Random process autocorrelation function

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

A modified Langevin equation can be derived for any property 0t. In addition the memory function will be related to the autocorrelation function of the random force in this equation. These results can be extended to multivariate processes. 

Data generally are classified as either deterministic or random. Deterministic means that the process under study can be described by an explicit mathematical relationship. Random means that the phenomenon under study cannot be described by an explicit mathematical function because each observation of the phenomenon is unique. A single representation of a random phenomenon is called a sample function. If, as in all experiments, the sample function is of finite length, then it is called a sample record. The set of all possible sample functions, y r), which the random process might produce, defines the random or stochastic process. The mean value and autocorrelation function for a random process are defined by... 

A random process is weakly stationary if its mean value and autocorrelation function are independent of r. Thus, for a weakly stationary random process, the mean value is a constant [fJiy r) = fiy] and the autocorrelation function depends only on the spatial lag 6 [e.g., Ryir, r + 6) = Ry d)]. A random process is strongly stationary if the infinite collection of higher order statistical moments and joint moments are space invariant. Most geophysical phenomena are not strongly stationary. However, the random process under study must be at least weakly stationary, otherwise the results of the space- or time-series analysis can be suspect. An extensive treatment of these statistical concepts is available 45, 46). A detailed re-... 

A Levy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the fluctuations in the fluid velocity [62]. In the present example the continuum form of the Levy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is... 

Correlated Data A random time series with low correlations between observations provides an autocorrelation function as shown in Figure 3.24. In the case of a stationary process of the first order, the function can be described by the following exponential model ... 

Random Processes with Drift and Periodicities Theoretically, fluctuation about zero is expected for a random process after a decrease of r t). In the case of a drifting process, the autocorrelation function remains at a positive or negative correlation level (cf. Figure 3.25). 

On the basis of the Viner-Hinchin theorem the autocorrelation function of the stationary random process is represented by the Fourier integral ... 

A stationary stochastic process can be characterized by the autocorrelation function of velocity i (r), where r is a time delay. The autocorrelation function measures the persistence of a given value of the random variable concerned. For particle diffusing velocities, when the particle possesses a given y-directed velocity component v t), a short time later it is likely to have a veloeity of similar magnitude and sign, v(t +1). The velocity covariance can be formed for the particle as the mean of the product of the two velocities. 

A random motion of nanoparticles in aqueous suspension changes the time intensity of the scattered light and the fluctuating signal is processed by forming the autocorrelation function G(t). For a monodisperse suspension of globular particles in Brownian motion (Dahneke 1983, Chu 1991, Brown 1993, Hunter 1993), it can be written as follows ... 

The velocity autocorrelation function tells you how fast a particle forgets its initial velocity, owing to Brownian randomization. When the time t is short relative to the correlation time of the physical process, the particle velocity will be nearly unchanged from time 0 to t, and v(t) will nearly equal v(0) so (u(O)v(t)) greater than the system s correlation time. Brownian motion will have had time to randomize the particle velocity relative to its initial velocity, so v it) will be uncorrelated with u (0). This means that v(0)v(t) will be negative just as often as it is positive, so the ensemble average will be zero, (u(O)u(t)) = 0. 

The methods for calculating A/Zexc( ). an equilibrium property of the system, have already been discussed. The dynamic aspects of the permeation process are captured in D (2 ). The fluctuation-dissipation theorem provides the connection between the local diffusion constant and the time autocorrelation function of the random force acting on the solute. 

As said, the Fourier transform of a stationary random process X t) usually does not exist. However, the Fourier transform of the autocorrelation function R x) always exists. To understand what is an autocorrelation function consider a generic random spectrum like that shown in Fig. 8.39 and sample the value of the variable X(f) at two different times ti and 2 = fi -I- t. For digital data the autocorrelation function R x) is a function of the time interval x — t2 — ti defined as... 

Then, the autocorrelation function R x) of a random process is the mean value of the product X(fi) X(t2) that can be approximated by the average value of the same product which can be obtained by sampling the random variable cr at times ti... 

However, in real life situations, wind flow is not deterministic but has random fluctuations about a mean value. Hence, this randomness must be included in the model in order to gain better insights on the behavior of system. Therefore, the nondimensional free stream velocity U is assumed to be a stationary Gaussian random process with the autocorrelation function as given in Eq. 55 with oy= 1 and Cq = 1 x 10 This implies a correlation length of 2.628 x 10 s. It must be... 

The stochastic analysis of structural vibrations deals with the description and characterization of structural loads and responses that are modeled as stochastic processes. The probabilistic characterization of the input process could be extremely complex in time domain where the probability density functions depend on the autocorrelation functions which experimentally have to be specified over given set points. Since this approach is difficult to be used in applications, stochastic vibration analysis of structural linear systems subjected to Gaussian input processes is quite often performed in the frequency domain by means of the spectral analysis. This analysis is a very powerful tool for the analytical and experimental treatment of a large class of physical as well as structural problems subjected to random excitations. The main reasons are... 

This relationship shows that the stationary counterpart of the multi-correlated stochastic process is a vector process, N(( ) (of order AT), with orthogonal increments. Furthermore, Gnn(Hermitian matrix function which describes the one-sided PSD function matrix of the so-called embedded stationary counterpart vector process, N(m). After some algebra it can be proved that the autocorrelation function matrix of the zero-mean Gaussian nonstationary random vector process F(t) can be obtained as... 

A point that has not been investigated is the possibility of considering u(k) a coloured noise instead of white noise, and therefore a non diagonal E. For example, the choice of a tridiagonal Ey would imply the assumption of u(k) a random walk process. On the one hand, by imposing a correlation among successive values of u(k), the flexibility of the output is reduced, and for example a delta function could not be recuperated. On the other hand, smoother outputs and better solutions could be obtained if good "a priori" estimations of the real autocorrelations of u(k) could be provided. 

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