Bayesian Sensor Calibration

A hands-on tutorial in Python for sensor engineers

Bayesian sensor calibration is an emerging technique combining statistical models and data to optimally calibrate sensors - a crucial engineering procedure. This tutorial provides the Python code to perform such calibration numerically using existing libraries with a minimal math background. As an example case study, we consider a magnetic field sensor whose sensitivity drifts with temperature.

Magnetic field sensor

Glossary

The bolded terms are defined in the International Vocabulary of Metrology (known as the VIM definitions).

Code availability

An executable Jupyter notebook for the tutorial is available on Github. It can be accessed via nbviewer.

Introduction

Physical sensors provide the primary inputs that enable systems to make sense of their environment. They measure physical quantities such as temperature, electric current, power, speed, or light intensity. A measurement result is an estimate of the true value of the measured quantity (the so-called measurand). Sensors are never perfect. Non-idealities, part-to-part variations, and random noise all contribute to the sensor error. Sensor calibration and the subsequent adjustment are critical steps to minimize sensor measurement uncertainty. The Bayesian approach provides a mathematical framework to represent uncertainty.

Objective

In this article, we aim to reproduce a simplified case inspired by (Berger 2002) and illustrated in the figure below. A sensor is intended to measure the current flowing through a wire via the measurement of the magnetic field B which is directly proportional to the current. We focus on the magnetic sensor and consider the following non-idealities. (1) The temperature Bis a parasitic influence quantity disturbing the measurement. (2) The sensor response varies from one sensor to another due to part-to-part manufacturing variations. (3) The sensed data is polluted by random errors. Using a Bayesian approach and the high-level PyMC Python library, we seek to calculate the optimum set of calibration parameters for a given calibration dataset.

Bayesian approach

Mathematical formulation

We assume that the magnetic sensor consists of a magnetic and temperature transducer, which can be modeled as polynomial transfer functions with coefficients varying from one sensor to the next according to normal probability distributions.

Python formulation

We use the code convention introduced in (Close, 2021). We define a data dictionary dd to store the nominal value of the parameters. In addition, we define the probability density functions capturing the variability of the parameters.

Classic approaches

We first consider two well-known classic calibration approaches that don’t rely on the Bayesian framework.

Full regression

The first calibration approach is a brute-force approach. A comprehensive dataset is collected for each sensor, with more calibration points than unknowns. The calibration parameters w for each sensor (4 unknowns) are determined by a regression fit.

Blind calibration

A blind calibration represents the other extreme. In this approach, a first reference set of sensors is fully calibrated as above. The following sensors are not individually calibrated. Instead, the average calibration parameters w0 of the reference set is reused blindly.

Bayesian calibration

The Bayesian calibration is an interesting trade-off between the previous two extremes. A reference set of sensors is fully calibrated as above. The calibration parameters for this reference set constitute some prior knowledge. The average w0 and the covariance matrix encode some relevant knowledge about the sensor response.

Conclusion

This article presented a Python workflow for simulating Bayesian sensor calibration, and compared it against well-known classic approaches. The mathematical and Python formulations are valid for a wide class of sensors, enabling sensor design to explore various approaches.

Sensor design