
Cross-correlation
Covariance and correlation / From Wikipedia, the free encyclopedia
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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.
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In probability and statistics, the term cross-correlations refers to the correlations between the entries of two random vectors and
, while the correlations of a random vector
are the correlations between the entries of
itself, those forming the correlation matrix of
. If each of
and
is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of
are known as autocorrelations of
, and the cross-correlations of
with
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 and
are two independent random variables with probability density functions
and
, respectively, then the probability density of the difference
is formally given by the cross-correlation (in the signal-processing sense)
; however, this terminology is not used in probability and statistics. In contrast, the convolution
(equivalent to the cross-correlation of
and
) gives the probability density function of the sum
.
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