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

Expecting the Unexpected: Entropy and Multifractal Systems in Finance

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.

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

The fractal dimension (D) of the active site of hAChE in the unliganded state and as part of complexes with hyperzine A, galantamine, and donepezil is calculated using molecular interatomic-distance histograms. Fractal matrices of structural changes (FMSCs) are formed by pairwise comparison of the values of D and by revealing the significance of their differences. FMSCs are found to be used to quantitatively estimate the changes in the structures of the molecules in various states. When analyzing FMSCs, we found that the most significant structural changes are related to the Glu202 amino acid residue. No structural perturbations are revealed in the case of Trp86, Gly122, Ala204, Phe338, Tyr341, Gly448, and Ile451.