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Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
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Applying Monte Carlo Method for Straight-Line Model Sensor Calibration.

Pedro M Ramos1, Fernando M Janeiro2

  • 1Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal.

Sensors (Basel, Switzerland)
|May 13, 2026
PubMed
Summary
This summary is machine-generated.

This study reviews methods for sensor calibration, focusing on linear regression when both input and output measurements have uncertainties. The Monte Carlo Method (MCM) is highlighted for its ability to handle complex uncertainties and optimize calibration measurements.

Keywords:
Monte Carlo methodcalibrationmeasurement uncertaintysensor straight-line modeluncertainty evaluationweighted total least squares regression

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Area of Science:

  • Measurement Science
  • Metrology
  • Instrumentation Engineering

Background:

  • Sensor calibration is crucial for accurate physical parameter estimation in measurement systems.
  • Traditional linear regression (least squares) fails to account for uncertainties in input measurements.
  • Accurate calibration requires methods that consider uncertainties in both sensor inputs and outputs.

Purpose of the Study:

  • To review and present methods for estimating straight-line model parameters under conditions of uncertainty in both input and output measurements.
  • To explore the application of the Monte Carlo Method (MCM) for sensor calibration, particularly in heteroscedastic cases.
  • To propose an MCM-based strategy for optimizing measurement input selection to minimize slope uncertainty.

Main Methods:

  • Review of existing methods for linear regression with input and output uncertainties.
  • Application of the Monte Carlo Method (MCM) to model uncertainties and covariances in sensor measurements.
  • Development of an MCM-based strategy for optimizing the selection of input values in heteroscedastic calibration scenarios.

Main Results:

  • The Monte Carlo Method (MCM) effectively handles various measurement uncertainties and covariances.
  • An MCM-based strategy was proposed and demonstrated to optimize measurement input selection.
  • The proposed strategy significantly reduced the number of required measurements in the presented case study.

Conclusions:

  • The Monte Carlo Method (MCM) offers a robust approach for sensor calibration when uncertainties exist in both input and output data.
  • Optimizing measurement input selection using MCM can substantially improve calibration efficiency.
  • This work provides a framework for more accurate and efficient sensor calibration, especially in complex measurement environments.