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Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient.

Katarzyna Maraj1, Dawid Szarek1, Grzegorz Sikora1

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This summary is machine-generated.

A new method using the time-averaged mean squared displacement (TAMSD) ratio effectively tests anomalous diffusion. This approach is independent of diffusion coefficients and outperforms traditional TAMSD methods, especially with limited data.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Financial Mathematics

Background:

  • Anomalous diffusion processes exhibit non-linear, power-law characteristics, differing from normal diffusion.
  • The time-averaged mean squared displacement (TAMSD) is a common statistic for analyzing diffusion, but its application can be limited.
  • Distinguishing between sub- and super-diffusive processes requires robust analytical tools.

Purpose of the Study:

  • To propose and validate a novel statistical approach for testing Gaussian anomalous diffusion models.
  • To develop a method that is independent of the diffusion coefficient, simplifying analysis.
  • To demonstrate the superiority of the new method over existing TAMSD-based approaches, particularly for small datasets.

Main Methods:

  • Development of a TAMSD ratio statistic based on different time lags.
  • Utilizing the quadratic form representation of the TAMSD ratio to calculate its characteristics.
  • Proposing a step-by-step testing procedure applicable to any Gaussian process.
  • Testing the methodology using the fractional Brownian motion model and real financial market data.

Main Results:

  • The TAMSD ratio statistic shows distinct behavior in the anomalous diffusion regime.
  • The proposed method is independent of the diffusion coefficient, removing a key limitation of the TAMSD approach.
  • The TAMSD ratio-based method demonstrates superior performance compared to the standard TAMSD method, especially for small sample sizes.
  • The methodology was successfully applied to real-world financial market data.

Conclusions:

  • The TAMSD ratio offers a more robust and versatile tool for analyzing anomalous diffusion.
  • This new approach simplifies the testing procedure by eliminating the need for prior knowledge of the diffusion coefficient.
  • The findings have significant implications for the analysis of complex systems, including financial markets.