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Abstract：
Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point\'s position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor\'s point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance.
摘要：
熵通过表示关于相点在吸引子上的位置的平均信息损失率来作为系统中混沌的度量。当处理多重分形系统时，单个指数不能完全描述它的动态，需要连续的指数谱，称为奇点光谱。从投资者的角度来看，熵的上升是异常回报的信号，可能是负回报。这意味着他必须期待意想不到的事情并为此做好准备。为了探索这个，我们分析纽约证券交易所(NYSE)美国指数及其成分股。通过这次考试,我们评估它们的多重分形特征，并使用熵来识别市场条件(看跌/看涨市场)，一种识别波动分形市场的有效方法。我们的发现挑战了传统的信念，证明价格下跌会导致熵增加，与文献中的一些研究相反，这些研究表明，市场危机中熵的减少意味着更多的确定性。相反，我们认为熊市可能会表现出更高的熵，表明意外极端事件的可能性更大。此外，我们的研究揭示了幂律行为，并表明不存在方差。
