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
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 correlation matrix 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 time series, 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 independent random variables with probability density functions
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 convolution
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}
.
