اظهار نظر در مورد 'تجزیه و تحلیل آنتروپی انتشار مقیاس چندگانه بر روی نوسانات سهام در بازارهای مالی

In their recent article ‘multifractal diffusion entropy analysis on stock volatility in financial markets’ Huang, Shang and Zhao (2012) [6] suggested a generalization of the diffusion entropy analysis method with the main goal of being able to reveal scaling exponents for multifractal times series. The main idea seems to be replacing the Shannon entropy by the Rényi entropy, which is a one-parametric family of entropies. The authors claim that based on their method they are able to separate long range and short correlations of financial market multifractal time series. In this comment I show that the suggested new method does not bring much valuable information in obtaining the correct scaling for a multifractal/mono-fractal process beyond the original diffusion entropy analysis method. I also argue that the mathematical properties of the multifractal diffusion entropy analysis should be carefully explored to avoid possible numerical artefacts when implementing the method in analysis of real sequences of data.

Revealing scale invariant properties of complex time series is an important area of research with hundreds of publications emerging every year. In 2002, N. Scafetta and colleagues have developed an interesting method of determining the scaling exponent of stochastic process based on computation of the diffusion entropy. An important particularity of the diffusion entropy analysis (DEA) method is that it allows us to obtain the correct scaling even in the case of strange kinetics, where the second moment is diverging [1] and [2]. On the other hand, it is well known, that in many cases the real-world time series show a multifractal scaling behaviour [3], [4] and [5], which cannot be described by a single scaling exponent. Recently, Huang and colleagues [6] have elaborated a technique based on extension of the initial DEA with the aim of being able to reveal multi scaling exponents both for long and short ranged correlations. This idea seems to be inspired by the method of multifractal detrended fluctuation analysis which is an extension of the classical detrended fluctuation analysis with weighting of detrended standard deviations according to their magnitudes [3]. The main idea of Huang and colleagues [6] is now to replace the Shannon entropy with the Rényi entropy when scaling the probability density function of diffusion process constructed using time series. The Rényi entropy is actually a one-parameter family of entropies defined in the continuous case by equation(1) View the MathML sourceS(q)=11−qln∫[p(x)]qdx, Turn MathJax on where qq has the meaning of a weight of different probabilities [7] (similarly, one can define the Rényi entropy in a discrete case); for q=1q=1 we obtain the Shannon entropy. The authors allow the parameter qq to vary within a large region (e.g. −1<q<4−1<q<4) and they suggest that the new method should potentially provide information about the correct scaling behavior across various scales. In other words, the authors claim that if the probability density of a diffusion process admits scaling, we should have equation(2) S(t,q)∝δ(q)log(t),S(t,q)∝δ(q)log(t), Turn MathJax on where tt is the diffusion time; the scaling exponent can now depend on qq. Based on the introduced method, they further try to reveal short term correlations between very small and extremely large oscillations of volatility of some financial time series. They call the new method the ‘multifractal diffusion entropy analysis method’, which I will refer to as MF DEA further for brevity. Below I will show that although the method suggested by Huang et al. might look sound at first glance, due to a number of conceptual drawbacks its implementation can be misleading. In particularly, some of the conclusions made by the authors regarding the volatility of the financial market may simply be artefacts of numerical simulations.

Summarizing the conceptual drawbacks listed in Section 2, one can conclude that the newly created method of multifractal diffusion entropy analysis (MF DEA) by Huang, Shang and Zhao needs to be carefully interpreted and its mathematical properties should be carefully understood to be able to provide the correct scaling for the case of a multifractal process. This concerns both artificial and real time series. In particular, the method might fail when comparing the difference between scaling of the tails and the central part of the underlying pdf, or when revealing correlations between extremely large and small fluctuations. Overall, real-world time series of a different nature varying from the financial market indexes to biomedical signals are often multifractal [4], [5] and [11]. Since scaling exponent δδ of the underlying pdf of the diffusion process constructed based on time series can be different from the Hurst exponent HH (even in the case of a finite standard deviation), we need to estimate both δδ and HH. The diffusion entropy analysis is currently the best method to estimate correctly the scaling exponent δδ and thus we definitely need to extend this method for the case of multifractal time series. However, it seems that this can hardly been done via using different parametric families of entropies as the Rényi entropy applied to the resultant pdf which includes different phases of fluctuations. In case the diffusion process exhibits different scaling exponents, this can be better addressed using some direct methods by reconstructing the underlying pdf. Another way to detect the scaling exponents of a multifractal process would be treating pdf separately for different phases of long time series (similarly to considering the local Holder exponents). Developing such a method would be of great importance for understanding the complex nature of multifractal time series.