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On precision limits for derivatives numerically calculated from noisy data

H Lanshammar

    Journal of Biomechanics
    |January 1, 1982
    PubMed
    Summary
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    This study verifies an error formula for calculating maximal precision in derivatives from noisy data and introduces the Relative Noise Amplification (RNA) index. The RNA helps assess differentiating filter quality, guiding selection for minimal noise in derivative calculations.

    Area of Science:

    • Numerical Analysis
    • Signal Processing
    • Data Science

    Background:

    • Calculating derivatives from noisy measurement data is crucial in many scientific and engineering fields.
    • Existing methods often struggle with noise amplification, leading to inaccurate derivative estimates.
    • A need exists for reliable methods to quantify and minimize noise in derivative calculations.

    Purpose of the Study:

    • To verify an error formula for predicting maximal precision in derivatives from noisy data.
    • To introduce and define the Relative Noise Amplification (RNA) as a quality index for differentiating filters.
    • To present bandwidth and RNA for local least squares polynomial fitting filters.

    Main Methods:

    • Verification of the error formula by comparing predicted noise with actual noise from local least squares polynomial fitting.

    Related Experiment Videos

  • Comparison with differentiation using Fourier series expansion to highlight noise exceeding minimum values.
  • Definition and calculation of the Relative Noise Amplification (RNA) for differentiating filters.

    • Main Results:

    • The error formula accurately predicts maximal precision in derivative calculations.
    • Differentiation via Fourier series expansion results in noise exceeding the calculated minimum.
    • The Relative Noise Amplification (RNA) index effectively quantifies filter performance, with lower RNA indicating less noise.
    • Analysis of 0th, 1st, and 2nd order derivatives for polynomials up to 8th order using local least squares fitting.

    Conclusions:

    • The verified error formula and the RNA index provide valuable tools for assessing and improving derivative calculations from noisy data.
    • Local least squares polynomial fitting offers a method for obtaining derivatives with controlled noise levels.
    • The presented results aid in selecting optimal filter parameters for various applications requiring accurate derivative estimation.