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This study reframes transfer operator inference using statistical density estimation, offering a new way to analyze bias and variance. Kernel density estimation (KDE) generally outperforms histogram density estimation (HDE) for accuracy in estimating Frobenius-Perron operator eigenvectors.

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Area of Science:

  • Dynamical Systems and Ergodic Theory
  • Statistical Inference
  • Numerical Analysis

Background:

  • Transfer operator inference is crucial for analyzing dynamical systems.
  • The Ulam method, specifically the Ulam-Galerkin approach, is a conventional technique.
  • This method can be viewed as density estimation using histograms.

Purpose of the Study:

  • To recast the inference of transfer operators within statistical density estimation.
  • To enable rigorous analysis of bias, variance, and mean square error.
  • To evaluate the performance of different density estimation techniques.

Main Methods:

  • Formulating transfer operator inference as a statistical density estimation problem.
  • Applying histogram density estimation (HDE) and kernel density estimation (KDE).
  • Analyzing bias-variance trade-offs and mean square error.

Main Results:

  • Kernel density estimation (KDE) generally shows higher accuracy than histogram density estimation (HDE).
  • KDE demonstrates limitations near boundary points and discontinuities.
  • The study validates the effectiveness of density estimation for Frobenius-Perron operator eigenvector estimation.

Conclusions:

  • Statistical density estimation provides a powerful framework for transfer operator inference.
  • KDE is a promising, though not perfect, method for this task.
  • Future research should explore other density estimation methods and high-dimensional applications.