Machado José A Tenreiro
Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, 4249-015 Porto, Portugal.
Entropy (Basel). 2020 Oct 8;22(10):1138. doi: 10.3390/e22101138.
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.
金融时间序列具有分形性质，这给它们的动态表征带来了挑战。道琼斯工业平均指数（DJIA）是最具影响力的金融指数之一，由于其重要性，它被用作本研究的试验台。本文通过结合多维缩放（MDS）计算工具以及距离、熵、分形维数和分数阶微积分的概念，探索了一种替代标准时间分析的策略。首先，考虑几种距离来测量所研究对象之间的相似性，并为MDS提供适当的输入信息。然后，MDS根据对象的相似性构建一种表示，其中时间可以被视为一个参数变量。所得的图显示出一种复杂的结构，通过香农熵和分形维数对其进行进一步分析。在最后一步中，通过将分数阶微积分和熵的概念联系起来，实现了更深入、更详细的评估。实际上，分数阶熵突出了其他工具所获得的结果，即DJIA的分形性质在不同时间尺度上是可见的，并且具有贯穿时间序列的分数阶记忆。
