Introduction of Error Estimation for Clarifying Fractal Phonomena

本研究的目的是通过提出误差估计技术，在关于分形现象的测量中引入可靠性的概念。1.在Mandelbrot提出的实现“自相似循环事件过程”的“分维”测量中，有两种方法被认为是重要的。这项研究通过这些方法得出了所得值的解析误差估计。结果表明，相对较少的数据足以获得理想工艺的精确值。提出了一种检验异方差时间序列数据随机性的新方法，并成功地应用于金融数据。这种方法检测到“均值回归”的存在，这是由长期记忆引起的分形的一个特征。这一结果还检测到了其他类型的序列相关性，这从金融角度来看是重要的。在对“均值回归”机制的分析中，人们发现股市指数在宏观经济分析中的表现可能有很大的不同。在他们的分析中，我们必须谨慎地选择它们。对于布朗运动路径的多边形近似，对观测“长度”的误差进行了解析计算。这一结果有望说明随着采样点数量的增加，观测到的“分维”的偏差如何收敛到真值。研究发现，“数字直度”和“数字凸度”是用作者提出的“长度”和“绝对曲率”来表征的。这一事实表明，这种方法中使用的“数字几何”和“长度”之间的关系是非常密切的。

The purpose of this study is to introduce the concept of reliability in the measurement concerning fractal phenomena, by proposing techniques of error estimation. The following outcomes are obtained :1. In the measurement of the "fractal dimension" of a realization "self-similar recurrent event process" proposed by Mandelbrot, two approaches are considered to be important. This study derived analytical error estimations of the obtained values by these approaches. The results suggest that a relatively small number of data are enough for obtaining accurate values for an ideal process.2. A new approach for testing the randomness of heteroskedastic time series data is proposed, and is successfully applied to finacial data. This approach detects existence of "mean reversion, " which is a feature of fractal caused by long term memory. This result also detects other types of serial correlations, which is important from financial point of view.3. In the analysis of the mechanism of "mean reversion, " it was found that stock market indices may behave in substantially different ways in the macroeconomic analysis. One must be careful in selecting them in their analysis.4. The error of observed "length" is analytically calculated for a polygonal approximation to a path of Brownian motion. This result is expected to show how the bias of observed "fractal dimension" convergers to the true value when the number of sampling points increases.5. It was found that the "digital straightness" and "digital convexity" are characterized in terms of the "length" and the "absolute curvature" proposed in the author. This fact shows that a relation of "digital geometry" and the "length" used in this approach is very close.