Resolution objective function

Rather than the separation function, resolution between individual peak pairs is used in most automated optimization procedures because it is easier to calculate, although non-Gaussian pe2dcs and overlapping peaks can present problems due to the difficulty of estimating peak widths. A simple objective function would be to consider only the separation between the worst separated peak pair, ignoring all others. If a set of... 

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. 

One of the most popular refinement programs is the state-of-the-art package Refmac (Murshudov et ah, 1997). Refmac uses atomic parameters (xyz, B, occ) but also offers optimization of TLS and anisotropic displacement parameters. The objective function is a maximum likelihood derived residual that is available for structure factor amplitudes but can also include experimental phase information. Refmac boasts a sparse-matrix approximation to the normal matrix and also full matrix calculation. The program is extremely fast, very robust, and is capable of delivering excellent results over a wide range of resolutions. 

Like CRF-1, CRF-2 also favors short analysis times and well resolved peaks. However, there is no threshold value for resolution, and the compromise between resolution and analysis time is not as well defined as in CRF-1. Inclusion of analysis time in the denominator of an objective function may result in the loss of some resolution, compensated for by a rapid analysis time (51). This is likely to occur to some extent when the resolution threshold is defined in excess of 1 to 1.25, since the peak-valley ratio utilized by CRF-2 does not diminish to an appreciable extent until the resolution falls below this range. Note that as the resolution drops below 1 to 1.25, however, CRF-2 decreases rapidly, and it is unlikely that a short analysis time will compensate for such poor resolution (33). Nevertheless, if a user-specified minimum resolution is an absolute requirement, it is probably better to use a threshold criterion such as CRF-1 in which the desired resolution is stipulated by the user. 

SOLUTION The optimization code is given in Code for Example 6.9. The outer loop of this program has no end. Given a very slow computer, you could watch the program run and terminate it when the value of the objective function ceases to change. On a modern computer it almost instantly reaches x 0 and y 3 with the value of the function approximately 1.0. The Basic function Rnd as implemented in Excel 2003 has a resolution of about 1 part in 10. This limits the accuracy ofx and y in this example to about 1 part in 10 . Better random number generators do exist but are rarely needed. 

Objective functions are used primarily for computer ranking of separation quality in approaches for automated method development. For this purpose the separation quality throughout the chromatogram must be expressed by a single-valued and easily calculated mathematical function [202-209], This has proven to be a difficult problem and no universal solution has emerged. There is no straightforward manner to uniquely define the resolution of all peaks simultaneously by a single number. More likely, for... 

The idea is that an object function can be approximately reconstructed by summing the contributions from all the tilted fans. This means that the back-projection is applied within a volume rather than across a plane. The volume elemental cell is a voxel and has the same implication for resolution that the pixel has in the planar representation. 

In previous work [9,11] it has been common practice to average over a large range of trial PSDs, as the problem was underconstrained and there existed a large number of local minima, so that the final PSD was dependent on the starting point. We have databases with a higher resolution and as a result the objective function appears to be mueh smoother and the global... 

The objectives of the inflammatory response can be viewed as a hierarchical ordered panel of events. The most successful consequence of an inflammatory response is the complete restoration of function and structure of the affected tissue, also denoted as resolution. If this is not possible, inflammation aims for healing by repair and replacement of lost tissue by scar tissue. 

Mom of the functions considered so far specifically tedce into account the nuitber of peaks found in the chroBatograa. If the object of the analysis is to detect the maximum nuaber of peaks, even if the resolution between individual peak pairs was poorer, then equations (4.47) and (4.48) would be inferior. A chroaatographic response function, CRF, that takes into account the siaultaneous iaportance of resolution, separation tiae, and the total nuaber of detectable pe2dcs can be e]q>ressed as follows... 

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