标题： 多分形扩散熵分析：概率直方图的最佳分箱宽度 显示英文标题

标题： Multifractal Diffusion Entropy Analysis: Optimal Bin Width of Probability Histograms

摘要： 在多分形扩散熵分析的框架下，我们提出了一种方法，用于选择从感兴趣的底层概率分布生成的直方图中的最优区间宽度。 所提出的方法使用了Rényi熵和均方误差分析的技术，讨论了在多分形谱估计误差最小的条件。 我们通过关注金融时间序列的标度行为来说明我们方法的实用性。 特别是，我们分析了在1950-2013时间段内以每日频率采样的S&P500股票指数。 为了展示所提出方法的优势，我们比较了不同区间宽度下的多分形$\delta$-谱，并展示了该方法的稳健性，尤其是在$q$的较大值情况下。 对于这些值，目前使用的其他方法，例如基于矩估计的方法，在重尾数据或具有长相关性的数据中往往会失效。 还讨论并阐明了$\delta$-谱与Rényi的$q$参数之间的关系，并通过一个多尺度时间序列的简单示例进行了说明。 显示英文摘要

摘要： In the framework of Multifractal Diffusion Entropy Analysis we propose a method for choosing an optimal bin-width in histograms generated from underlying probability distributions of interest. The method presented uses techniques of R\'{e}nyi's entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal. We illustrate the utility of our approach by focusing on a scaling behavior of financial time series. In particular, we analyze the S&P500 stock index as sampled at a daily rate in the time period 1950-2013. In order to demonstrate a strength of the method proposed we compare the multifractal $\delta$-spectrum for various bin-widths and show the robustness of the method, especially for large values of $q$. For such values, other methods in use, e.g., those based on moment estimation, tend to fail for heavy-tailed data or data with long correlations. Connection between the $\delta$-spectrum and R\'{e}nyi's $q$ parameter is also discussed and elucidated on a simple example of multiscale time series.