Correlation matrices

The correlation matrix, C, can be retrieved from the variance matrix according to the following equation  

Each component in is within the range -1-1. If the correlation equals 0, there would be no correlation between the parameters, and if the correlation is close to 1 or -1, the parameters are highly correlated. This can be a consequence of  

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All the other infonnation needed for this process is contained in the two time correlation matrix because the process is Gaussian. A somewhat involved calculation [18] results (for 2 > in... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The essential degrees of freedom are found by a principal component analysis of the position correlation matrix Cy of the cartesian coordinate displacements Xi with respect to their averages xi), as gathered during a long MD run ... 

Kelkar and McCarthy (1995) proposed another method to use the feedforward experiments to develop a kinetic model in a CSTR. An initial experimental design is augmented in a stepwise manner with additional experiments until a satisfactory model is developed. For augmenting data, experiments are selected in a way to increase the determinant of the correlation matrix. The method is demonstrated on kinetic model development for the aldol condensation of acetone over a mixed oxide catalyst. 

It can be shown that all symmetric matrices of the form X X and XX are positive semi-definite [2]. These cross-product matrices include the widely used dispersion matrices which can take the form of a variance-covariance or correlation matrix, among others (see Section 29.7). 

The matrix Cp contains the variances of the columns of X on the main diagonal and the covariances between the columns in the off-diagonal positions (see also Section 9.3.2.4.4). The correlation matrix Rp is derived from the column-standardized matrix Zp ... 

The covariances between the parameters are the off-diagonal elements of the covariance matrix. The covariance indicates how closely two parameters are correlated. A large value for the covariance between two parameter estimates indicates a very close correlation. Practically, this means that these two parameters may not be possible to be estimated separately. This is shown better through the correlation matrix. The correlation matrix, R, is obtained by transforming the co-variance matrix as follows... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

Evidently in the linear regime the probability is Gaussian, and the correlation matrix is therefore given by... 

This last point may be seen more clearly by defining a time-correlation matrix related to the inverse of this,... 

Here we have used the symmetry and commuting properties of the matrices to obtain the final line. This shows that the correlation matrix goes like... 

Correlation matrix, linear thermodynamics, regression theorem, 17-20 Coupled cluster (CC), ab initio calculations, P,T-odd interactions, 254-259 Coupled continuum, two-pathway excitation, coherence spectroscopy isolated resonances, 168-169 structureless excitation, 167 CPT theorem ... 

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

In contrast to correlation matrix the covariance matrix is scale-dependent. In case of autoscaled variables the covariance matrix equals the correlation matrix. 

Using standard statistical notation (Malinowski, 2003) the correlation matrix C is given by... 

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