多重分形系统

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模板:AnchorA Strange attractor that exhibits multifractal scaling
文件:WF111-Anderson transition-multifractal.jpeg
Example of a multifractal electronic eigenstate at the Anderson localization transition in a system with 1367631 atoms.

A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.[1]

A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.

多重分形系统是一个分形系统的推广,在这个系统中,单个指数(分形维数)不足以描述其动态,相反,需要一个连续的指数谱(所谓的奇异谱)。

Multifractal systems are common in nature. They include the length of coastlines, mountain topography,[2] fully developed turbulence, real-world scenes, heartbeat dynamics,[3] human gait[4]模板:Failed verification and activity,[5] human brain activity,[6][7][8][9][10][11][12] and natural luminosity time series.[13] Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more.[citation needed] The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models,[14] as well as the geometric Tweedie models.[15] The first convergence effect yields monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.[16]

Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometric Tweedie models. The first convergence effect yields monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.

多重分形系统在自然界是很常见的。它们包括海岸线的长度、山地地形、充分发展的湍流、现实世界的场景、心跳动力、人体步态和活动、人脑活动以及自然光度时间序列。从流体动力学中的湍流到互联网交通、金融、图像建模、纹理合成、气象学、地球物理学等各种各样的领域都已经提出了模型。序列(时间序列)数据的多重分形性质的起源归因于数学收敛效应,这种收敛效应与 Tweedie 指数分散模型和几何 Tweedie 模型的统计分布族有关。第一种收敛效应产生单分形维数序列,第二种收敛效应产生单序列的变异。

Multifractal analysis is used to investigate datasets, often in conjunction with other methods of fractal and lacunarity analysis. The technique entails distorting datasets extracted from patterns to generate multifractal spectra that illustrate how scaling varies over the dataset. Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.[17][18][19]

Multifractal analysis is used to investigate datasets, often in conjunction with other methods of fractal and lacunarity analysis. The technique entails distorting datasets extracted from patterns to generate multifractal spectra that illustrate how scaling varies over the dataset. Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.

多重分形分析是用来研究数据集,往往与其他方法的分形和空隙度分析。这项技术需要通过扭曲从模式中提取的数据集来生成多重分形谱,从而说明数据集的尺度变化。多重分形分析技术已经应用于多种实际情况,如地震预测和医学图像解释。

Definition

In a multifractal system [math]\displaystyle{ s }[/math], the behavior around any point is described by a local power law:

In a multifractal system s, the behavior around any point is described by a local power law:

= = 定义 = = 在多重分形系统中,任何点周围的行为都用局部幂定律来描述:

[math]\displaystyle{ s(\vec{x}+\vec{a})-s(\vec{x}) \sim a^{h(\vec{x})}. }[/math]
s(\vec{x}+\vec{a})-s(\vec{x}) \sim a^{h(\vec{x})}.

S (vec { x } + vec { a })-s (vec { x }) sim a ^ { h (vec { x })}.

The exponent [math]\displaystyle{ h(\vec{x}) }[/math] is called the singularity exponent, as it describes the local degree of singularity or regularity around the point [math]\displaystyle{ \vec{x} }[/math].[citation needed]

The exponent h(\vec{x}) is called the singularity exponent, as it describes the local degree of singularity or regularity around the point \vec{x}.

指数 h (vec { x })称为奇点指数,它描述了奇点{ x }周围的局部奇异度或正则度。

The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension [math]\displaystyle{ D(h): }[/math] the singularity spectrum. The curve [math]\displaystyle{ D(h) }[/math] versus [math]\displaystyle{ h }[/math] is called the singularity spectrum and fully describes the statistical distribution of the variable [math]\displaystyle{ s }[/math].[citation needed]

The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension D(h): the singularity spectrum. The curve D(h) versus h is called the singularity spectrum and fully describes the statistical distribution of the variable s.

