Editing Cross-correlation (section)

{{Short description|Covariance and correlation}}
{{Correlation and covariance}}
[[File:Comparison convolution correlation.svg|thumb|400px|Visual comparison of [[convolution]], cross-correlation and [[autocorrelation]].  For the operations involving function {{mvar|f}}, and assuming the height of {{mvar|f}} is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point.  Also, the vertical symmetry of {{mvar|f}} is the reason <math>f*g</math> and <math>f \star g</math> are identical in this example.]]

In [[signal processing]], '''cross-correlation''' is a [[Similarity measure|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 for searching a long signal for a shorter, known feature. It has applications in [[pattern recognition]], [[single particle analysis]], [[electron tomography]], [[averaging]], [[cryptanalysis]], and [[neurophysiology]]. The cross-correlation is similar in nature to the [[convolution]] of two functions.  In an [[autocorrelation]], which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.

In [[probability]] and [[statistics]], the term ''cross-correlations'' refers to the [[covariance and correlation|correlations]] between the entries of two [[Multivariate random variable|random vectors]] <math>\mathbf{X}</math> and <math>\mathbf{Y}</math>, while the ''correlations'' of a random vector <math>\mathbf{X}</math> are the correlations between the entries of <math>\mathbf{X}</math> itself, those forming the [[correlation matrix]] of <math>\mathbf{X}</math>. If each of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> is a scalar random variable which is realized repeatedly in a [[time series]], then the correlations of the various temporal instances of <math>\mathbf{X}</math> are known as ''autocorrelations'' of <math>\mathbf{X}</math>, and the cross-correlations of <math>\mathbf{X}</math> with <math>\mathbf{Y}</math> across time are temporal cross-correlations. In probability and statistics, the definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1.

If <math>X</math> and <math>Y</math> are two [[independent (probability)|independent]] [[random variable]]s with [[probability density function]]s <math>f</math> and <math>g</math>, respectively, then the probability density of the difference <math>Y - X</math> is formally given by the cross-correlation (in the signal-processing sense) <math>f \star g</math>; however, this terminology is not used in probability and statistics. In contrast, the [[convolution]] <math>f * g</math> (equivalent to the cross-correlation of <math>f(t)</math> and <math>g(-t)</math>) gives the probability density function of the sum <math>X + Y</math>.