相关函数

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卷积、互相关性和自相关性的视觉比较。

Visual comparison of convolution, cross-correlation and autocorrelation.

视觉比较[卷积，互相关和自相关]

A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations.

A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations.

相关函数(量子场论)是一个给出随机变量之间的统计相关性的函数，其统计相关性取决于这些变量之间的空间或时间距离。如果我们认为随机变量之间的相关函数(量子场论) 代表在两个不同点测量的相同数量，那么这通常指的是由自相关组成的自相关函数。不同随机变量的相关函数有时被称为互相关函数，因为它们是由互相关组成的，该种函数主要强调由互相关组成的不同变量。

Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations.

相关函数作为在时间或空间距离上的函数，是一种有效用的依赖性指标，可以用来评估样本点之间所需的距离，以使值有效地不相关。此外，在没有观测值的点上，它们进行了插值处理，提供了相关计算基础。

Correlation functions used in astronomy, financial analysis, econometrics, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions.

天文学、金融分析、计量经济学和统计力学中所使用到的相关函数的区别仅在于它们所应用的特定随机过程的不同。在量子场论中，量子分布上存在相关函数。

定义

For possibly distinct random variables X(s) and Y(t) at different points s and t of some space, the correlation function is

[math]\displaystyle{ C(s,t) = \operatorname{corr} ( X(s), Y(t) ) , }[/math]

对于在某些空间上不同点 s 和 t的可能的不同随机变量 x (s)和 y (t)，其相关函数(量子场论)为：

[math]\displaystyle{ C(s,t) = \operatorname{corr} ( X(s), Y(t) ) , }[/math]

where [math]\displaystyle{ \operatorname{corr} }[/math] is described in the article on correlation. In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with [math]\displaystyle{ i,j }[/math] element

其中 [math]\displaystyle{ \operatorname{corr} }[/math] 在相关性的文章中有描述。在这个定义中，我们假设随机变量是标量。如果不是，则可以定义更复杂的相关函数。例如，若 X(s) 是一个 n 维元素的随机向量， Y(t) 是一个q 维元素的向量，则用 [math]\displaystyle{ i,j }[/math] 元素定义相关函数的 n×q 矩阵：

[math]\displaystyle{ C_{ij}(s,t) = \operatorname{corr}( X_i(s) ，Y_j(t) ) 。 }[/math]

When n=q, sometimes the trace of this matrix is focused on. If the probability distributions have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —

When n=q, sometimes the trace of this matrix is focused on. If the probability distributions have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —

当 n=q 时，有时该矩阵的迹会集聚。如果概率分布具有目标空间对称性，即在随机变量的值空间中存在对称性(也称为内对称性) ，则相关矩阵将具有诱导对称性。类似地，如果随机变量所存在的空间(或时间)域具有对称性(也称为时空对称性) ，则相关函数(量子场论)将具有相应的空间或时间对称性。重要的时空对称的例子有 --

平移对称 场域中C(s,s') = C(s − s')，其中s和 s' 被解释为给出点的坐标的向量

rotational symmetry in addition to the above gives C(s, s') = C(|s − s'|) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).

旋转对称 除上面提到的以外，还给出了C(s,s') = C(s − s')，其中|x|表示向量“ x”的标准值(对于实际的旋转，这是欧几里得或2-范数)。

Higher order correlation functions are often defined. A typical correlation function of order n is

[math]\displaystyle{ C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle. }[/math]模板:Clarification needed

[math]\displaystyle{ C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle. }[/math]

高阶相关函数经常被定义。一个典型的 n 阶相关函数(量子场论)为：

[math]\displaystyle{ C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle。 }[/math]

If the random vector has only one component variable, then the indices [math]\displaystyle{ i,j }[/math] are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime.

如果随机向量只有一个分量变量，那么指数[math]\displaystyle{ i,j }[/math]是冗余的。如果存在对称性，那么相关函数(量子场论)可以被分解成对称性的不可约表示 — 包括内对称性和时空对称性。

概率分布的性质

With these definitions, the study of correlation functions is similar to the study of probability distributions. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of Gaussian processes.

在这些定义下，相关函数的研究类似于概率分布的研究。许多随机过程可以用它们自有的相关函数完全表征；其中最著名的例子就是一类高斯过程。

Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of random walks and led to the notion of the Itō calculus.

在有限个点上定义的概率分布总是可以规范化的，但是当它们定义在连续空间上时，则需要格外注意。对这种分布的研究始于对随机游动的研究，并引出了 Itō 演算的概念。

The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime ( see Osterwalder-Schrader axioms ). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.

欧几里德空间中的 Feynman 路径积分将这个问题推广到了统计力学所着眼的其他问题中。任何满足被称为反射正性的相关函数条件的概率分布，在威克转动到闵可夫斯基时空之后，就引导出了局域量子场论（见Osterwalder-Schrader公理）。重整化运算是从概率分布空间到其本身的一组特定的映射。如果一个量子场论的映射有一个可以给出量子场论的不动点，则这个量子场论是可重整化的。

相关函数

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概率分布的性质

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