Correlated/uncorrelated

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. 

Correlated events are related in time and this time relation can be measured either with respect to an external clock or to the events themselves. Random or uncorrelated events bear no fixed time relation to each other but, on the other... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

I Principal Component Analysis (PCA) transforms a number of correlated variables into a smaller number of uncorrelated variables, the so-called principal components. 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

The correlation function is a number between -1 and 1, where 1 indicates that the two quantities are completely correlated, -1 that they are (completely) anti-correlated and 0 means that they are independent (uncorrelated). 

We recall that the MFT assumes that does not induce any correlation between separated sites if all of the sites are mutually uncorrelated at i = 0, MFT assumes that they remain uncorrelated at all later times t > 0. A virtue of this approach is that it permits an easy derivation of the limiting value density, pt->oo- Because the underlying assumption is generally not valid, however, we should hardly be surprised to learn that the limiting densities obtained for most of the interesting (i.e, nonlinear) rules differ significantly from those obtained by Monte Carlo simulations of those same rules. 

Setting p = p = Pf, at equilibrium, we find that the only real stable nonzero solution for 0 < pe < 1 is Pe 0.370. This uncorrelated approximation actually describes the infinite temperature limit (effectively, T >> 1) rather well, since as the temperature increases, local correlations of the basic Life rule steadily decrease. 

Of course, the effect of excluded volume is opposite and greatly exceeds that shown in Fig. 1.10, which is produced by uncorrelated collective interaction. Unfortunately, neither of them results in sign-alternating behaviour of angular or translational momentum correlation functions. This does not have a simple explanation either in gas-like or solid-like models of liquids. As is clearly seen from MD calculations, even in... 

Correlated or geminate radical pairs are produced in unimolecular decomposition processes (e.g. peroxide decomposition) or bimolecular reactions of reactive precursors (e.g., carbene abstraction reactions). Radical pairs formed by the random encounter of freely diffusing radicals are referred to as uncorrelated or encounter (P) pairs. Once formed, the radical pairs can either collapse, to give combination or disproportionation products, or diffuse apart into free radicals (doublet states). The free radicals escaping may then either form new radical pairs with other radicals or react with some diamagnetic scavenger... 

The measurement being differential, the noise which in uncorrelated between both arms is much more dangerous than correlated noise. 

The next pair of canonical variates, t2 and U2 also has maximum correlation P2, subject, however, to the condition that this second pair should be uncorrelated to the first pair, i.e. t t2 = u U2 = 0. For the example at hand, this second canonical correlation is much lower p2 = 0.55 R = 0.31). For larger data sets, the analysis goes on with extracting additional pairs of canonical variables, orthogonal to the previous ones, until the data table with the smaller number of variables has been... 

Here, f(xj x2) is sometimes called the correlation factor. Consequently, IT x, x2) =0 defines the completely uncorrelated case. However, note that in this case, i. e., for f(x2 x2) = 0, p2(x1 x2) is normalized to the wrong number of pairs, since JJ p2llconditional probability l(x2 x1). This is the probability of finding any electron at position 2 in coordinate-spin space if there is one already known to be at position 1... 

The difference between Q(x2 jq) and the uncorrelated probability of finding an electron at x2 describes the change in conditional probability caused by the correction for selfinteraction, exchange and Coulomb correlation, compared to the completely uncorrelated situation ... 

For the parameters used to obtain the results in Fig. 3, X 0.6 so the mean free path is comparable to the cell length. If X -C 1, the correspondence between the analytical expression for D in Eq. (43) and the simulation results breaks down. Figure 4a plots the deviation of the simulated values of D from Do as a function of X. For small X values there is a strong discrepancy, which may be attributed to correlations that are not accounted for in Do, which assumes that collisions are uncorrelated in the time x. For very small mean free paths, there is a high probability that two or more particles will occupy the same collision volume at different time steps, an effect that is not accounted for in the geometric series approximation that leads to Do. The origins of such corrections have been studied [19-22]. 

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