Autocorrelation functions statistical analysis

Setup and principle of FCS and FCCS have been reviewed extensively previously [12,13]. The technique is based on the statistical analysis of equilibrium fluorescence fluctuations induced by, e.g., the d3mamics of fluorescent molecules in a tiny observation volume. By correlating these fluctuations with itself at a later time r, an autocorrelation curve is obtained, which can be fitted to an appropriate model function to extract the characteristic time scales of the system. The two basic parameters of a FCS autocorrelation curve are the decay time, reflecting time scales of molecules dynamics, and the amplitude, indicating the average number of particles in the detection volume. 

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-... 

Both statistical methods and artifical neural networks need a fixed number of descriptors for the analysis of a set of molecules independent of their size and number of atoms. In order to transform the information from the structural diagram into a representation with a fixed number of components an autocorrelation function can be used ... 

Uncorrelated Data In the first step of data analysis, it should be checked whether the data are uncorrelated or correlated. Uncorrelated data do not show any trends in their autocorrelation function (Figure 3.23). Note how small the r(r) values are for the empirical autocorrelations in Figure 3.23. Such data can be described by the methods discussed in Chapter 2. In other words, uncorrelated data are a prerequisite to apply the methods of descriptive statistics discussed in Chapter 2. 

Rigorous statistical mechanical analysis indicates that the dielectric relaxation function 0(f) of an isotropic system in the linear response regime is equivalent to an autocorrelation function of a microscopic polarization p(f) fluctuating through the molecular motion at equilibrium (Cole, 1967 Kubo, 1957) ... 

Statistical analysis—fitting models to measured autocorrelation functions... 

On-line estimation of the second-order statistics of a speech signal from a sample function of the noisy signal has proven to be a better choice. Since the analysis frame length is usually relatively small, the covariance matrices of the speech signal maybe assumed Toeplitz. Thus, the autocorrelation function of the clean signal in each analysis frame must be estimated. The Fourier transform of a windowed autocorrelation function estimate provides estimates of the variances of the clean signal spectral components. 

The methods used for expressing the data fall into two categories, time domain techniques and frequency domain techniques. The two methods are related because frequency and time are the reciprocals of each other. The analysis technique influences the data requirements. Reference 9 provides a brief overview of the various mathematical methods and a multitude of additional references. Specialized transforms (Fourier) can be used to transfer information between the two domains. Time domain measures include the normal statistical measures such as mean, variance, third moment, skewness, fourth moment, kurto-sis, standard deviation, coefficient of variance, and root mean squEire eis well as an additional parameter, the ratio of the standard deviation to the root mean square vtJue of the current (when measuring current noise) used in place of the coefficient of variance because the mean could be zero. An additional time domain measure that can describe the degree of randonmess is the autocorrelation function of the voltage or current signal. The main frequency domain... 

The analysis is based on averages of four realizations of reconstructions of two different porous media with c=0.42 and 8=0.355, respectively. Typical cross sections for each medium are shown in Fig. 2. A comparison between the experimental autocorrelation functions (SANS data) and the autocorrelation functions measured on the reconstructed media is presented in Fig. 3a and 3b. It is evident that the stochastic reconstructions exhibit nearly identical autocorrelation functions with the experimentally observed ones. This indicates that the reconstructed materials respect the basic statistical content of the actual porous materials. 

The analysis involved deconvolution by iterative reconvolution, background subtraction, and optional correction for shift of the instrument response function. Statistical tests included chi-square, the Durbin-Watson test, the covariance matrix, a runs test, and the autocorrelation function [6]. An alternative form of data analysis involves distributions of lifetimes rather than a series of exponentials. Differentiation of systems obeying a decay law made up of three discrete components from systems where there exists a continuous distribution of lifetimes, or a distribution plus one or more discrete components, is a nontrivial analytical problem. Methods involving the minimization of the chi-square parameter are commonly used, but recently the maximum entropy method (MEM) has gained popularity [7]. Inherent in the MEM method is the theoretical lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Recent work has compared the MEM with a new version of the exponential series method (ESM) which allows use of the same size probe function as the MEM and found that the two methods gave comparable results [8]. 

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