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

José A Tenreiro Machado

doi:10.3390/e22101138

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.

Highlights

The Dow Jones Industrial Average (DJIA), or Dow Jones, is a stock market index that reflects the stock performance of 30 relevant companies included in the U.S stock exchanges
We find all sorts of financial indices for capturing the dynamics of markets and stock exchange institutions
We consider the fractal dimension and entropy measures for analyzing the 3-dim portraits produced by the multidimensional scaling (MDS)

Summary

Introduction

The Dow Jones Industrial Average (DJIA), or Dow Jones, is a stock market index that reflects the stock performance of 30 relevant companies included in the U.S stock exchanges. A number of techniques have been proposed to investigate the financial indices and to unravel the embedded complex dynamics [14–18] Such studies adopt the underlying concept of linear time flow and consider that the fractal nature of the index is intrinsic to its own artificial nature. The Gedankenexperiment in the follow-up addresses the controversy about the texture of time [19–22], but just in the limited scope of financial indices For this purpose, the concepts of multidimensional scaling (MDS), fractional dimension, entropy, and fractional calculus are brought up as useful tools to tackle complex systems. In the last few decades, FC became a popular tool for analyzing phenomena with long-range memory and non-locality [49–57] The association of these mathematical and computational tools yields relevant viewpoints when analyzing financial indices [7–9,11,58–61].

The DJIA Dataset

Distances

The MDS Loci

Data Pre-Processing Using P1

Data Pre-Processing Using P2

Conclusions