A plot showing 100 random numbers with a " hidden " sine function, and an autocorrelation ( correlogram ) of the series on the bottom.

In statistics, the autocorrelation of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference.

* The Small-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the electron density.

* Partial autocorrelation function

* GPU accelerated calculation of autocorrelation function.

A well-known result of Fourier transforms is the autocorrelation theorem, which states that the autocorrelation c ( r ) of a function f ( r )

Therefore, the autocorrelation function c ( r ) of the electron density ( also known as the Patterson function ) can be computed directly from the reflection intensities, without computing the phases.

The Wiener – Khinchin theorem, ( or Wiener – Khintchine theorem or Khinchin – Kolmogorov theorem ), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

The autocorrelation function of the 10Hz Doppler Rayleigh fading channel.

The normalised autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind:

The autocorrelation function of the Rayleigh fading channel shown above with 10 Hz maximum Doppler shift is shown in the figure.

With its strong autocorrelation function, the forward pilot allows mobiles to determine system timing and distinguish different BTS's for handoff.

Using such formal reasoning, one may already guess that for a stationary random process, the power spectral density and the autocorrelation function of this signal, should be a Fourier pair.

If the signal is not wide-sense stationary, or strictly stationary, then the autocorrelation function must be a function of two variables.

* The spectral density of and the autocorrelation of form a Fourier transform pair ( for PSD versus ESD, different definitions of autocorrelation function are used ).

A similar result holds for the total power in a power spectral density being equal to the corresponding mean total signal power, which is the autocorrelation function at zero lag.

In this case, the measure of correlation is the autocorrelation function ( sometimes called self-coherence ).

By default, correlation function refers to the two-point autocorrelation function.

To a complex electric field corresponds an intensity and an intensity autocorrelation function defined by

Spatial patterns, such as the distribution of a species, are the result of either true or induced spatial autocorrelation.

After the spatial distribution of the variables is determined through discrete sampling, statistical methods are used to quantify the magnitude, intensity, and extent of spatial autocorrelation present in the data ( such as correlograms, variograms, and peridograms ), as well as to map the amount of spatial variation.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

* In statistics, spatial autocorrelation between sample locations also helps one estimate mean value uncertainties when sampling a heterogeneous population.

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 function being made up of autocorrelations.

Sometimes, algorithms can be used to determine the amount of autocorrelation between samples and the value of ( the period between samples that are actually used ) computed from this, but in practice there is a fair amount of " black magic " involved.

* The process of simulated annealing is often used to reduce the " random walk " behavior in the early part of the sampling process ( i. e. the tendency to move slowly around the sample space, with a high amount of autocorrelation between samples, rather than moving around quickly, as is desired ).

The goal of these variations is to reduce the autocorrelation between samples sufficiently to overcome any added computational costs.

There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function ( which is itself obtained from the covariances ).

In addition, there has been a flurry of activity extending FCS in various ways, for instance to laser scanning and spinning-disk confocal microscopy ( from a stationary, single point measurement ), in using cross-correlation ( FCCS ) between two fluorescent channels instead of autocorrelation, and in using Förster Resonance Energy Transfer ( FRET ) instead of fluorescence.

Other techniques that may reduce autocorrelation are collapsed Gibbs sampling, blocked Gibbs sampling, and ordered overrelaxation ; see below.

There is lot of spatial autocorrelation in urban land uses ; it ’ s driven by historical path dependence: this sort of thing got started here and seeds more of the same.

This is because the autocorrelation of a MLS is 1 for zero-lag, and nearly zero (− 1 / N where N is the sequence length ) for all other lags ; in other words, the autocorrelation of the MLS can be said to approach unit impulse function as MLS length increases.

" All short-range dependent processes are characterized by an autocorrelation function which decays exponentially fast ; processes with long-range dependence exhibit a much slower decay of the correlations-their autocorrelation functions typically obey some power law.

The simplest FCS experiment is thus normal 3D diffusion, for which the autocorrelation is:

The complementary codes first discussed by Golay were pairs of binary complementary codes and he noted that when the elements of a code of length N were either or 1 it followed immediately from their definition that the sum of their respective autocorrelation sequences was zero at all points except for the zero shift where it is equal to K * N. ( K being the number of code words in the set ).

Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum ( the intensity of each frequency ) to its autocorrelation.

The optical transfer function T ( w ) of an optical system is given by the autocorrelation of its pupil function f ( x, y ):

At long times the flux at time t, J ( t ), is uncorrelated with its value a long time earlier J ( 0 ) and the autocorrelation function decays to zero.

The value of the autocorrelation function at lag is the power of, or its variance if the mean value of is zero:

This type of GRF is completely described by its power spectral density, and hence, through the Wiener-Khinchin theorem, by its two-point autocorrelation function, which is related to the power spectral density through a Fourier transformation.

A BS is pseudo-random ( PRBS ) if its autocorrelation function:

As an example, raw FCS data and its autocorrelation for freely diffusing Rhodamine 6G are shown in the figure to the right.