Cross-correlation matrix

Table 3 Cross-Correlation Matrix for Neutral Solute Data in Table... 

Table 4 Cross-Correlation Matrix for Ionic Solute Data in Table 2 ...

Cross-correlation matrix for the MEKC database. Data from Poole and Poole (30). 

One can (at one s peril) freeze the elements of the cross correlation matrix at their initial value (which is the frozen Gaussian approximation) or evaluate their time dependence from the closed set of equations of motion... 

Cross-correlation matrix (r ) Model Statistics System constants ... 

TABLE IV. Cross-correlation Matrix for Some of the Substituent Parameters Used in Generating Equations 7-11 (where n-25)... 

The concept of the autoassociative memory was extended to bidirectional associative memories (BAM) by Kosko (1987,1988). This memory, shown in Fig. 19.30, is able to associate pairs of the patterns a and b. This is the two-layer network with the output of the second layer connected directly to the input of the first layer. The weight matrix of the second layer is and W for the first layer. The rectangular weight matrix W is obtained as a sum of the cross-correlation matrixes... 

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 adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

Linear representations are by far the most frequently used descriptor type. Apart from the already mentioned structural keys and hashed fingerprints, other types of information are stored. For example, the topological distance between pharmacophoric points can be stored [179, 180], auto- and cross-correlation vectors over 2-D or 3-D information can be created [185, 186], or so-called BCUT [187] values can be extracted from an eigenvalue analysis of the molecular adjacency matrix. 

The matrix Rxx (x) is not a symmetrical matrix. It contains cross-correlation coefficients between the variables x, x2, and x3 as off-diagonal elements. A symmetrical matrix is, therefore, produced by averaging the cross-correlation elements ... 

PCA is a statistical technique that has been used ubiquitously in multivariate data analysis." Given a set of input vectors described by partially cross-correlated variables, the PCA will transform them into a set that is described by a smaller number of orthogonal variables, the principle components, without a significant loss in the variance of the data. The principle components correspond to the eigenvectors of the covariance matrix, m, a symmetric matrix that contains the variances of the variables in its diagonal elements and the covariances in its off-diagonal elements (15) ... 

A tensor is a mathematical representation of a three dimensional shape using a three by three matrix (Figure 50.9, Eq. 8). The matrix represents the total magnitude of the object in each of nine dkections, three of which are equal due to symmetry (e.g. Dxy = Dyx) of the other three cross components, reducing the problem to six dimensions. Hence, six directions of diffusion measurement are the minimum needed to define the diffusion tensor. More measures can be used to improve the accuracy of the measurement through cross correlation of multiple measures. 

Let us consider now more in detail the simplest nontrivial case of a multi-component mixture that is a binary fluid. We first simplify some expressions presented above, and let us start from the elements of the memory functions matrix. Taking into account the relations (35) and (36), we can introduce the normalized generalized mutual diffusion coefficient D(k,z), the normalized generalized thermal diffusion coefficient L)- (k. z), and the cross-correlation coefficient ((k, z) as follows,... 

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