Model Complexity Reduction in Bayesian Sensor Calibration and Its Relation to Principal Component Analysis

Calibration is a costly, necessary part of sensor manufacturing. Bayesian sensor calibration leverages prior knowledge in the form of a sample of sensor response functions from a sensor ensemble such as a fabrication lot. Based on a response model with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> parameters, it allows to infer the measurand of interest and its predictive uncertainty from output signals of specimens of the ensemble after their lean calibration with a small number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> of measurements, possibly with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}\lt {M}$ </tex-math></inline-formula>. This article answers the question of whether it is possible to reduce model complexity by choosing models with parameter number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widetilde {\textit {M}}\lt {M}$ </tex-math></inline-formula>, while guaranteeing a required predictive accuracy. Reduced models are obtained by projections of a high-dimensional model, termed full model, onto a subspace of its parameter space. An I-optimality-based loss function implementing the Bayesian calibration approach is derived. For given <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widetilde {\textit {M}}$ </tex-math></inline-formula>, it allows to find an optimal reduced model and, at the same time, the optimal experimental design of the calibration. The approach is applied to 48 specimens of a CMOS Hall sensor system cross-sensitive to temperature and mechanical stress. Starting from an 11-parameter full model, reduced models with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widetilde {\textit {M}}={2},{4}$ </tex-math></inline-formula>, and 5 are identified for calibration routines with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}={2},{4}$ </tex-math></inline-formula>, and 6 measurements, respectively. Despite the significantly smaller parameter numbers, the resulting root mean square (rms) predictive uncertainties of 102, 77.0, and 64.8 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>T, respectively, are increased by less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${1}\%$ </tex-math></inline-formula> from the full-model values. A general conclusion is that model order reduction (MOR) in the present Bayesian framework invariably entails an uncertainty increase, similar to principal component analysis (PCA).