由所有具有相同奇异性指数的点构成的集合称为指数 h 的奇异流形,是分形维数 d (h)的分形集合: 奇异谱。曲线 d (h)与 h 称为奇异谱,它完全描述了变量 s 的统计分布。

In practice, the multifractal behaviour of a physical system [math]\displaystyle{ X }[/math] is not directly characterized by its singularity spectrum [math]\displaystyle{ D(h) }[/math]. Rather, data analysis gives access to the multiscaling exponents [math]\displaystyle{ \zeta(q),\ q\in{\mathbb R} }[/math]. Indeed, multifractal signals generally obey a scale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale [math]\displaystyle{ a }[/math]. Depending on the object under study, these multiresolution quantities, denoted by [math]\displaystyle{ T_X(a) }[/math], can be local averages in boxes of size [math]\displaystyle{ a }[/math], gradients over distance [math]\displaystyle{ a }[/math], wavelet coefficients at scale [math]\displaystyle{ a }[/math], etc. For multifractal objects, one usually observes a global power-law scaling of the form:[citation needed]

In practice, the multifractal behaviour of a physical system X is not directly characterized by its singularity spectrum D(h). Rather, data analysis gives access to the multiscaling exponents \zeta(q),\ q\in{\mathbb R}. Indeed, multifractal signals generally obey a scale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale a. Depending on the object under study, these multiresolution quantities, denoted by T_X(a), can be local averages in boxes of size a, gradients over distance a, wavelet coefficients at scale a, etc. For multifractal objects, one usually observes a global power-law scaling of the form:

实际上,物理系统 x 的多重分形行为并不直接影响其奇异谱 d (h)的拥有属性。相反,数据分析提供了对{ mathbb r }中的多尺度指数 zeta (q) ,q 的访问。实际上,多重分形信号通常遵循一种尺度不变性特性,即对于多分辨率量,根据其尺度 a 产生幂律行为。根据研究对象的不同,这些多分辨率量,用 t _ x (a)表示,可以是大小为 a 的盒子中的局部平均值,距离为 a 的梯度,尺度为 a 的小波系数,等等。对于多重分形对象,人们通常观察到形式的全局幂律缩放:

[math]\displaystyle{ \langle T_X(a)^q \rangle \sim a^{\zeta(q)}\ }[/math]
\langle T_X(a)^q \rangle \sim a^{\zeta(q)}\
langle t _ x (a) ^ q rangle sim a ^ { zeta (q)}

at least in some range of scales and for some range of orders [math]\displaystyle{ q }[/math]. When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.[20]

at least in some range of scales and for some range of orders q. When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.

至少在一定的尺度范围内,在一定的阶数范围内。当这种行为被观察到时,人们会谈到尺度不变性、自相似性或者多尺度。

Estimation

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Using so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum [math]\displaystyle{ D(h) }[/math] and the multi-scaling exponents [math]\displaystyle{ \zeta(q) }[/math] through a Legendre transform. While the determination of [math]\displaystyle{ D(h) }[/math] calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the [math]\displaystyle{ \zeta(q) }[/math] relies on the use of statistical averages and linear regressions in log-log diagrams. Once the [math]\displaystyle{ \zeta(q) }[/math] are known, one can deduce an estimate of [math]\displaystyle{ D(h), }[/math] thanks to a simple Legendre transform.[citation needed]

Using so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum D(h) and the multi-scaling exponents \zeta(q) through a Legendre transform. While the determination of D(h) calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the \zeta(q) relies on the use of statistical averages and linear regressions in log-log diagrams. Once the \zeta(q) are known, one can deduce an estimate of D(h), thanks to a simple Legendre transform.

