Weighted cross-correlation

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. 

There are, however, limits in the resolution of heterogeneous populations. In the particular case of multiple diffusing fluorescent species, for example in binding studies, the diffusion times must differ by a factor of 1.6-2.0 to be resolved, corresponding to a 4-8-fold increase in molecular weight. This limitation is alleviated in dual-color fluorescence cross-correlation spectroscopy (DC-FCCS) [116], in which the fluorescence fluctuation is simultaneously recorded in the same observation volume for two different species labeled with distinguishable fluorophores. The fraction of double-labeled and hence associated species is derived from the cross-correlation function between the two fluorescences (fig. 3.4B). 

Tugarinov V, Hwang PM, Ollerenshaw JE, Kay LE (2003) Cross-correlated relaxation enhanced H- C NMR spectroscopy of methyl groups in very high molecular weight proteins and protein complexes. J Am Chem Soc 125 10420-10428... 

The weighting function of the system g(9) can be estimated with Eq. (31) from the cross correlation function of the PRBS tracer experiment. The weighting function can be used for the validation of the model and the determination of the model parameters. Generally it is sufficient to measure for two PRBS periods. Stationary conditions are adjusted dining the first period. The measured tracer output concentrations of the second period are used for the calculation of the cross correlation function. 

With the adaptation of NMR techniques for larger molecules, it becomes possible to analyze proteins with molecular weights reaching 50 kDa. These techniques include H-, N-TROSY (transverse relaxation-optimized spectroscopy, with the mutual cancellation of H-, N-dipole-dipole coupling and the N chemical shift anisotropy) and CRINEPT (Cross-correlated Relaxation-Enhanced Polarization Transfer, combining insensitive nuclei enhanced by polarization transfer (INEPT) transfer with cross-correlated relaxation-induced polarization transfer). They are used in conjunction with the N-, c-labeling of the protein for increased sensitivity. 

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... 

The mass weighted average velocity is not determined in the vast majority of cases and, for example, in the case of cross correlation the velocity measured in the above hypothetical example would be vo. Whilst the example is merely illustrative it does demonstrate that the flow patterns within the pipeline may have a major effect on the performance of the measuring instrument. 

Dielectric relaxation thus resembles self-diffusion. Both processes observe the motion of single macromolecules through a uniform albeit fluctuating background. In a two-component polymer-solvent system, dielectric spectroscopy reveals the effect of intermacromolecular interactions on single-molecule size and reorientation. Dielectric measurements on a three-component polymer-polymer-solvent mixture, in which a tracer polymer has a nonzero type-A dipole and a potentially nondilute matrix polymer has none, can be used for example to separate the effects of probe and matrix molecular weights on dielectric relaxation. This motif in the comparative study of binary and ternary solutions appears repeatedly below. Finally, dielectric measurements on block copolymers in which some copolymer subchains have been inverted end-to-end or have no dipole moment allow one to observe internal motions and dynamic cross-correlations of subchains. 

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