
A correlation function is a
function that gives the statistical
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...
between
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s, 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
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
, which is made up of
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
s. 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-correlation
In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
s.
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.
Correlation functions used in
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
financial analysis
Financial analysis (also known as financial statement analysis, accounting analysis, or analysis of finance) refers to an assessment of the viability, stability, and profitability of a business, sub-business or project.
It is performed by profe ...
,
econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...
, and
statistical mechanics differ only in the particular stochastic processes they are applied to. In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
there are
correlation functions over quantum distributions.
Definition
For possibly distinct random variables ''X''(''s'') and ''Y''(''t'') at different points ''s'' and ''t'' of some space, the correlation function is
:
where
is described in the article on
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...
. 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
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
with ''n'' elements and ''Y''(t) is a vector with ''q'' elements, then an ''n''×''q'' matrix of correlation functions is defined with
element
:
When ''n''=''q'', sometimes the
trace of this matrix is focused on. If the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s 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
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ...
), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —
*translational symmetry yields ''C''(''s'',''s''
') = ''C''(''s'' − ''s''
') where ''s'' and ''s''
' are to be interpreted as vectors giving coordinates of the points
*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).
Higher order correlation functions are often defined. A typical correlation function of order ''n'' is (the angle brackets represent the
expectation value)
:
If the random vector has only one component variable, then the indices
are redundant. If there are symmetries, then the correlation function can be broken up into
irreducible representations of the symmetries — both internal and spacetime.
Properties of probability distributions
With these definitions, the study of correlation functions is similar to the study of
probability distributions
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...
. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of
Gaussian processes.
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
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb ...
s and led to the notion of the
Itō calculus.
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
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
after
Wick rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that su ...
to
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
(see
Osterwalder-Schrader axioms). The operation of
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering ...
is a specified set of mappings from the space of probability distributions to itself. A
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
See also
*
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
*
Correlation does not imply causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
*
Correlogram
In the analysis of data, a correlogram is a chart of correlation statistics.
For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram.
If cross-correlation is plotte ...
*
Covariance function
*
Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficien ...
*
Correlation function (astronomy)
*
Correlation function (statistical mechanics)
*
Correlation function (quantum field theory)
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where the ...
*
Mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as ...
*
Rate distortion theory
Rate or rates may refer to:
Finance
* Rates (tax), a type of taxation system in the United Kingdom used to fund local government
* Exchange rate, rate at which one currency will be exchanged for another
Mathematics and science
* Rate (mathem ...
*
Radial distribution function
In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.
I ...
{{Statistical mechanics topics
Covariance and correlation
Time series
Spatial analysis