利用所谓的多重分形理论,我们可以证明,在一些适当的假设下,通过勒让德变换,奇异谱 d (h)和多重标度指数 zeta (q)之间存在对应关系。虽然 d (h)的确定需要对数据进行一些详尽的局部分析,这将导致难以和数值上不稳定的计算,但 zeta (q)的估计依赖于使用对数对数图中的统计平均数和线性回归。一旦 zeta (q)被知道,我们可以推导出 d (h)的估计,这要感谢一个简单的勒让德变换。

Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The [math]\displaystyle{ \zeta(q) }[/math] are statistically interpreted, as they characterize the evolution of the distributions of the [math]\displaystyle{ T_X(a) }[/math] as [math]\displaystyle{ a }[/math] goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models.[citation needed]

Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The \zeta(q) are statistically interpreted, as they characterize the evolution of the distributions of the T_X(a) as a goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models.

多重分形系统通常用乘性级联等随机过程来建模。Zeta (q)的统计解释,因为他们刻画了 t _ x (a)分布的演变为从大到小的尺度。这种演变通常被称为统计间歇性,它背离了高斯模型。

Modelling as a multiplicative cascade also leads to estimation of multifractal properties.脚本错误:没有“Footnotes”这个模块。 This methods works reasonably well, even for relatively small datasets. A maximum likely fit of a multiplicative cascade to the dataset not only estimates the complete spectrum but also gives reasonable estimates of the errors.[21]

Modelling as a multiplicative cascade also leads to estimation of multifractal properties. This methods works reasonably well, even for relatively small datasets. A maximum likely fit of a multiplicative cascade to the dataset not only estimates the complete spectrum but also gives reasonable estimates of the errors.

作为一个乘性级联模型也导致多重分形特性的估计。即使对于相对较小的数据集,这种方法也能很好地工作。一个最大可能适合的乘性级联的数据集不仅估计完整的频谱,而且给出合理的误差估计。

Estimating multifractal scaling from box counting

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Multifractal spectra can be determined from box counting on digital images. First, a box counting scan is done to determine how the pixels are distributed; then, this "mass distribution" becomes the basis for a series of calculations.[22][23][24] The chief idea is that for multifractals, the probability [math]\displaystyle{ P }[/math] of a number of pixels [math]\displaystyle{ m }[/math], appearing in a box [math]\displaystyle{ i }[/math], varies as box size [math]\displaystyle{ \epsilon }[/math], to some exponent [math]\displaystyle{ \alpha }[/math], which changes over the image, as in Eq.0.0 (NB: For monofractals, in contrast, the exponent does not change meaningfully over the set). [math]\displaystyle{ P }[/math] is calculated from the box-counting pixel distribution as in Eq.2.0.

[math]\displaystyle{ P_{[i,\epsilon]} \varpropto \epsilon^{-\alpha_i} \therefore\alpha_i \varpropto \frac{\log{P_{[i,\epsilon]}}}{\log{\epsilon^{-1}}} }[/math]

 

 

 

 

(Eq.0.0)

Multifractal spectra can be determined from box counting on digital images. First, a box counting scan is done to determine how the pixels are distributed; then, this "mass distribution" becomes the basis for a series of calculations. The chief idea is that for multifractals, the probability P of a number of pixels m, appearing in a box i, varies as box size \epsilon, to some exponent \alpha, which changes over the image, as in (NB: For monofractals, in contrast, the exponent does not change meaningfully over the set). P is calculated from the box-counting pixel distribution as in .


多重分形谱可以通过对数字图像的盒子计数来确定。首先,进行盒计数扫描以确定像素的分布情况; 然后,这种“质量分布”成为一系列计算的基础。其主要思想是,对于多重分形来说,出现在方框 i 中的若干像素 m 的概率 p 随方框大小 ε 变化,随着图像的变化而变化,如下所示(注: 对于单分形,相反,指数在整个集合中没有意义的变化)。P 是根据盒计数像素分布计算出来的。

[math]\displaystyle{ \epsilon }[/math] = an arbitrary scale (box size in box counting) at which the set is examined
\epsilon = an arbitrary scale (box size in box counting) at which the set is examined

检查盒子的任意尺度(盒子大小)

