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correlation function wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 properties of probability distributions 3 see also 4 references toggle the table of contents correlation function 9 languages العربية català italiano 日本語 қазақша 한국어 русский українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia correlation as a function of distance for other uses see correlation function disambiguation this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources correlation function news newspapers books scholar jstor december 2023 learn how and when to remove this message visual comparison of convolution cross correlation and autocorrelation a correlation function is a function that gives the statistical correlation between random variables contingent on the spatial or temporal distance between those variables 1 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 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 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 definition edit for possibly distinct random variables x s and y t at different points s and t of some space the correlation function is c s t corr x s y t displaystyle c s t operatorname corr x s y t where corr displaystyle operatorname corr 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 i j displaystyle i j element c i j s t corr x i s y j t displaystyle c_ ij s t operatorname corr x_ i s y_ j t 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 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 c i 1 i 2 i n s 1 s 2 s n x i 1 s 1 x i 2 s 2 x i n s n displaystyle c_ i_ 1 i_ 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 if the random vector has only one component variable then the indices i j displaystyle i j 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 edit 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 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 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 see also edit autocorrelation correlation of a signal with a time shifted copy of itself as a function of shift correlation does not imply causation refutation of a logical fallacy correlogram chart of correlation statistics covariance function function in probability theory pearson product moment correlation coefficient measure of linear correlation pages displaying short descriptions of redirect targets correlation function astronomy function describing the distribution of galaxies in the universe correlation function statistical mechanics measure of a system s order correlation function quantum field theory expectation value of time ordered quantum operators mutual information measure of dependence between two variables rate distortion theory theory about lossy data compression pages displaying short descriptions of redirect targets radial distribution function description of particle density in statistical mechanics references edit pal manoranjan bharati premananda 2019 introduction to correlation and linear regression analysis applications of regression techniques springer singapore pp 1 18 doi 10 1007 978 981 13 9314 3_1 retrieved december 14 2023 v t e statistical mechanics theory principle of maximum entropy ergodic theory statistical thermodynamics ensembles partition functions equations of state thermodynamic potential u h f g maxwell relations models ferromagnetism models ising potts heisenberg percolation particles with force field depletion force lennard jones potential mathematical approaches boltzmann equation h theorem vlasov equation bbgky hierarchy stochastic process mean field theory and conformal field theory critical phenomena phase transition critical exponents correlation length size scaling entropy boltzmann gibbs shannon tsallis rényi von neumann applications statistical field theory elementary particle superfluidity condensed matter physics complex system chaos information theory boltzmann machine authority control databases gnd retrieved from https en wikipedia org w index php title correlation_function oldid 1221076171 categories covariance and correlation time series spatial analysis hidden categories articles with short description short description is different from wikidata articles needing additional references from december 2023 all articles needing additional references pages displaying short descriptions of redirect targets via module annotated link this page was last edited on 27 april 2024 at 18 38 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents correlation function 9 languages add topic
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