Crosscorrelation
Covariance and correlation / From Wikipedia, the free encyclopedia
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In signal processing, crosscorrelation 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 innerproduct. 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 crosscorrelation is similar in nature to the convolution of two functions. In an autocorrelation, which is the crosscorrelation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.
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In probability and statistics, the term crosscorrelations refers to the correlations between the entries of two random vectors $\mathbf {X}$ and $\mathbf {Y}$, while the correlations of a random vector $\mathbf {X}$ are the correlations between the entries of $\mathbf {X}$ itself, those forming the correlation matrix of $\mathbf {X}$. If each of $\mathbf {X}$ and $\mathbf {Y}$ is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of $\mathbf {X}$ are known as autocorrelations of $\mathbf {X}$, and the crosscorrelations of $\mathbf {X}$ with $\mathbf {Y}$ across time are temporal crosscorrelations. 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 $X$ and $Y$ are two independent random variables with probability density functions $f$ and $g$, respectively, then the probability density of the difference $YX$ is formally given by the crosscorrelation (in the signalprocessing sense) $f\star g$; however, this terminology is not used in probability and statistics. In contrast, the convolution $f*g$ (equivalent to the crosscorrelation of $f(t)$ and $g(t)$) gives the probability density function of the sum $X+Y$.
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