[math]\displaystyle{ i }[/math] = the index for each box laid over the set for an [math]\displaystyle{ \epsilon }[/math]
i = the index for each box laid over the set for an \epsilon
i = 在集合上放置的每个盒子的索引为 ε
[math]\displaystyle{ m_{[i,\epsilon]} }[/math] = the number of pixels or mass in any box, [math]\displaystyle{ i }[/math], at size [math]\displaystyle{ \epsilon }[/math]
m_{[i,\epsilon]} = the number of pixels or mass in any box, i, at size \epsilon
m {[ i,epsilon ]} = 任何方框中像素或质量的数量,i,大小为 epsilon
[math]\displaystyle{ N_\epsilon }[/math] = the total boxes that contained more than 0 pixels, for each [math]\displaystyle{ \epsilon }[/math]
N_\epsilon = the total boxes that contained more than 0 pixels, for each \epsilon
n epsilon = 每个 epsilon 包含超过0像素的总盒子
[math]\displaystyle{ M_\epsilon = \sum_{i=1}^{N_\epsilon}m_{[i,\epsilon]} = }[/math] the total mass or sum of pixels in all boxes for this [math]\displaystyle{ \epsilon }[/math]

 

 

 

 

(Eq.1.0)

[math]\displaystyle{ P_{[i,\epsilon]} = \frac{m_{[i,\epsilon]}}{M_\epsilon} = }[/math] the probability of this mass at [math]\displaystyle{ i }[/math] relative to the total mass for a box size

 

 

 

 

(Eq.2.0)

[math]\displaystyle{ P }[/math] is used to observe how the pixel distribution behaves when distorted in certain ways as in Eq.3.0 and Eq.3.1:

P is used to observe how the pixel distribution behaves when distorted in certain ways as in and :

P 用于观察像素分布在以下特定方式扭曲时的表现:

[math]\displaystyle{ Q }[/math] = an arbitrary range of values to use as exponents for distorting the data set
Q = an arbitrary range of values to use as exponents for distorting the data set
q = 一个任意范围的值,用作扭曲数据集的指数
[math]\displaystyle{ I_{{(Q)}_{[\epsilon]}} = \sum_{i=1}^{N_\epsilon} {P_{[i,\epsilon]}^Q} = }[/math] the sum of all mass probabilities distorted by being raised to this Q, for this box size

 

 

 

 

(Eq.3.0)

  • When [math]\displaystyle{ Q=1 }[/math], Eq.3.0 equals 1, the usual sum of all probabilities, and when [math]\displaystyle{ Q=0 }[/math], every term is equal to 1, so the sum is equal to the number of boxes counted, [math]\displaystyle{ N_\epsilon }[/math].
[math]\displaystyle{ \mu_{{(Q)}_{[i,\epsilon]}} = \frac{P_{[i,\epsilon]}^Q}{I_{{(Q)}_{[\epsilon]}}} = }[/math] how the distorted mass probability at a box compares to the distorted sum over all boxes at this box size

 

 

 

 

(Eq.3.1)


  • When Q=1, equals 1, the usual sum of all probabilities, and when Q=0, every term is equal to 1, so the sum is equal to the number of boxes counted, N_\epsilon.


当 q = 1,等于1,通常所有概率的和,当 q = 0,每个项都等于1,所以这个和等于计算的盒子数,n。

These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of [math]\displaystyle{ \epsilon }[/math]-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of \epsilon-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

这些扭曲的方程进一步用于解决当缩放或分解成一系列 epsilon 大小的片段并被 q 扭曲时集合的行为,以便为集合的维度找到不同的值,如下所示:

  • An important feature of Eq.3.0 is that it can also be seen to vary according to scale raised to the exponent [math]\displaystyle{ \tau }[/math] in Eq.4.0:
[math]\displaystyle{ I_{{(Q)}_{[\epsilon]}} \varpropto \epsilon^{\tau_{(Q)}} }[/math]

 

 

 

 

(Eq.4.0)

  • An important feature of is that it can also be seen to vary according to scale raised to the exponent \tau in :


