分形分析的修正函数法

IF 0.2 Q4 ENGINEERING, ELECTRICAL & ELECTRONIC
Oleg V. Lazorenko, A. A. Onishchenko, L. Chernogor
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引用次数: 1

摘要

在许多分形分析方法中使用的主要数值特征之一是相应的分形维数。在绝大多数情况下,这些维度的估计精度相当小,这首先不能满足研究人员-实践者。本文提出了一种校正函数的方法,它可以补偿分形维数真实值与其估计之间的依赖关系一直存在的非线性,这种非线性是用信号和过程的单分形分析方法对已知数量的被研究信号的离散数据向量样本进行的。该方法的主要思想是利用一组具有已知分形维数的模型分形信号来建立和应用一个特殊的校正函数。概述了新方法的数学基础。以正则化、装箱、变分和赫斯特分维的评价为例,分析了修正函数法在实际应用中的特点。对于他们来说,定义了所研究信号的离散数据向量的样本数的最小值,在这个最小值上这些维数仍然可以被估计。以一组模型单分形和多重分形信号为例,以动态分形分析方法为例,验证了纠偏函数创建方法的有效性。结果表明,由于修正函数法的应用,分形维数估算值与真实已知值的最大偏差在给定维数下由25 ~ 55%减小到5 ~ 7%。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Corrective Function Method for the Fractal Analysis
One of the main numerical characteristics used in numerous methods of fractal analysis is the corresponding fractal dimensions. The accuracy of estimating these dimensions in the vast majority of cases is quite small, which cannot satisfy, first of all, researchers-practitioners. The method of the corrective function is put forward, which makes it possible to compensate for the ever-existing nonlinearity of the dependence between the true value of the fractal dimension and its estimation, performed using the selected method of monofractal analysis of signals and processes for a known number of samples of the discrete data vector of the investigated signal. The main idea of the method is to build and apply a special correction function using a set of model fractal signals with previously known values of the fractal dimension. The mathematical bases of the new method are outlined. Features of the practical application of the corrective function method are considered on the example of the evaluation of regularization, boxing, variation and Hurst fractal dimensions. For them, the minimum values of the number of samples of the discrete data vector of the investigated signal, at which these dimensions can still be estimated, are defined. Using a set of model monofractal and multifractal signals on the example of the dynamical fractal analysis method, the effectiveness of the created method of the corrective function is shown. It is proven that due to the application of the correction function method, the maximum deviation of the estimated fractal dimension from the true known value for the specified dimensions is reduced from 25 – 55% to 5 – 7%.
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Visnyk NTUU KPI Seriia-Radiotekhnika Radioaparatobuduvannia
Visnyk NTUU KPI Seriia-Radiotekhnika Radioaparatobuduvannia ENGINEERING, ELECTRICAL & ELECTRONIC-
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