Content deleted Content added
No edit summary |
This looks as if a physicist wrote it. The author assumed all stochastic processes have expection zero. No distinction was made between covariance and correlation. Wikipeida conventions were flouted. |
||
Line 1:
{{cleanup}}
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'''.
Line 5 ⟶ 7:
==Definition==
:<math>C(s,s') = \langle X(s) X(s')\rangle</math>▼
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<sub>i</sub>(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>▼
▲In this definition, it has been assumed that the stochastic variable is
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 —▼
*'''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).▼
▲or a scalar, which is the trace of this matrix. If the [[probability
▲*'''translational symmetry''' yields
▲*'''rotational symmetry''' in addition to the above gives
''n'' 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 <math>i_j</math> are redundant. If there are symmetries, then the correlation function can be broken up into [[irreducible representation]]s of the symmetries — both internal and spacetime.
| |||