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The definition of the autocorrelation coefficient of a stochastic ... The traditional test for the presence of first-order autocorrelation is the Durbin ...
A smaller focus of the laser beam yields a coarser speckle pattern, a lower number of speckle on the detector, and thus a larger second-order autocorrelation. The most important use of the autocorrelation function is its use for size determination.
The first order correlation function, measured at the same time and position gives us the intensity i.e. () (,) =. The classical nth order normalized correlation function is defined by dividing the n-th order correlation function by all corresponding intensities:
Correlogram example from 400-point sample of a first-order autoregressive process with 0.75 correlation of adjacent points, along with the 95% confidence intervals (plotted about the correlation estimates in black and about zero in red), as calculated by the equations in this section.
Classification of the different kinds of optical autocorrelation. In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort pulses produced by modelocked lasers.
Most systems with chemical relaxation also show measurable diffusion as well, and the autocorrelation function will depend on the details of the system. If the diffusion and chemical reaction are decoupled, the combined autocorrelation is the product of the chemical and diffusive autocorrelations.
In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with , element
It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.