Maximum-entropy processing

Figure 4.19 Electrospray spectra of a protein (a) after transformation, and (b) after maximum entropy processing. From applications literature published by Micromass UK Ltd, Manchester, UK, and reproduced with permission.

The raw electrospray spectrum obtained is shown in Figure 5.14. Maximum entropy processing of these data yielded the spectrum shown in Figure 5.15, which shows the presence of two species with molecular masses of 14293.6 and 14 309.6 Da, with the latter being attributed to partial oxidation of the parent protein. 

Green B.N., Hutton T., and Vinogradov S.N. (1996), Analysis of complex protein and glycoprotein mixtures by electrospray ionization mass spectrometry with maximum entropy processing, Method. Mol. Biol. 61, 279-294. 

Fig. 11.3. The Z-contrast image of the YBCO asymmetric 30° [001] tilt boundary shown in Fig. 11.2 after maximum entropy processing.

Another class of methods such as Maximum Entropy, Maximum Likelihood and Least Squares Estimation, do not attempt to undo damage which is already in the data. The data themselves remain untouched. Instead, information in the data is reconstructed by repeatedly taking revised trial data fx) (e.g. a spectrum or chromatogram), which are damaged as they would have been measured by the original instrument. This requires that the damaging process which causes the broadening of the measured peaks is known. Thus an estimate g(x) is calculated from a trial spectrum fx) which is convoluted with a supposedly known point-spread function h(x). The residuals e(x) = g(x) - g(x) are inspected and compared with the noise n(x). Criteria to evaluate these residuals are Maximum Entropy (see Section 40.7.2) and Maximum Likelihood (Section 40.7.1). 

As indicated before, the maximum entropy approach does not process the measurements themselves. Instead, it reconstructs the data by repeatedly taking revised trial data (e.g. a spectrum or chromatogram), which are artificially corrupted with measurement noise and blur. This corrupted trial spectrum is thereafter compared with the measured spectrum by a x -test. From all accepted spectra the maximum entropy approach selects that spectrum, f with minimal structure (which is equivalent to maximum entropy). The maximum entropy approach applied for noise elimination consists of the following steps ... 

The central engine of this data workflow is the process of spectral deconvolution. During spectral deconvolution, sets of multiply charged ions associated with particular proteins are reduced to a simplified spectrum representing the neutral mass forms of those proteins. Our laboratory makes use of a maximum entropy-based approach to spectral deconvolution (Ferrige et al., 1992a and b) that attempts to identify the most likely distribution of neutral masses that accounts for all data within the m/z mass spectrum. With this approach, quantitative peak intensity information is retained from the source spectrum, and meaningful intensity differences can be obtained by comparison of LC/MS runs acquired and processed under similar conditions. 

It should be indicated that a probability density function has been derived on the basis of maximum entropy formalism for the prediction of droplet size distribution in a spray resulting from the breakup of a liquid sheet)432 The physics of the breakup process is described by simple conservation constraints for mass, momentum, surface energy, and kinetic energy. The predicted, most probable distribution, i.e., maximum entropy distribution, agrees very well with corresponding empirical distributions, particularly the Rosin-Rammler distribution. Although the maximum entropy distribution is considered as an ideal case, the approach used to derive it provides a framework for studying more complex distributions. 

In many atomization processes, physical phenomena involved have not yet been understood to such an extent that mean droplet size could be expressed with equations derived directly from first principles, although some attempts have been made to predict droplet size and velocity distributions in sprays through maximum entropy principle.I252 432] Therefore, the correlations proposed by numerous studies on droplet size distributions are mainly empirical in nature. However, the empirical correlations prove to be a practical way to determine droplet sizes from process parameters and relevant physical properties of liquid and gas involved. In addition, these previous studies have provided insightful information about the effects of process parameters and material properties on droplet sizes. 

This chapter will illustrate the process of solving crystal structures using the maximum entropy (ME) method. In the first section the theory is described this is followed by a practical example of the method in action, and there is then a brief review of other applications. 

Diffusion in general, not only in the case of thin films, is a thermodynamically irreversible self-driven process. It is best defined in simple terms, such as the tendency of two gases to mix when separated by a porous partition. It drives toward an equilibrium maximum-entropy state of a system. It does so by eliminating concentration gradients of, for example, impurity atoms or vacancies in a solid or between physically connected thin films. In the case of two gases separated by a porous partition, it leads eventually to perfect mixing of the two. 

Among the various processing options available to improve the quality of FIDs and the corresponding spectra, Linear Prediction (LP) and the Maximum Entropy Method (MEM) - not available with WIN-NMR - are probably the most exciting and powerful, even though they are not widely used. 

Fig. 6.8. A Principle of frequency-multiplexed CARS microspectroscopy A narrow-bandwidth pump pulse determines the inherent spectral resolution, while a broad-bandwidth Stokes pulse allows simultaneous detection over a wide range of Raman shifts. The multiplex CARS spectra shown originate from a 70 mM solution of cholesterol in CCI4 (solid line) and the nonresonant background of coverglass (dashed line) at a Raman shift centered at 2900 cm-1. B Energy level diagram for a multiplex CARS process. C Schematic of the multiplex CARS microscope (P polarizer HWP/QWP half/quarter-wave plate BC dichroic beam combiner Obj objective lens F filter A analyzer FM flip mirror L lens D detector S sample). D Measured normalized CARS spectrum of the cholesterol solution. E Maximum entropy method (MEM) phase spectrum (solid line) retrieved from (D) and the error background phase (dashed line) determined by a polynomial fit to those spectral regions without vibrational resonances. F Retrieved Raman response (solid line) calculated from the spectra shown in (E), directly reproducing the independently measured spontaneous Raman response (dashed line) of the same cholesterol sample...

Lord Kelvin (1824—1907), a Scottish mathematician and physicist, did the pioneering work on the second law of thermodynamics, arguing that it was the explanation of irreversible processes. He noted that the continual increase of entropy would lead to a universe with a uniform temperature and maximum entropy. 

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