Cross-correlation

In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, cross-correlation is a <a href="/facts/Similarity_measure/FmmFco8c">measure of similarity</a> of two series as a function of the displacement of one relative to the other. This is also known as a sliding <a href="/facts/Dot_product/tNz8MLjT">dot product</a> or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in <a href="/facts/Pattern_recognition/o1s38opj">pattern recognition</a>, <a href="/facts/Single_particle_analysis/nrJAQtkq">single particle analysis</a>, <a href="/facts/Electron_tomography/Po9Htj8d">electron tomography</a>, <a href="/facts/Averaging/etawWjw4">averaging</a>, <a href="/facts/Cryptanalysis/7CPEQlJI">cryptanalysis</a>, and <a href="/facts/Neurophysiology/XLkEyW8c">neurophysiology</a>. The cross-correlation is similar in nature to the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of two functions. In an <a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a>, 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 <a href="/facts/Probability/fgYvMhwb">probability</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the term cross-correlations refers to the <a href="/facts/Covariance_and_correlation/ARwVYuSW">correlations</a> between the entries of two <a href="/facts/Multivariate_random_variable/qMfooyVf">random vectors</a> 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 and 
 
 
 
 
 Y
 
 
 
 {\displaystyle \mathbf {Y} }
 
, while the correlations of a random vector 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 are the correlations between the entries of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 itself, those forming the <a href="/facts/Correlation_matrix/egkluAEm">correlation matrix</a> of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
. If each of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 and 
 
 
 
 
 Y
 
 
 
 {\displaystyle \mathbf {Y} }
 
 is a scalar random variable which is realized repeatedly in a <a href="/facts/Time_series/fSXPR817">time series</a>, then the correlations of the various temporal instances of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 are known as autocorrelations of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
, and the cross-correlations of 
 
 
 
 
 X
 
 
 
 {\displaystyle \mathbf {X} }
 
 with 
 
 
 
 
 Y
 
 
 
 {\displaystyle \mathbf {Y} }
 
 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 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are two <a href="/facts/Independent_(probability)/NUzQtnUL">independent</a> <a href="/facts/Random_variable/TwTBXnLT">random variables</a> with <a href="/facts/Probability_density_function/zvfybna4">probability density functions</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
, respectively, then the probability density of the difference 
 
 
 
 Y
 −
 X
 
 
 {\displaystyle Y-X}
 
 is formally given by the cross-correlation (in the signal-processing sense) 
 
 
 
 f
 ⋆
 g
 
 
 {\displaystyle f\star g}
 
; however, this terminology is not used in probability and statistics. In contrast, the <a href="/facts/Convolution/PVPrdz9J">convolution</a> 
 
 
 
 f
 ∗
 g
 
 
 {\displaystyle f*g}
 
 (equivalent to the cross-correlation of 
 
 
 
 f
 (
 t
 )
 
 
 {\displaystyle f(t)}
 
 and 
 
 
 
 g
 (
 −
 t
 )
 
 
 {\displaystyle g(-t)}
 
) gives the probability density function of the sum 
 
 
 
 X
 +
 Y
 
 
 {\displaystyle X+Y}
 
.

Cross-correlation open-in-new

Cross-correlation