Introduction of Error Estimation for Clarifying Fractal Phonomena

引入误差估计以澄清分形现象

• 批准号: 06650070
• 负责人: KISHIMOTO Kazuo
• 金额: $ 0.38万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06650070/
• 关键词: Fractal; Error Estimation; Length; Point Pattern; Fractal Dimension; Brownian Motion; Coastal Lines; Stock Price Changes; 曲率; Fractal; Error Estimation; Length; Point Pattern; Fractal Dimension; Brownian Motion; Coastal Lines; Stock Price Changes; 曲率

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.

这项研究的目的是通过提出误差估计技术来介绍有关分形现象的可靠性概念。获得以下结果：1。在测量Mandelbrot提出的实现“实现的“分形维度””中，两种方法被认为很重要。这项研究得出了通过这些方法对获得值的分析误差估计。结果表明，相对较少的数据足以获得理想过程的准确值2。提出了一种测试异质时间序列数据随机性的新方法，并成功地应用于最佳数据。这种方法检测到“平均恢复”的存在，这是由长期记忆引起的分形特征。该结果还检测到其他类型的串行相关性，从财务角度来看，这很重要3。在对“平均恢复”机制的分析中，发现股票市场指数在宏观经济分析中的行为可能会大不相同。必须在分析中选择它们。4。分析观察到的“长度”的误差是针对布朗运动路径的多边形近似值计算的。预计该结果将显示观察到的“分形维度”收敛器的偏置如何在抽样点增加时与真实值。5。已经发现，“数字笔直度”和“数字凸度”的特征是作者提出的“长度”和“绝对曲率”。这一事实表明，在这种方法中使用的“数字几何”和“长度”的关系非常接近。