Fractal and Entropy Analysis of the Dow Jones Index Using Multidimensional Scaling

Area of Science:

• Quantitative Finance
• Complex Systems Analysis
• Time Series Modeling

Background:

• Financial time series exhibit complex dynamics challenging traditional analysis.
• The Dow Jones Industrial Average (DJIA) is a key financial index for studying market behavior.
• Fractal properties are increasingly recognized in financial markets.

Purpose of the Study:

• To explore an alternative dynamical characterization of financial time series.
• To investigate the fractal nature of the Dow Jones Industrial Average (DJIA).
• To integrate multidimensional scaling (MDS) with concepts of distance, entropy, and fractional calculus.

Main Methods:

• Utilized multidimensional scaling (MDS) as a computational tool.
• Applied various distance metrics to quantify similarities in time series data.
• Incorporated Shannon entropy and fractal dimension for structural analysis.
• Employed fractional calculus, specifically fractional-order entropy, for advanced assessment.

Main Results:

• MDS generated complex representations of the DJIA time series, revealing underlying structures.
• Shannon entropy and fractal dimension analysis confirmed the intricate patterns.
• Fractional-order entropy highlighted the fractal nature and fractional order memory of the DJIA.
• The study demonstrated fractal properties across multiple time scales.

Conclusions:

• The integration of MDS, entropy, and fractional calculus offers a robust framework for financial time series analysis.
• The DJIA exhibits a persistent fractal nature with memory effects evident at different time scales.
• Fractional calculus provides deeper insights into the complex dynamics of financial markets.