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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
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164:, 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
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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
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353:. 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
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851:{\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 .}
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547:), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —
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have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called
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Pal, Manoranjan; Bharati, Premananda (2019). "Introduction to
Correlation and Linear Regression Analysis".
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is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
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Higher order correlation functions are often defined. A typical correlation function of order
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are redundant. If there are symmetries, then the correlation function can be broken up into
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to emphasize that different variables are being considered and because they are made up of
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is a specified set of mappings from the space of probability distributions to itself. A
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With these definitions, the study of correlation functions is similar to the study of
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differ only in the particular stochastic processes they are applied to. In
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If the random vector has only one component variable, then the indices
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in
Euclidean space generalizes this to other problems of interest to
511:{\displaystyle C_{ij}(s,t)=\operatorname {corr} (X_{i}(s),Y_{j}(t)).}
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1036: – Description of particle density in statistical mechanics
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319:{\displaystyle C(s,t)=\operatorname {corr} (X(s),Y(t)),}
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Pages displaying short descriptions of redirect targets
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Pages displaying short descriptions of redirect targets
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of the symmetries — both internal and spacetime.
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1021: – Measure of dependence between two variables
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184:points for which there are no observations.
1013:Correlation function (quantum field theory)
243:of some space, the correlation function is
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116:Learn how and when to remove this message
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982: – Image of correlation statistics
976: – Refutation of a logical fallacy
895:Properties of probability distributions
219:For possibly distinct random variables
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1060:. Springer, Singapore. pp. 1–18.
988: – Function in probability theory
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1057:Applications of Regression Techniques
994: – Measure of linear correlation
632:is (the angle brackets represent the
533:of this matrix is focused on. If the
342:{\displaystyle \operatorname {corr} }
21:Correlation function (disambiguation)
16:Correlation as a function of distance
974:Correlation does not imply causation
54:adding citations to reliable sources
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1009: – Measure of a system's order
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1001:Correlation function (astronomy)
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150:correlation function
50:improve this article
19:For other uses, see
1400:elementary particle
1165:partition functions
986:Covariance function
944:Minkowski spacetime
880:{\displaystyle i,j}
612: −
590:rotational symmetry
574: −
539:internal symmetries
404:{\displaystyle i,j}
1427:information theory
1334:correlation length
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1019:Mutual information
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106:December 2023
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67: –
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61:Find sources:
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39:This article
37:
33:
28:
27:
22:
1388:Applications
1339:size scaling
1333:
1071:December 14,
1069:. Retrieved
1056:
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922:The Feynman
921:
917:ItĹŤ calculus
913:random walks
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48:Please help
43:verification
40:
1460:Time series
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1249:force field
1241:percolation
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616:'|) where |
351:correlation
158:correlation
134:convolution
1449:Categories
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1041:References
215:Definition
207:there are
76:newspapers
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