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Analyzing Systematic Errors and Biases in Measurement: A Comprehensive Review of Least Squares

This article explores the intricate relationship between systematic errors, biases, and the method of least squares in measurement. It discusses the impact of true values and biases on least squares adjustments, emphasizing traceability, the Gauß-Markoff theorem, and uncertainty assessment. Furthermore, it examines empirical variance-covariance matrices, the mean of means, and the need for appropriate weightings to minimize uncertainties. Key comparisons and the calibration of standards reveal insights into aligning true values with measured data, shedding light on the challenges in metrology.

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Analyzing Systematic Errors and Biases in Measurement: A Comprehensive Review of Least Squares

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Presentation Transcript

  1. Unknown systematic errors and the method of least squares Michael Grabe alternative error model: true values and biases

  2. Quantity to be measuredtruevalue First Principle Does metrology exist without a net of true values? Not likely!

  3. Impact of true values and biases in least squares • Traceability • Gauß-Markoff theorem • Assessment of uncertainties • Key Comparisons

  4. Traceability Least squares adjustment

  5. Mean of means averaging is permitted if and only if the respective true values are identical

  6. mass Adjustment ad hoc ?

  7. A different approach empirical variance-covariance matrix

  8. Mean of means

  9. Let the input data be arithmetic means

  10. Gauß-Markoff Theorem The uncertainties are minimal... ...if the system has been weighted appropriately

  11. according to the GUM we should have but we encounter biases abolish the theorem ...

  12. Consequences ... • no more test of consistency • how to weight the system to minimize uncertainties?

  13. more ... and of utmost importance: weightings • shift estimators and • reduce measurement uncertainties

  14. true value reduction shift before after a picture

  15. Traceability: vary the weights by trial and error ... Assessment of uncertainties

  16. true value true value ... true value ... Key Comparison National Standards

  17. ... T ... Round Robin Calibration of a Travelling Standard T

  18. Consider the grand mean KCRV and the differences Key comparisons do more ...

  19. and look for where „consistent“ with

  20. against Should we test when constributesto ? Problem: In some cases the GUM may localize the true value of the travelling standard, in others not ...

  21. true value KCRV Differences between KCRV and individual calibrations ...

  22. ... true value Individual Calibrations a horizontal line should intersect each of the uncertainties

  23. ... KCRV true value KCRV and individual calibrations

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