  • 一个重要的特点是,它也可以被视为根据提高到指数 τ 的比例而变化:

Thus, a series of values for [math]\displaystyle{ \tau_{(Q)} }[/math] can be found from the slopes of the regression line for the log of Eq.3.0 versus the log of [math]\displaystyle{ \epsilon }[/math] for each [math]\displaystyle{ Q }[/math], based on Eq.4.1:

Thus, a series of values for \tau_{(Q)} can be found from the slopes of the regression line for the log of versus the log of \epsilon for each Q, based on :

因此,可以从回归线的斜率中找到 tau _ {(q)}的一系列值,其依据是:

[math]\displaystyle{ \tau_{(Q)} = {\lim_{\epsilon\to0}{\left[ \frac {\log{I_{{(Q)}_{[\epsilon]}}}} {\log{\epsilon}} \right ]}} }[/math]

 

 

 

 

(Eq.4.1)

[math]\displaystyle{ D_{(Q)} = {\lim_{\epsilon\to0} { \left [ \frac{\log{I_{{(Q)}_{[\epsilon]}}}}{\log{\epsilon^{-1}}} \right ]}} {(1-Q)^{-1}} }[/math]

 

 

 

 

(Eq.5.0)

[math]\displaystyle{ D_{(Q)} = \frac{\tau_{(Q)}}{Q-1} }[/math]

 

 

 

 

(Eq.5.1)

[math]\displaystyle{ \tau_{{(Q)}_{}} = D_{(Q)}\left(Q-1\right) }[/math]

 

 

 

 

(Eq.5.2)

[math]\displaystyle{ \tau_{(Q)} = \alpha_{(Q)}Q - f_{\left(\alpha_{(Q)}\right)} }[/math]

 

 

 

 

(Eq.5.3)


  • For the generalized dimension:




  • 关于一般维度:
  • [math]\displaystyle{ \alpha_{(Q)} }[/math] is estimated as the slope of the regression line for log A[math]\displaystyle{ \epsilon }[/math],Q versus log [math]\displaystyle{ \epsilon }[/math] where:
  • \alpha_{(Q)} is estimated as the slope of the regression line for versus where:


  • alpha _ {(q)}估计为回归线的斜率,其中:
[math]\displaystyle{ A_{\epsilon,Q} = \sum_{i=1}^{N_\epsilon}{\mu_{{i,\epsilon}_{Q}}{P_{{i,\epsilon}_{Q}}}} }[/math]

 

 

 

 

(Eq.6.0)

  • Then [math]\displaystyle{ f_{\left(\alpha_{{(Q)}}\right)} }[/math] is found from Eq.5.3.


  • Then f_{\left(\alpha_模板:(Q)\right)} is found from .
  • 然后 f _ { left (alpha _ {{(q)} right)}从。
  • The mean [math]\displaystyle{ \tau_{(Q)} }[/math] is estimated as the slope of the log-log regression line for [math]\displaystyle{ \tau_{{(Q)}_{[\epsilon]}} }[/math] versus [math]\displaystyle{ \epsilon }[/math], where:
  • The mean \tau_{(Q)} is estimated as the slope of the log-log regression line for \tau_{{(Q)}_{[\epsilon]}} versus \epsilon, where:


  • 平均 tau _ {(q)}被估计为 tau _ {{(q)} _ {[ epsilon ]}}相对于 epsilon 的对数对数回归线的斜率,其中:
[math]\displaystyle{ \tau_{(Q)_{[\epsilon]}} = \frac{\sum_{i=1}^{N_\epsilon} {P_{[i,\epsilon]}^{Q-1}}} {N_\epsilon} }[/math]

 

 

 

 

(Eq.6.1)

In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.[22]

In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.

在实践中,数据概率分布取决于数据集是如何取样的,所以优化算法已经开发,以确保足够的取样。

Applications

Multifractal analysis has been successfully used in many fields, including physical, information, and biological sciences.[25] For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.[26]

Multifractal analysis has been successfully used in many fields, including physical, information, and biological sciences. For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.

