Cross-correlation

Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
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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 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 correlations between the entries of two random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$, while the correlations of a random vector ${\displaystyle \mathbf {X} }$ are the correlations between the entries of ${\displaystyle \mathbf {X} }$ itself, those forming the correlation matrix of ${\displaystyle \mathbf {X} }$. If each of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of ${\displaystyle \mathbf {X} }$ are known as autocorrelations of ${\displaystyle \mathbf {X} }$, and the cross-correlations of ${\displaystyle \mathbf {X} }$ with ${\displaystyle \mathbf {Y} }$ 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are two independent random variables with probability density functions ${\displaystyle f}$ and ${\displaystyle g}$, respectively, then the probability density of the difference ${\displaystyle Y-X}$ is formally given by the cross-correlation (in the signal-processing sense) ${\displaystyle f\star g}$; however, this terminology is not used in probability and statistics. In contrast, the convolution ${\displaystyle f*g}$ (equivalent to the cross-correlation of ${\displaystyle f(t)}$ and ${\displaystyle g(-t)}$) gives the probability density function of the sum ${\displaystyle X+Y}$.