 sections, 3 theorems, 28 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 13 sections, 3 theorems, 28 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. System Model
  4. Algorithms
  5. Solution proposed by Gleser and Watson
  6. Multivariate Regression Approach
  7. Hybrid Approach
  8. Results
  9. Monte Carlo Simulations
  10. Eigenvector Denoising by Augmentation
  11. Case Study: 8 Sensor Board Scenario
  12. Discussion and Outlook
  13. Conclusion

Key Result

Lemma 1

Let $\boldsymbol{\lambda}$ be the diagonal matrix of the $p$ largest eigenvalues and $\mathbf{U}$ be the matrix of all corresponding eigenvectors of $\mathbf{X}^T\mathbf{X}+\mathbf{Y}^T\mathbf{Y}$, then the estimates of $\boldsymbol{\Theta}$ and $\mathbf{B}$ are given by (eigenTheta) and (eigenB):

Figures (8)

  • Figure 1: Concept of expert-based learning. Some model instances from sensor 1 are matched by experts with their representatives in sensor 2, the remaining instances from sensor 1 are calculated using the estimated AT, or vice versa.
  • Figure 2: Comparison of (a) raw and (b) feature-wise normalized data.
  • Figure 3: Multisensor board of the Bosch BME688 development kit, which provides 8 identical constructed sensors Bib:Bosch.
  • Figure 4: Simulated data of $\mathbf{X}$, $\boldsymbol{\Theta}$, $\mathbf{Y}$ and $\hat{\mathbf{Y}}$. The $x$-axis denotes the first component of a data vector $\mathbf{v}$ and the $y$-axis the second component.
  • Figure 5: Simulation result, mean errors $\bar{e}_{x}$ and $\bar{e}_{y}$ for the augmented implementation of Gleser and Watson compared to the augmented Alg. \ref{['Alg1']} as a function of the standard deviation $\sigma$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • proof