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第43行：
第43行： − − In mathematics, one can consider the scaling properties of a function or curve under rescalings of the variable . That is, one is interested in the shape of for some scale factor , which can be taken to be a length or size rescaling. The requirement for to be invariant under all rescalings is usually taken to be − :f(\lambda x)=\lambda^{\Delta}f(x) − for some choice of exponent , and for all dilations . This is equivalent to being a homogeneous function of degree . 第55行：
第51行： − − Examples of scale-invariant functions are the monomials f(x)=x^n, for which , in that clearly 第122行：
第116行： − 如果是平均值，期望功率的频率，那么噪声标度为: p (f) = lambda ^ {-Delta } p (lambda f) ，其中白噪声 = 0，粉红噪声 =-1，布朗噪声 =-2(更一般的是布朗运动)。+ + + + + 第128行：
第126行： − 更准确地说，随机系统的标度关系到从所有可能的随机配置集中选择一个特定配置的可能性。这里需要更多的背景知识。概率和熵肯定与选择一个特定的配置有关，但是尺度不变性和这个联系并不明显。这种可能性是由美国概率分布研究所给出的。+ 第134行：
第132行： − 尺度不变分布的例子有帕累托分布分布和 Zipfian 分布。+
→Scale invariance in stochastic processes 随机过程中的标度不变性
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
for some choice of exponent {{mvar|Δ}}, and for all dilations {{mvar|λ}}. This is equivalent to {{mvar|f}} being a [[homogeneous function]] of degree {{mvar|Δ}}.
for some choice of exponent {{mvar|Δ}}, and for all dilations {{mvar|λ}}. This is equivalent to {{mvar|f}} being a [[homogeneous function]] of degree {{mvar|Δ}}.
在数学中，我们会考虑函数或曲线在变量{{mvar|x}}重新标度下的标度性质。也就是说，人们对某些标度因子{{mvar|λ}} 对应下{{math|''f'' (''λx'')}}的形状感兴趣，这些标度因子可以被视为长度或大小的重新标度。对于某些选择的指数{{mvar|Δ}}和所有的膨胀{{mvar|λ}}，要求{{math|''f'' (''x'')}} 在所有重新标度下保持不变需要满足：
在数学中，我们会考虑函数或曲线在变量{{mvar|x}}重新标度下的标度性质。也就是说，人们对某些标度因子{{mvar|λ}} 对应下{{math|''f'' (''λx'')}}的形状感兴趣，这些标度因子可以被视为长度或大小的重新标度。对于某些选择的指数{{mvar|Δ}}和所有的膨胀{{mvar|λ}}，要求{{math|''f'' (''x'')}} 在所有重新标度下保持不变需要满足：
Examples of scale-invariant functions are the [[monomial]]s <math>f(x)=x^n</math>, for which {{math|Δ {{=}} ''n''}}, in that clearly
Examples of scale-invariant functions are the [[monomial]]s <math>f(x)=x^n</math>, for which {{math|Δ {{=}} ''n''}}, in that clearly
许多标变函数的实例是单项式：
许多标变函数的实例是单项式：
with = 0 for white noise, = −1 for pink noise, and = −2 for Brownian noise (and more generally, Brownian motion).
with = 0 for white noise, = −1 for pink noise, and = −2 for Brownian noise (and more generally, Brownian motion).
如果{{math|''P''(''f'' )}}是频率{{mvar|f }}处的平均期望幂，那么噪声依下式标度变化：
<math>P(f) = \lambda^{-\Delta} P(\lambda f)</math>
当{{mvar|Δ}}= 0时对应'''White noise 白噪声'''，{{mvar|Δ}}=-1时对应'''Pink noise 粉红噪声'''，{{mvar|Δ}}=-2时对应'''Brownian noise 布朗噪声'''（更一般的是'''Brownian motion 布朗运动'''）。
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the [[probability distribution]].
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the [[probability distribution]].
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the probability distribution.
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the probability distribution.
更准确地说，随机系统中的标度变化涉及从所有可能的随机排列中选择一个特定排列的可能性。这一可能性可由概率分布给出。此外还需要更多的背景内容。概率和熵必然与一个特定排列的选定有关，但标度不变性与它们之间的相互联系还不明显。
Examples of scale-invariant distributions are the [[Pareto distribution]] and the [[Zipfian distribution]].
Examples of scale-invariant distributions are the [[Pareto distribution]] and the [[Zipfian distribution]].
Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.
Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.
标度不变分布的例子还有'''Pareto distribution 帕累托分布'''和'''Zipfian distribution 齐夫分布'''。
===Scale invariant Tweedie distributions 标度不变的Tweedie分布===
===Scale invariant Tweedie distributions 标度不变的Tweedie分布===