What is "cross-correlation"

n. A notion used across various fields of scientific study that shows the relation between members of two or more groups of data.

In signal processing, cross-correlation is a measure of similarity of two series as a function of the lag 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.

For continuous functions f and g, the cross-correlation is defined as:

$(f \star g)(\tau)\ \stackrel{\mathrm{def}}{=} \int_{-\infty}^{\infty} f^*(t)\ g(t+\tau)\,dt,$

where f denotes the complex conjugate of f and τ is the lag.

Similarly, for discrete functions, the cross-correlation is defined as:

$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$

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 power.

In probability and statistics, the term cross-correlations is used for referring to the correlations between the entries of two random vectors X and Y, while the autocorrelations of a random vector X are considered to be the correlations between the entries of X itself, those forming the correlation matrix (matrix of correlations) of X. This is analogous to the distinction between autocovariance of a random vector and cross-covariance of two random vectors. One more distinction to point out is that 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 Y − X is formally given by the cross-correlation (in the signal-processing sense) f ⋆ g; however this terminology is not used in probability and statistics. In contrast, the convolution f * g (equivalent to the cross-correlation of f(t) and g(−t) ) gives the probability density function of the sum X + Y.