Autocorrelation function definition

In several papers, only the time-dependent part of the autocorrelation function is considered, and the definition is then... 

Autocorrelation function of a power signal, definition, 103 Automatic processing of standard data, outlier processing, 38-43... 

For reference, we provide brief definitions and discussions of basic statistical quantities the mean, variance, autocorrelation function, and autocorrelation time. 

In considering random fluctuations or signals, it is found from the definition of autocorrelation function that Equations IV and V also hold and further that... 

It is possible to derive an equation which describes the time evolution of the time-correlation function Cn(t) where C stands for different autocorrelation functions depending on the definition of the scalar product (i), (ii), or (iii) of Eq. (73) adopted. 

According to this equation Cn t) depends only on the values of the memory function Kn(i) for all times t prior to t. Since the autocorrelation function C//(x) is real the memory function must also be real. This can also be deduced directly from the definition of the memory function, Eq. (99). 

Experimentally the density correlations fire most important, and we therefore exemplify the construction of grand canonical correlation functions with the density autocorrelation function or the density cimiulant. Recall the definition (3.14) of the local segment density of the m-th chain ... 

One definition of the autocorrelation function, rxx, using autoscaled values (mean = 0, standard deviation = 1) is ... 

There are several technical details in a rigorous definition of the autocorrelation function for velocity. First, one has to remember the vectorial character of velocity, because clearly the direction in which the particle is knocked is important to its subsequent dynamic history. Then, according to the way it is defined, one has to take the product of the velocity at f = 0, Vg, and that at the later chosen time, v,. However, it is not as simple as just multiplying together the two veetors, Vg and v,. One has to allow for the distribution of positions and momenta of the particle in the system at the beginning, that is, at i = 0. To allow for this, one can introduce symbolically a probability distribution coefficient, g. Therefore, the expression for the autocorrelation function will involve the product gVgV,. 

The fluorescence intensity trajectories of the donor (/d(f)) and acceptor (/a(t)) give autocorrelation times (Fig. 24.2b) indistinguishable from fitting an exponential decay to the autocorrelation functions, (A/d (0) A/d (t)) and (A/a (0) A/a (t)), where A/d(t) is /d(t) — (Id), (Id) is the mean intensity of the overall trajectory of a donor, and A/a(t) has the same definition for an intensity trajectory of an acceptor. In contrast, the cross-correlation function between the donor and acceptor trajectories, (A/d (0) A/d (t)), is anticorrelated with the same decay time (Fig. 24.2b) which supports our assignment of anticorrelated fluctuations of the fluorescence intensities of the donor and acceptor to the spFRET process. 

Consider Eq. (6.84). This result was obtained for a harmonic system of identical and equivalent atoms. We could however reverse our reasoning and define a vibrational spectrum for a dense atomic system from the velocity autocorrelation function according to Eq. (6.84). Since this function can be computed for all systems, including liquids and disordered solids, we may use (6.84) as a definition of a spectrum that may be interpreted as density of modes fimction for such media. We can then use it in expressions such as (4.33), and (6.92). Is this approach to dynamics in condensed phases any good ... 

We begin with the definition of the velocity autocorrelation function of a monatomic solute... 

Note the autocorrelation function looks for correlations between residuals separated by various amounts of time. Specifically, its ith value expresses the amount of correlation between pairs of residuals separated by i - 1 time points (or recorder channels). If there is no correlation amongst the residuals, then the autocorrelation function values should be completely random. There are typically only half as many (plus one) autocorrelation function values as there are residuals, owing to the way the former are calculated. Also, by definition, the first value is equal to unity, as each residual is completely correlated with itself. Typically, autocorrelation values have not been calculated in the case of non-linear least-squares fitting of phase/modulation data. 

From Eq. 109 we find the integrodifferential equation for the autocorrelation function of the normal modes (for a definition see Eq. 49)... 

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