Data Analysis Cross-correlation

Butz T, Ceolin M, Ganal P, Schmidt PC, Taylor MA and Troger W (1996) A new approach in nuclear quadrupole interaction data analysis cross-correlation. Physica Scripta 54 234-239. 

Cross-Correlation Analysis Cross-correlation analysis is an established methodology for determining the likelihood that two spectral data sets are related to each other (Owens, 1991). Where novel analyte-specific peaks are identified in... 

A theorem, which we do not prove here, states that the nonzero eigenvalues of the product AB are identical to those of BA, where A is an nxp and where B is a pxn matrix [3]. This applies in particular to the eigenvalues of matrices of cross-products XX and X which are of special interest in data analysis as they are related to dispersion matrices such as variance-covariance and correlation matrices. If X is an nxp matrix of rank r, then the product X X has r positive eigenvalues in A and possesses r eigenvectors in V since we have shown above that ... 

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. 

It should be mentioned that rotational anisotropy of the molecule will result in an increase in the R2 values for NH vectors having particular orientation with respect to the diffusion tensor frame [46]. This increase could be misinterpreted as conformational exchange contributions, and, vice versa, small values of Rex, usually of the order or 1 s 1 or less, could be mistaken for the manifestation of the rotational anisotropy. Therefore, identification of residues subjected to conformational exchange is critical for accurate analysis of relaxation data. Additional approaches are necessary to distinguish between the two effects. As suggested earlier [27] (see also Ref. [26]), a comparison between R2 and the cross-correlation rate r]xy could serve this purpose, as tjxy contains practically the same combination of spec-... 

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). 

The fitting function used was constructed from functions derived for use in the analysis of pump-probe dynamics data [7]. The function accounts for the cross correlation of the pump and probe laser pulses and deconvolutes the laser pulse width. A pulse width of 120 fs was used for the analysis of all of the data as this width was found to best represent the cross correlation of the second and third harmonic laser pulses of the laser system. 

It has been possible by means of multivariate cross-correlation analysis to include the interaction between the metals which arise as a result of emission and transformation. The assumption is that the metals are transported at the same rate. The multivariate cross-correlation function Rxy(f) expresses a broad maximum at t = 3, i.e. 1.5 h (Fig. 6-20). This means that the mean transport rate of the metals is 3 km h Hydrological data gathered on the same day renders this result plausible ... 

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) ... 

The PLS multivariate data analysis of the training set was carried out on the descriptors matrix to correlate the complete set of variables with the activity data. From a total of 710 variables, 559 active variables remained after filtering descriptors with no variability by the ALMOND program. The PLS analysis resulted in four latent variables (LVs) with / = 0.76. The cross validation of the model using the leave-one-out (LOO) method yielded values of 0.72. As shown in Table 9.2, the GRIND descriptors 11-36, 44-49, 12-28, 13-42, 14-46, 24-46 and 34-45 were found to correlate with the inhibition activity in terms of high coefficients. 

Tomographic results. Upper-mantle velocity structure (James et al. 20016) was determined by tomographic techniques based on the analysis of delay times from teleseismic broadband waveform data. Relative arrival times of phases P, PKPdf, S and SKS were retrieved via a multichannel cross-correlation procedure using all possible pairs of waveforms (VanDecar Crosson 1990). This procedure produces highly accurate delay times, with typical standard errors for the... 

Although the pulse sequences used to study phase transitions are usually quite simple in the examples presented in this review (one to maximum four pulses), the interpretation may be subtle. Solid-state NMR nevertheless remains a difficult technique since quantitative interpretation of the spectra rely on a profound knowledge of the chemical composition and structure of the sample analysis of NMR results also requires a model to relate the observed NMR spectral shapes or relaxation behavior to hypothesis concerning the structure and dynamics of the atoms or molecules carrying spins. That NMR motionally average the atomic and molecular displacements that occur on a time-scale faster than 10—8 10—9s is an important point that should be considered in the interpretation of data. In particular, the difference in perception between NMR and X-ray diffraction with regard to fast and slow dynamical disorder in molecular crystals undergoing phase transitions between different polymorphs was illustrated. In fact, the interpretation of NMR data almost always needs the support of other data obtained by different techniques. Therefore, we emphasized the different complementarities with X-ray (or neutron) diffraction, IQNS and other spectroscopic methods to provide, by cross-correlation of the different data, consistent picture of the phase transition. 

Reverse engineering has been successfully applied to relatively simple biomolecular systems. Using a combination of cross-correlation analysis and multidimensional scaling, the glycolytic pathway was reconstructed from metabolite activity data from an in vitro enzymatic reactor system [21]. A complete spatio-temporal model of developmental gene expression in Drosophila was constructed for a small gene set based on models of differential equations and protein expression data [22],... 

Table 2 General relationships between the clay mineral distribution and the three main Earth s orbital frequency bands according to latitude, from cross-correlation spectral analysis of X-ray diffraction data on North Atlantic cores...

The direct demodulation algorithm provides a general approach to handle a large variety of image restoration or reconstruction problems. Computer simulations and analysis results for COS-B and CGRO 7-ray data show that in comparison with traditional techniques, e.g. maximum entropy method, cross-correlation deconvolution or likelihood approach, the direct demodulation method has high sensitivity, high resolution ability and capability to effectively reduce the effect of statistical fluctuations and noise in data and to simultaneously restore both the extended and discrete features in the object. 

The time critical part of this analysis is the segment averaging in the frequency domain. Figure 2 shows a flow diagram of the computer program for the determination of cross-correlations. Data have been sampled simultaneously from two anemometer probes and stored in alternating elements of an input buffer. 

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