分形理论中若干问题的小波解法
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摘要
现代分形理论中存在有许多问题尚待解决，尤其关于分维数的计算、分形反问题及奇异性分析等方面的研究为重点。小波在处理非平稳性和局域分析方面有着独特的优越性，可被用来提取有关分形尺度性质的细微信息。本文以小波理论为工具围绕分形中若干问题进行了探讨，主要提出并解决了如下问题：
1．分析了分形曲线维数的传统计算方法及其缺陷，提出了多分维分形曲线的模式。基于小波阈值去噪方法，给出曲线上点的局部分形维数的计算公式及算法。仿真计算结果及图示均表明该法是有效的。
2．基于小波变换在零点具有的的保相似形特征，在已知分形插值函数图象情况下，从有限数目个零点出发，给出寻找一组仿射压缩变换使该函数图象成为未知迭代函数系统吸引子的一种构造方法。
3．提出了分形时变维数的概念，将其引入两类具有局部自相似性的有偏随机过程中。借助小波基固有的尺度特性很适合分析非平稳性和局部自相似性这一特点，给出赫斯特指数的小波估算公式及算法，证明了小波估算值和真实值是相容的，并用仿真结果加以验证，同时对上海股票市场周指数变化进行了实证分析。
4．借助小波变换对自相似过程所起的白化作用和最小二乘法，建立了无须知道1/f过程中参数分布的谱参数估计方法。
5．提出在最优尺度上跟踪小波模极大值线的参数方法，为研究多分形谱奇异性提供了一种途径。探讨了分形测度小波变换极大值线的拓扑分岔情况，并提出了一种基于小波变换模极大值的多重分形谱估算方法。
There are many problems to be solved in modern fractal theory, especially focuses on studies of computation on fractal dimension,inverse problem and sigularities. The scale property of wavelet is well suited for analyzing non-stationarity and localization and used to extract microscopic information about the scaling properties of fractals. In this dissertation, based on wavelet theory,the following problems are discussed:
1. Traditional computional methods and limitation of dimension of fractal curve being analyzed ,A model of fractal curve having multidimension is proposed. Based on noise reduction for wavelet softhresholding, computional formula and algorithm about point-wise local dimension on the curve are given. Results from simulations show the validity of he algorithm.
2. Given a graph of a fractal interpolation function which is the attractor of an unknown IPS with affine constration maps, the maps are found based on the self-similarity of the zero-crossing points of wavelet transform. The effectiveness of method is shown in an example.
3. The concept of time-varying dimension are proposed and introduced in two kinds of biased stochastic process with locally self-similarity. The estimation formula and algorithm of Hurst Index are based on Daubechies wavelet analysis of samples dada . Simulation result indicates effectiveness of estimation method and proves that estimation value is a consistent result of true value. Week-index analysis of Shanghai
stock market is taken as a real example by time-varying dimension.
4. Based on correlation-decaying charecteristics of discrete wavelet coefficients of
lf processes and the least-square algorithm estimation for spectral parameter of yf
processes is presented without knowing the distribution of the parameter.
5. A parameterization method for tracing wavelet maxima lines at the finest scale continuously is presented, by means of which the sigularity of multifractal spectrum can be analyzed well. Topologic bifurcation situation of maxima lines is also discussed. In the end ,an estimation for multifractal spectrum is proposed in terms of wavelet module maxima lines.
1．分析了分形曲线维数的传统计算方法及其缺陷，提出了多分维分形曲线的模式。基于小波阈值去噪方法，给出曲线上点的局部分形维数的计算公式及算法。仿真计算结果及图示均表明该法是有效的。
2．基于小波变换在零点具有的的保相似形特征，在已知分形插值函数图象情况下，从有限数目个零点出发，给出寻找一组仿射压缩变换使该函数图象成为未知迭代函数系统吸引子的一种构造方法。
3．提出了分形时变维数的概念，将其引入两类具有局部自相似性的有偏随机过程中。借助小波基固有的尺度特性很适合分析非平稳性和局部自相似性这一特点，给出赫斯特指数的小波估算公式及算法，证明了小波估算值和真实值是相容的，并用仿真结果加以验证，同时对上海股票市场周指数变化进行了实证分析。
4．借助小波变换对自相似过程所起的白化作用和最小二乘法，建立了无须知道1/f过程中参数分布的谱参数估计方法。
5．提出在最优尺度上跟踪小波模极大值线的参数方法，为研究多分形谱奇异性提供了一种途径。探讨了分形测度小波变换极大值线的拓扑分岔情况，并提出了一种基于小波变换模极大值的多重分形谱估算方法。
There are many problems to be solved in modern fractal theory, especially focuses on studies of computation on fractal dimension,inverse problem and sigularities. The scale property of wavelet is well suited for analyzing non-stationarity and localization and used to extract microscopic information about the scaling properties of fractals. In this dissertation, based on wavelet theory,the following problems are discussed:
1. Traditional computional methods and limitation of dimension of fractal curve being analyzed ,A model of fractal curve having multidimension is proposed. Based on noise reduction for wavelet softhresholding, computional formula and algorithm about point-wise local dimension on the curve are given. Results from simulations show the validity of he algorithm.
2. Given a graph of a fractal interpolation function which is the attractor of an unknown IPS with affine constration maps, the maps are found based on the self-similarity of the zero-crossing points of wavelet transform. The effectiveness of method is shown in an example.
3. The concept of time-varying dimension are proposed and introduced in two kinds of biased stochastic process with locally self-similarity. The estimation formula and algorithm of Hurst Index are based on Daubechies wavelet analysis of samples dada . Simulation result indicates effectiveness of estimation method and proves that estimation value is a consistent result of true value. Week-index analysis of Shanghai
stock market is taken as a real example by time-varying dimension.
4. Based on correlation-decaying charecteristics of discrete wavelet coefficients of
lf processes and the least-square algorithm estimation for spectral parameter of yf
processes is presented without knowing the distribution of the parameter.
5. A parameterization method for tracing wavelet maxima lines at the finest scale continuously is presented, by means of which the sigularity of multifractal spectrum can be analyzed well. Topologic bifurcation situation of maxima lines is also discussed. In the end ,an estimation for multifractal spectrum is proposed in terms of wavelet module maxima lines.
引文
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