Pair correlation coefficients

Given that Wj. r) we finally obtain for the pair correlation coefficient... 

Let s consider a related question What does a twin pair-correlation coefficient tell us anyway If the correlation is high, does that mean the cognitive performance of the twins is very similar Well, sometimes yes and sometimes no. I want to demonstrate here how it s possible for the answer to be no. We... 

Quadruply excited configuration Coefficient from 37-configuration wave function by full C.I. Unlinked four-cluster coefficient calculated from pair correlation coefficients ... 

Al. Statistical-mechanical computations of the second virial coefficient and pair-correlation-coefficients of liquid N2, O2,... 

Intercorrelation coefficients are then computed. These tell when one descriptor is redundant with another. Using redundant descriptors increases the amount of fitting work to be done, does not improve the results, and results in unstable fitting calculations that can fail completely (due to dividing by zero or some other mathematical error). Usually, the descriptor with the lowest correlation coefficient is discarded from a pair of redundant descriptors. 

In addition to analyzing the residuals, it may be desirable to determine the degree of agreement between sets of paired measurements and estimates. The linear correlation coefficient is... 

The temporal correlation coefficient at each monitoring location can be calculated by analysis of the paired values over a time period of record. The spatial correlation coefficient at a given time can be calculated by analysis of the paired values from each station. For the spatial correlation... 

Comparisons (49) of measured concentrations of SFg tracer released from a 36-m stack, and those estimated by the PTMPT model for 133 data pairs over PasquiU stabilities varying from B through F, had a linear correlation coefficient of 0.81. Here 89% of the estimated values were within a factor of 3 of the measured concentrations. The calculations were most sensitive to the selection of stability class. Changing the stability classification by one varies the concentration by a factor of 2 to 4. 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

When these are close together, most of the simultaneously measured velocities will relate to fluid in the same eddy and the correlation coefficient will be high. When the points are further apart the correlation coefficient will fall because in an appreciable number of the pairs of measurements the two velocities will relate to different eddies. Thus, the distance apart of the measuring stations at which the correlation coefficient becomes very poor is a measure of scale of turbulence. Frequently, different scales of turbulence can be present simultaneously. Thus, when a fluid in a tube flows past an obstacle or suspended particle, eddies may form in the wake of the particles and their size will be of the same order as the size of the particle in addition, there will be larger eddies limited in size only by the diameter of the pipe. 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

At the simplest level the orientational correlation of molecular pairs can be characterised by the averages of the even Legendre polynomials Pl(cos J ij) where is the angle between the symmetry axes of molecules i and j separated by a distance r. This correlation coefficient is denoted by... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

Elliott et al. utilized a clenbuterol immunoassay to determine clenbuterol residues in cattle tissues and fluids. The LOD was 0.25 ug for liver. Animals were dosed with medicated feed (1.6 ug kg per day), and pairs were slaughtered during the medication phase and at 14,28, and 42 days after withdrawal. Clenbuterol concentrations in liver and retina/choroid samples were confirmed by GC/MS. Correlation coefficients between the ELISA and GC/MS were = 0.92 for retina/choroid samples and... 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

The correlation coefficient ranges between -1 and +1. An error correlation of+1 indicates that the errors in both x and y for any particular measurement of an x, y pair will always be in the same direction (either both greater than, or both less than the true values). An error correlation of -1 indicates that the x error will always be in the opposite direction than the y error. An error correlation of zero indicates that the x- andy -errors are independent of one another. 

Increasing the excess of ethanol increases the conversion of acetic acid to ethyl acetate. To carry out the calculation more accurately would require activity coefficients to be calculated for the mixture (see Poling, Prausnitz and O Connell6 and Chapter 4). The activity coefficients depend on correlating coefficients between each binary pair in the mixture, the concentrations and temperature. 

The final key point to note about the MND, which can also be seen from Figure 1-2, is the fact that when the MND is projected onto the plane defined by any two axes of the coordinate system the data may show some correlation (as does the data in Figure 1-2). In fact, the projection onto any of the planes defined by two of the axes will have some value for the correlation coefficient between the corresponding pair of variables. The amount of correlation between projections along any pair of axis can vary from zero, in which case the data would lie in a circular blob, to unity, in which case the data would all lie exactly on a straight line. 

Thus a comparison of the correlation coefficient between two or more sets of X, Y data pairs cannot be adequately performed unless the standard deviations of the two data sets are nearly identical or unless the correlation coefficient confidence limits for the data sets are compared. In summary, if one Set A of X, Y paired data has a correlation of 0.95, this does not necessarily indicate that it is more highly correlated than a second Set B of X, Y paired data with a correlation of say 0.90. The meaning of this will be described in greater detail later. 

The confidence limits for a correlation coefficient for a given number of X, Y pairs (n) at a specified confidence limit is calculated as Z2 (Equation 60-16). 

The following description and corresponding MathCad Worksheet allows the user to test if two correlation coefficients are significantly different based on the number of sample pairs (N) used to compute each correlation. For the Worksheet, the user enters the confidence level for the test (e.g., 0.95), two comparative correlation coefficients, r, and r2, and the respective number of paired (X, Y) samples as N and N2. The desired confidence level is entered and the corresponding z statistic and hypothesis test is performed. A Test result of 0 indicates a significant difference between the correlation coefficients a Test result of 1 indicates no significant difference in the correlation coefficients at the selected confidence level. 

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