Bayesian Sensor Calibration
Moritz Berger, Christian Schott, Oliver Paul
Abstract
The calibration of multisensor systems can cause significant costs in terms of time and resources, in particular when cross-sensitivities to parasitic influences are to be compensated. Successful calibration ensures the trustworthy subsequent operation of a sensor system, guaranteeing that one or several measurands of interest can be inferred from its output signals with specified uncertainty. As shown in the present study, this goal can be reached by reduced calibration procedures with fewer calibration conditions than parameters that are needed to model the device response. This is achieved using Bayesian inference by combining the calibration data of a sensor system with statistical prior information about the ensemble to which it belongs. Optimal reduced sets of calibration conditions are identified by the method of Bayesian experimental design. The method is demonstrated on a Hall–temperature sensor system whose nonlinear response model requires seven parameters in the temperature range between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {-}30$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$150 ^{\circ} \text{C}$ </tex-math></inline-formula> and for magnetic field values <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${B}$ </tex-math></inline-formula> between −25 and 25 mT. For the prior, a multivariate normal distribution of the model parameters is acquired using 14 specimens of the sensor ensemble. I-optimal calibration at one, two, and three temperatures reduces the root-mean-square (rms) standard deviation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${B}$ </tex-math></inline-formula> inferred from sensor output signals from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$203 \boldsymbol {\mu } \text{T}$ </tex-math></inline-formula> before calibration down to 78, 41, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$34 \boldsymbol {\mu }\text{T}$ </tex-math></inline-formula> . Similar conclusions apply to G-optimal calibration. This article describes how to implement the Bayesian prior acquisition, inference, and experimental design. The proposed approach can help save resources and cut costs in sensor calibration.