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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.
Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices. For example, for discrete-time signals f [ k ] {\displaystyle f[k]} and g [ k ] {\displaystyle g[k]} the cross-covariance is defined as
If cross-correlation is plotted, the result is called a cross-correlogram. The correlogram is a commonly used tool for checking randomness in a data set . If random, autocorrelations should be near zero for any and all time-lag separations.
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. [1] If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an ...
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution () differs from cross-correlation only in that either () or () is reflected about the y-axis in convolution; thus it is a cross-correlation of () and (), or () and ().
where G xy (f) is the Cross-spectral density between x and y, and G xx (f) and G yy (f) the auto spectral density of x and y respectively. The magnitude of the spectral density is denoted as |G|. Given the restrictions noted above (ergodicity, linearity) the coherence function estimates the extent to which y(t) may be predicted from x(t) by an ...