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

Abstract

Financial time series have a fractal nature that poses challenges for their dynamical characterization. The Dow Jones Industrial Average (DJIA) is one of the most influential financial indices, and due to its importance, it is adopted as a test bed for this study. The paper explores an alternative strategy to the standard time analysis, by joining the multidimensional scaling (MDS) computational tool and the concepts of distance, entropy, fractal dimension, and fractional calculus. First, several distances are considered to measure the similarities between objects under study and to yield proper input information to the MDS. Then, the MDS constructs a representation based on the similarity of the objects, where time can be viewed as a parametric variable. The resulting plots show a complex structure that is further analyzed with the Shannon entropy and fractal dimension. In a final step, a deeper and more detailed assessment is achieved by associating the concepts of fractional calculus and entropy. Indeed, the fractional-order entropy highlights the results obtained by the other tools, namely that the DJIA fractal nature is visible at different time scales with a fractional order memory that permeates the time series.

Expecting the Unexpected: Entropy and Multifractal Systems in Finance

“…Mandelbrot et al [33] suggested fractional Brownian motions [34,35] as applicable to natural phenomena and finance. Machado et al [36] empirically confirmed a strong correlation between the entropy and the value of the fractional order.…”

Structural Fractal Analysis of the Active Site of Acetylcholinesterase in Complexes with Huperzine A, Galantamine, and Donepezil

“…The methods of fractal geometry are now widely used in scientific research. The key quantitative characteristic of fractal objects is fractal dimension D. Therefore, a significant number of modern works are devoted to the calculation and analysis of this quantity in various fields of knowledge, including physics, [22,23] geophysics, [24,25] chemistry, [26,27] biology, [28] medicine, [29,30] economics, [31,32] materials science, [33] and computer science. [34,35] Another important characteristic of fractals is Hurst exponent H, which is used as the measure of a long-range dependence in time and spatial series and changes from 0 to 1.…”

Fractional Calculus and the Future of Science

“…Machado [ 3 ] chose to explore the fractal nature of financial time series using the Dow Jones industrial average (DJIA) but avoided using standard time series analysis. Instead, he uses multidimensional scaling (MDS) together with the concepts of distance, entropy, fractal dimension, and the FC.…”