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Initial Parameter Estimation for Non-Linear Optimization -- Trigonometric Function

Tilo Strutz

TL;DR

This technical report outlines a new strategy for selecting suitable initial parameters for a trigonometric model and unevenly sampled data, ensuring that the optimisation procedure starts sufficiently close to the global minimum.

Abstract

Nonlinear optimisation techniques are commonly employed to minimise complex cost functions, with their effectiveness determined largely by the structure of the underlying error landscape. These methods require initial parameter values, and in the presence of multiple local minima, they are prone to becoming trapped in suboptimal regions. The likelihood of locating the global minimum increases substantially when the initialisation lies within its corresponding basin of attraction. Consequently, high-quality initial parameters are critical for successful optimisation. This technical report outlines a new strategy for selecting suitable initial parameters for a trigonometric model and unevenly sampled data, ensuring that the optimisation procedure starts sufficiently close to the global minimum. The proposed parameter estimation approach is strictly NI-based, interpretable, and explainable. It targets at complicated cases which include: samples with strong random noise, samples with only few covered periods, and samples which cover only a fraction of one period. Special attention is put on the frequency estimation. It can be shown that an estimation of initial parameters with sufficient accuracy is possible down to a signal-noise-ratio of 1.4 dB at much lower computational costs than the Lomb-Scargle-periodogram method requires.

Initial Parameter Estimation for Non-Linear Optimization -- Trigonometric Function

TL;DR

This technical report outlines a new strategy for selecting suitable initial parameters for a trigonometric model and unevenly sampled data, ensuring that the optimisation procedure starts sufficiently close to the global minimum.

Abstract

Nonlinear optimisation techniques are commonly employed to minimise complex cost functions, with their effectiveness determined largely by the structure of the underlying error landscape. These methods require initial parameter values, and in the presence of multiple local minima, they are prone to becoming trapped in suboptimal regions. The likelihood of locating the global minimum increases substantially when the initialisation lies within its corresponding basin of attraction. Consequently, high-quality initial parameters are critical for successful optimisation. This technical report outlines a new strategy for selecting suitable initial parameters for a trigonometric model and unevenly sampled data, ensuring that the optimisation procedure starts sufficiently close to the global minimum. The proposed parameter estimation approach is strictly NI-based, interpretable, and explainable. It targets at complicated cases which include: samples with strong random noise, samples with only few covered periods, and samples which cover only a fraction of one period. Special attention is put on the frequency estimation. It can be shown that an estimation of initial parameters with sufficient accuracy is possible down to a signal-noise-ratio of 1.4 dB at much lower computational costs than the Lomb-Scargle-periodogram method requires.
Paper Structure (25 sections, 21 equations, 13 figures, 7 tables, 8 algorithms)

This paper contains 25 sections, 21 equations, 13 figures, 7 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Initial parameter estimates for trigonometric functions
  3. Estimating the offset $a_1$ and the amplitude $a_2$
  4. Frequency estimation
  5. Analysis and presumptions regarding the signal properties
  6. Proposed determination of the frequency parameter $a_3$
  7. Determination of a reference distance:
  8. Derivation of the typical distance:
  9. Refinement of the typical distance:
  10. Final estimate:
  11. Case of a single crossing:
  12. Final remarks:
  13. Estimation of the phase shift
  14. Example
  15. Validation of initial parameter estimation
  16. ...and 10 more sections

Figures (13)

  • Figure 1: Error landscape $\chi^2(\mathbf{a})$ of a trigonometric function as a function of frequency and phase (offset $a_1$ and amplitude $a_2$ are kept constant)
  • Figure 2: Example of a trigonometric function and corresponding erroneous observations
  • Figure 3: Example of a trigonometric function with dense sampling and strong noise. The magnified section shows how the noisy curve crosses the centre line $a_1\approx 10.5$ several times (spurious crossings).
  • Figure 4: Example of a trigonometric function with two different noise levels $\sigma$
  • Figure 5: Sorted values dependent on the noise level (standard deviation): (a) distances between crossings, (b) corresponding averaged absolute deviation of observation values $y_i$ from estimated mean $\hat{a}_1$
  • ...and 8 more figures