多重分形分析已成功地应用于许多领域,包括物理、信息和生物科学。例如,钢筋混凝土剪力墙表面残余裂缝模式的量化。

Dataset distortion analysis

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文件:Distort.png
Multifractal analysis is analogous to viewing a dataset through a series of distorting lenses to home in on differences in scaling. The pattern shown is a Hénon map.

Multifractal analysis has been used in several scientific fields to characterize various types of datasets.[27][5][8] In essence, multifractal analysis applies a distorting factor to datasets extracted from patterns, to compare how the data behave at each distortion. This is done using graphs known as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration.[22] Several types of multifractal spectra are used in practise.

Multifractal analysis has been used in several scientific fields to characterize various types of datasets. In essence, multifractal analysis applies a distorting factor to datasets extracted from patterns, to compare how the data behave at each distortion. This is done using graphs known as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration. Several types of multifractal spectra are used in practise.

多重分形分析已经应用于多个科学领域来表征各种类型的数据集。本质上,多重分形分析将一个扭曲因子应用于从模式中提取的数据集,以比较数据在每次扭曲时的表现。这是使用图称为多重分形谱,类似于通过“扭曲镜头”查看数据集,如图所示。多重分形谱在实际应用中有几种类型。

DQ vs Q

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文件:Dqvsq.png
DQ vs Q spectra for a non-fractal circle (empirical box counting dimension = 1.0), mono-fractal Quadric Cross (empirical box counting dimension = 1.49), and multifractal Hénon map (empirical box counting dimension = 1.29).

One practical multifractal spectrum is the graph of DQ vs Q, where DQ is the generalized dimension for a dataset and Q is an arbitrary set of exponents. The expression generalized dimension thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension using box counting are described below).

One practical multifractal spectrum is the graph of DQ vs Q, where DQ is the generalized dimension for a dataset and Q is an arbitrary set of exponents. The expression generalized dimension thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension using box counting are described below).

一个实用的多重分形谱是 DQ vs q 图,其中 DQ 是数据集的广义维数,q 是任意指数集。因此,表达式 generalized dimension 指的是一组数据集的维(下面将描述使用盒计数确定广义维的详细计算)。

Dimensional ordering

The general pattern of the graph of DQ vs Q can be used to assess the scaling in a pattern. The graph is generally decreasing, sigmoidal around Q=0, where D(Q=0) ≥ D(Q=1) ≥ D(Q=2). As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns. The image shows D(Q) spectra from a multifractal analysis of binary images of non-, mono-, and multi-fractal sets. As is the case in the sample images, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals.

The general pattern of the graph of DQ vs Q can be used to assess the scaling in a pattern. The graph is generally decreasing, sigmoidal around Q=0, where D(Q=0) ≥ D(Q=1) ≥ D(Q=2). As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns. The image shows D(Q) spectra from a multifractal analysis of binary images of non-, mono-, and multi-fractal sets. As is the case in the sample images, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals.

= = = 维序 = = = = DQ 与 q 图的一般模式可以用来评估一个模式中的标度。图一般是递减的,在 q = 0左右,其中 d (q = 0)≥ d (q = 1)≥ d (q = 2)。如图所示,这个图形光谱的变化可以帮助区分模式。图像显示了 d (q)谱从非,单分形和多分形集的二进制图像的多重分形分析。正如在样本图像的情况,非和单分形往往比多分形有更平坦的 d (q)光谱。

The generalized dimension also gives important specific information. D(Q=0) is equal to the capacity dimension, which—in the analysis shown in the figures here—is the box counting dimension. D(Q=1) is equal to the information dimension, and D(Q=2) to the correlation dimension. This relates to the "multi" in multifractal, where multifractals have multiple dimensions in the D(Q) versus Q spectra, but monofractals stay rather flat in that area.[22][23]

The generalized dimension also gives important specific information. D(Q=0) is equal to the capacity dimension, which—in the analysis shown in the figures here—is the box counting dimension. D(Q=1) is equal to the information dimension, and D(Q=2) to the correlation dimension. This relates to the "multi" in multifractal, where multifractals have multiple dimensions in the D(Q) versus Q spectra, but monofractals stay rather flat in that area.

