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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.
[A] For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations. [1]
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 ...
The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), [5] and some value in the open interval (,) in all other cases, indicating the degree of linear dependence between the variables. As it ...
In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function. Correlation functions describe how microscopic variables, such as spin and density, at different positions are related.
The r* cross-correlation metric is based on the variance metrics of SSIM. It's defined as r*(x, y) = σ xy / σ x σ y when σ x σ y ≠ 0, 1 when both standard deviations are zero, and 0 when only one is zero. It has found use in analyzing human response to contrast-detail phantoms. [18] SSIM has also been used on the gradient of ...
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.