Pairwise correlation

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

These first components of the autocorrelation coefficient of the seven physicochemical properties were put together with the other 15 descriptors, providing 22 descriptors. Pairwise correlation analysis was then performed a descriptor was eliminated if the correlation coefficient was equal or higher than 0.90, and four descriptors (molecular weight, the number of carbon atoms, and the first component of the 2D autocorrelation coefficient for the atomic polarizability and n-charge) were removed. This left 18 descriptors. 

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

When interactions Uy in a binary alloy are purely pairwise, i,e, the rhs of Eq, (4) includes only the first term, one can derive an exact relation between F ci and its derivatives over c . Firstly we note that according to Eqs, (2)-(6) the fluctuation correlator / y =< (n — Ci)(iij — cj) > is related to the free energy as... 

Fig. 1.16. Angular momentum and kinetic energy correlation functions for compressed nitrogen. MD simulation from [62], T = 300 K. The lines are continuous (800 amagat), close dotted (600 amagat), sparse dotted (400 amagat), dashed (300 amagat) and pairwise dotted (200 amagat). Reduced time units are the same as in Fig. 1.15.

Suspected correlations are displayed by calling option (Display Correlation Graph Pairwise) the coordinates can be marked either with a circle (default), or with the appropriate index so that outliers can be readily identified. 

Taken globally, the results show a remarkable adaptability of acetylcholine which can be justified considering both its intrinsic flexibility, and the fact that its intramolecular interactions are not very strong and that almost all media can compete with them. Such adaptability finds a noteworthy implication in significant pairwise correlations between physicochemical properties and geometrical descriptors as well as among physicochemical properties. Thus, Fig. 1.6 shows the revealing 3D... 

The pairwise correlation of more than two variables X, x2,..., xm is characterized by the correlation matrix R... 

Gruneisen showed that a number of properties of monatomic solids could be correlated if the pairwise interaction between atoms was of the form... 

We have presumed a taxon and some indicators. Next, we take one of these indicators and assign scores to a group of individuals on each indicator (e.g., everyone gets a score from 1 to 7 on sadness, anhedonia, and suicidality), much as one does with the Beck Depression Inventory and similar self-report scales. Finally, we examine the pairwise intercorrelations of the indicators at all possible values of all other indicators. In the depression example, we would examine the correlation of sadness and anhedonia for those who score 1 on suicidality, those who score 2 on suicidality, and so forth, up the scale to those who score 7 on suicidality. Similarly, we would examine the correlation of sadness and suicidality for those who score 1 on anhedonia, those who score 2 on anhedonia, and so forth. This would be continued for all possible combinations of indicators. 

It tests all linear contrasts among the population means (the other three methods confine themselves to pairwise comparison, except they use a Bonferroni type correlation procedure). 

Kendall s tau correlation r Kendall) also measures the extent of monotonically increasing or decreasing relationships between the variables. It is also a nonparametric measure of association. It is computationally more intensive than the Spearman rank correlation because all slopes of pairs of data points have to be computed. Then Kendall s tau correlation is defined as the average of the signs of all pairwise slopes. The range of r is —1 to +1 the method is relatively robust against outliers for many applications p and r give similar answers. 

It should be noted that the existence (or nonexistence) of one type of correlation does not, in general, imply the existence (or nonexistence) of another type of correlation. For instance, a system can be pairwise correlated but not triply correlated. In Appendix A, we present two simple probabilistic examples where there exist pair correlations but not triple correlations, and vice versa. 

This is an extension of the model discussed in Section 4.3. The assumption is made that the binding of a ligand does not affect the state of the adsorbent molecule, hence all correlations are due to direct ligand-ligand interaction. For ligand-Ugand interaction we usually assume pairwise additivity, i.e.,... 

There is no pairwise additivity neither in the sense = gabSbcSac o " the sense = gatSbe when the direct correlation abc is strictly pairwise additive. 

We compare here the average correlation in the three models of the four-site system. In the case of direct interactions only, it is intuitively clear and easily proven that the average correlation depends only on the sign of 5(2) - 1 (assuming the subunits are identical, that direct interactions are pairwise additive, and neglecting long-range interactions). Hence, when 5(2) > 1 (positive direct correlation), we always have... 

Figure 6.2. Binding isotherms and the average correlation, g(C) - 1 for the tetrahedral (T), square (S), and linear (L) models. The sites are identical and all correlations are due to direct ligand-ligand pairwise additive interactions, (a) Curves for positive cooperativity, S(2) = 10 (b) curves for negative coopera-tivity, S(2) = 0.1. Note that in these systems the cooperativity increases in absolute magnitude from L to S to T.

Referring to -RT In a as interaction energy, it was only natural to assume that each connected pair would contribute a factor a. This is equivalent to the assumption of pairwise additivity of both the triplet and quadruplet correlation. In our language, these assumptions are equivalent for the square model to... 

In the C °o limit, all the sites are bound the average correlation g(C is determined by the mth-order correlation function, which is 5 for the cyclic and 5 for the open linear system. This is true within the pairwise additive approximation for direct interaction, and neglecting long-range correlations. 

The ASTM-EFA standard method of analyzing lead In gasoline requires extraction of alkyl lead Iodide complexes Into methyllsobutylketone and a subsequent flame atomic absorption analysis of the extract A more direct method has been proposed ( ) which uses Zeeman atomic absorption analysis after sample dilution. Both methods were used to analyze a set of five field collected samples. The results showed a bias (average difference between method results) of 0.0012 g/gal with the standard flame results higher. The correlation coefficient between the results was 0.9998 0.0009, and a pairwise t-test showed no difference between the methods (6). 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

Lastly, it is desirable that parameters are able to discriminate between positive and negative conditions in a variety of experimental conditions. In other words they should be robust and reproducible. For this purpose, the Pearson correlation coefficient between all experimental repeats using control wells is calculated. Robust parameters have high Pearson correlation coefficients (above 0.7) in pairwise comparisons of experimental repeats. For this analysis we have developed another R template in KNIME to calculate the Pearson correlation coefficient between experimental runs. 

In response, two dominant algorithms were developed initially for on-lattice spin systems [89,90]. Both revolve around determining an effective cluster of spins iteratively based on pairwise energies to construct a move type capable of overcoming the correlation problem in these systems, which were typically... 

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