Hurst entropy: A method to determine predictability in a binary series based on a fractal-related process

Area of Science:

• Information theory
• Time series analysis
• Statistical modeling

Background:

• Shannon's information entropy quantifies predictability based on event frequency.
• Existing entropy measures do not account for autocorrelation, a measure of memory in time series.
• Binary time series can exhibit short- and long-term memory, influencing their predictability.

Purpose of the Study:

• To establish a mathematical connection between information entropy and autocorrelation in binary time series.
• To investigate how information entropy is affected by the frequency of 0s and 1s and the Hurst exponent.
• To determine the relationship between predictability, information entropy, and autocorrelation.

Main Methods:

• Numerical simulations were employed to model binary time series.
• An analytical approach was used to derive mathematical relationships.
• The Hurst exponent was calculated to quantify autocorrelation.

Main Results:

• Information entropy was found to be dependent on both the frequency of 0s and 1s and the Hurst exponent.
• A method was developed to quantify how predictability changes with these parameters.
• The study successfully linked information entropy and autocorrelation.

Conclusions:

• Predictability in binary time series is influenced by both event frequency and memory effects (autocorrelation).
• The findings provide a more comprehensive understanding of information entropy in the presence of autocorrelation.
• This research has implications for fields utilizing binary time series, including neuroscience and econophysics.