Sparse data sets

The second step concerns distance selection and metrization. Bound smoothing only reduces the possible intervals for interatomic distances from the original bounds. However, the embedding algorithm demands a specific distance for every atom pair in the molecule. These distances are chosen randomly within the interval, from either a uniform or an estimated distribution [48,49], to generate a trial distance matrix. Unifonn distance distributions seem to provide better sampling for very sparse data sets [48]. 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

The Wilson plot is shown in Figure 4.At first sight all seems well except that the overall temperature factor is B=327 57a. This is clearly unlikely, but is a common feature when carrying out normalisation with such a sparse data set. To overcome the problem, a temperature factor of B=0.0A is imposed on the data. 

Smith, R. T., Zoltani, C. K., Klem, G. J., and M. W. Coleman, 1991, Reconstruction of the tomographic images from sparse data sets by a new finite element maximum entropy approach Appl. Opt., 30, 573-582. 

Deterioration in parameter estimation has been observed in simulation studies in which the value of the intersubject variability was greater than 60% and the residual variability was set at 15%. " A series of studies in which observations were randomly deleted from a data-rich set to create a sparse data set, and parameter estimation done using the FO, showed good performance of the FO approach when compared with the results obtained using the full data The... 

Different results may be observed under conditions that are ostensibly the same. To keep track of this variation, we must maintain records or statistics. There are two general strategies that we may employ. First, we may simply store the results. That is, if we have a thousand observations, we can maintain access to all the individual values. The record may then be employed as an empirical distribution function, in which particular percentiles may be identified on demand. Second, we may use a mathematical model to summarize the distribution. There are two very different reasons for doing this. First, a statistical model may be used to provide a concise summary. The facility with which an analyst can store and retrieve data makes this motivation less compelling than it once was. Second, when a sparse data set is not considered representative of a large population, a model may also be used to infer or predict values that are not represented in the data set. 

Nathanail, C.P. (1997) Expert knowledge to select model variogram parameters for geostatistical interpolation of sparse data sets. In Yong, R.N. Thomas, H.R. (eds), Geoenvironmental Engineering Contaminated Ground Fate of Pollutants and Remediation, Thomas Telford, pp. 240-247. 

The results are presented in Table 8.5. The dense data set resulted in accurate and precise estimates of the fixed effects and the BSV in CL, but overestimated the BSV in VI and Q. In contrast, the BSV in V2 and ka was underestimated. The sparse data set resulted in reasonable estimates of all the fixed effects, except V2 which was significantly overestimated (224 L versus a true value of 125 L). 

Further, BSV in Q and V2 were also overestimated. Hence, the sparse data set did not reproduce the data... 

New studies examining the textures, chemistry, zoning, and petrogenesis of phosphates in well-constrained metamorphic terranes to supplement the current, sparse data sets. Complete, accurate chemical analyses of natural phosphates from these studies would be especially revealing about the geochemical controls on monazite stability. 

Wallenstein, D. Kanz, B. Harding, R. H. Evaluation of Sparse Data Sets Obtained From Microactivity Testing of FCC Catalysts. Appl. Catal., A 1999 178, 117. 

In this review we will focus on applications of FT to processing of non-uniformly (sparsely) data sets devoted to the reconstruction of high-resolution multidimensional NMR spectra. 

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