Autocorrelation, definition

Autocorrelation Illustration. We choose a shape function Y (r) which describes a particle in 2D space (cf. Fig. 2.4a). Because of the definition of Y (r), T 2 (r) takes the value of the volume which is shared by the particle and its imagined ghost which is displaced by r. In any case the overlap integral becomes maximal for r = 0. Here the correlation is perfect. 

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. 

This may be easily proved by applying the definition of the Fourier transform. It is a version of the autocorrelation theorem. The authors elaborate in a later paper (Kawata and Ichioka, 1980b). 

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 ... 

Several types of autocorrelation are often used for landscapes. In several important papers, Weinberger and Stadler consider both autocorrelation between adjacent points along a random walk in the landscape and autocorrelation between points a given Hamming distance apart independent of any walk [67,77,78,82,83], Both definitions yield similar information about the landscape and can be computed from one another for stationary landscapes. Other types of autocorrelation are based on neighborhoods defined by complex mutation operations such as crossover [45-49,85],... 

Eq. (351) can be transformed to Eq. (359). Further identifying Ns with 2ti p( ), Eq. (346) becomes identical with Eq. (361). Hence, under certain circumstances the quantized ARRKM theory is equivalent to the rigorous quantum reaction rate theory. A number of remarks are in order. First, assumption (a) is automatically satisfied by definition. Second, assumption (b) implies that Fw in the quantized ARRKM theory be the direct analog of the quantum flux operator in the flux-flux autocorrelation formalism. Third, assumption (c) requires that the action of the operator 0jy(V5 v) at any particular time, say at time zero, is equivalent to the action of the projector P i) at time infinity. Regarding 0vi (V5 v) as the analog... 

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 ... 

In accord with the general definition [GT2, Eq. (22b)] the dipolar autocorrelator (spectral function) can be written in our notation as... 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

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

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