广义维数也给出了重要的具体信息。D (q = 0)等于容量维数,在本图所示的分析中,容量维数是盒数维数。D (q = 1)等于信息维度,d (q = 2)等于关联维数维度。这与多重分形中的“多重”有关,其中多重分形在 d (q)与 q 谱中具有多重维数,但单分形在该区域保持相当平坦。

[math]\displaystyle{ f(\alpha) }[/math] versus [math]\displaystyle{ \alpha }[/math]

Another useful multifractal spectrum is the graph of [math]\displaystyle{ f(\alpha) }[/math] versus [math]\displaystyle{ \alpha }[/math] (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like DQ versus Q spectra, they also show typical patterns useful for comparing non-, mono-, and multi-fractal patterns. In particular, for these spectra, non- and mono-fractals converge on certain values, whereas the spectra from multifractal patterns typically form humps over a broader area.

Another useful multifractal spectrum is the graph of f(\alpha) versus \alpha (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like DQ versus Q spectra, they also show typical patterns useful for comparing non-, mono-, and multi-fractal patterns. In particular, for these spectra, non- and mono-fractals converge on certain values, whereas the spectra from multifractal patterns typically form humps over a broader area.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =.这些曲线通常会上升到最大值,接近于 q = 0的分形维数,然后下降。与 DQ 和 q 光谱一样,它们也显示了用于比较非分形、单分形和多分形模式的典型模式。特别是,对于这些光谱来说,非分形和单分形集中在某些特定的值上,而多分形图案的光谱通常在更广的区域形成峰。

Generalized dimensions of species abundance distributions in space

One application of Dq versus Q in ecology is characterizing the distribution of species. Traditionally the relative species abundances is calculated for an area without taking into account the locations of the individuals. An equivalent representation of relative species abundances are species ranks, used to generate a surface called the species-rank surface,[28] which can be analyzed using generalized dimensions to detect different ecological mechanisms like the ones observed in the neutral theory of biodiversity, metacommunity dynamics, or niche theory.[28][29]

One application of Dq versus Q in ecology is characterizing the distribution of species. Traditionally the relative species abundances is calculated for an area without taking into account the locations of the individuals. An equivalent representation of relative species abundances are species ranks, used to generate a surface called the species-rank surface, which can be analyzed using generalized dimensions to detect different ecological mechanisms like the ones observed in the neutral theory of biodiversity, metacommunity dynamics, or niche theory.

= = = 空间物种多度分布的广义维数 = = = = Dq 与 q 在生态学中的一个应用是描述物种的分布。传统上计算一个地区的相对物种丰度时没有考虑个体的位置。相对物种丰度的等价表示是物种等级,用于生成一个称为物种等级面的表面,这个表面可以用广义维度来分析,以发现不同的生态机制,如生物多样性中性理论、协整动力学或生态位理论中观察到的机制。

See also

  • Fractional Brownian motion
  • Detrended fluctuation analysis
  • Tweedie distributions
  • Markov switching multifractal
  • Weighted planar stochastic lattice (WPSL)

= =

  • 分数布朗运动
  • 去趋势涨落分析
  • Tweedie 分布
  • Markov 切换多重分形
  • 加权平面随机晶格(WPSL)

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Further reading

= 进一步阅读 = =

External links

  • Movies of visualizations of multifractals

= = 外部链接 =

  • 多重分形的可视化电影

模板:Fractals


Category:Fractals Category:Dimension theory

范畴: 分形范畴: 维理论


This page was moved from wikipedia:en:Multifractal system. Its edit history can be viewed at 多重分形系统/edithistory