Cross-correlation matrix: Difference between revisions

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For [[stochastic process]]es, including those that arise in [[statistical mechanics]] and Euclidean [[quantum field theory]], a '''correlation function''' is the [[correlation]] between [[random variable]]s at two different points in space or time. If one considers the correlation function between random variables at the same point but at two different times then one refers to this as the '''autocorrelation function'''. If there are multiple random variables in the problem then correlation functions of the ''same'' random variable are also sometimes called autocorrelation. Correlation functions of different random variables are sometimes called '''cross correlations'''.

Correlation functions used in [[correlation function (astronomy)|astronomy]], [[financial analysis]], [[quantum field theory]] and [[statistical mechanics]] differ only in the particular stochastic processes they are applied to.

==Definitions==

==Definition==

Consider a [[probability density]] [[functional]] <b>P[X(s)]</b> for stochastic variables <b>X(s)</b> at different points <b>s</b> of some space, then the correlation function is

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In this definition, it has been assumed that the stochastic variable is a scalar. If it is not, then one can define more complicated correlation functions. For example, if one has a vector <b>X_i(s)</b>, then one can define the matrix of correlation functions

:<math>C_{ij}(s,s') = \langle X_i(s) X_j(s') \rangle</math>

or a scalar, which is the trace of this matrix. If the probability density <b>P[X(s)]</b> has any target space symmetries, ie, symmetries in the space of the stochastic variable (also called '''internal symmetries'''), then the correlation matrix will have induced symmetries. If there are symmetries of the space (or time) in which the random variables exist (also called '''spacetime symmetries''') then the correlation matrix will have special properties. Examples of important spacetime symmetries are &mdash;

*'''translational symmetry''' yields <b>C(s,s')=C(s-s')</b> where <b>s</b> and <b>s'</b> are to be interpreted as vectors giving coordinates of the points

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

Higher order correlation functions are often defined. A typical correlation function of order <b>n</b> is

:<math>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 variable has only one component, then the indices <b>i_i</b> etc are redundant. If there are symmetries, then the correlation function can be brken up into [[irreducible representation]]s of the symmetries &mdash; both internal and spacetime.

The case of correlations of a single random variable can be thought of as a special case of autocorrelation of a stochastic process on a space which contains a single point.

==Properties of probability distributions==

With these definitions, the study of correlation functions is equivalent to the study of probability distributions. 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 walk]]s and led to the notion of the [[Ito calculus]].

A caveat, though, in [[quantum field theory]], we sometimes have nonpositive states and in that case, a probabilistic interpretation makes no sense. At any rate, even in ordinary quantum field theory, we need to work with quantum probability instead of classical [[probability]].

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]] lead to a local [[quantum field theory]] after [[Wick rotation]] to [[Minkowski spacetime]]. 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.

==Overview==

''Ideally, it will make clear that the correlation functions in astronomy, financial market analysis, etc., are all instances of the '''same''' idea; therefore, a disambiguation page is not what this should be.''

==See also==