Matrix data resolution

Note also that the data resolution matrix is not a function of the data but only of the operator of the forward problem. It can therefore be studied without actually performing the geophysical observations and can therefore be used for planning the field experiment. 

Data matrices (two-way data). Electrophoretic data resulting from multiway detectors, such as in CE-DAD and CE-MS techniques, can be arranged in a table of values or a data matrix. Data are structured over the two domains of measurement, in which each column corresponds to a wavelength (or m/q ratio) and each row corresponds to a time point. Two-way data can be exploited for studies of peak purity and mixture resolution using curve resolution and related factor analysis methods. 

Time constraints ate an important factor in selecting nmr experiments. There are four parameters that affect the amount of instmment time requited for an experiment, A preparation delay of 1—3 times should be used. Too short a delay results in artifacts showing up in the 2-D spectmm whereas too long a delay wastes instmment time. The number of evolution times can be adjusted. This affects the F resolution. The acquisition time or number of data points in can be adjusted. This affects resolution in F. EinaHy, the number of scans per EID can be altered. This determines the SNR for the 2-D matrix. In general, a lower SNR is acceptable for 2-D than for 1-D studies. 

A method of resolution that makes a very few a priori assumptions is based on principal components analysis. The various forms of this approach are based on the self-modeling curve resolution developed in 1971 (55). The method requites a data matrix comprised of spectroscopic scans obtained from a two-component system in which the concentrations of the components are varying over the sample set. Such a data matrix could be obtained, for example, from a chromatographic analysis where spectroscopic scans are obtained at several points in time as an overlapped peak elutes from the column. 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

In previous methods no pre-knowledge of the factors was used to estimate the pure factors. However, in many situations such pre-knowledge is available. For instance, all factors are non-negative and all rows of the data matrix are nonnegative linear combinations of the pure factors. These properties can be exploited to estimate the pure factors. One of the earliest approaches is curve resolution, developed by Lawton and Sylvestre [7], which was applied on two-component systems. Later on, several adaptations have been proposed to solve more complex systems [8-10]. 

In their fundamental paper on curve resolution of two-component systems, Lawton and Sylvestre [7] studied a data matrix of spectra recorded during the elution of two constituents. One can decide either to estimate the pure spectra (and derive from them the concentration profiles) or the pure elution profiles (and derive from them the spectra) by factor analysis. Curve resolution, as developed by Lawton and Sylvestre, is based on the evaluation of the scores in the PC-space. Because the scores of the spectra in the PC-space defined by the wavelengths have a clearer structure (e.g. a line or a curve) than the scores of the elution profiles in the PC-space defined by the elution times, curve resolution usually estimates pure spectra. Thereafter, the pure elution profiles are estimated from the estimated pure spectra. Because no information on the specific order of the spectra is used, curve resolution is also applicable when the sequence of the spectra is not in a specific order. 

When the rows of a data matrix follow a certain pattern, e.g. the appearance and disappearance of compounds as a function of time, a fixed-size window EFA is applicable. This is the case, for instance, for data sets generated by hyphenated measurement techniques such as HPLC with DAD. Fixed-size window EFA [22] can be applied for detecting the presence of minor compounds (< 1 %) and for the resolution of a data set into its components (pure spectra and elution profiles). 

In Section 34.2 we explained that factor analysis consists of a rotation of the principal components of the data matrix under certain constraints. When the objects in the data matrix are ordered, i.e. the compounds are present in certain row-windows, then the rotation matrix can be calculated in a straightforward way. For non-ordered spectra with three or less components, solution bands for the pure factors are obtained by curve resolution, which starts with looking for the purest spectra (i.e. rows) in the data matrix. In this section we discuss the VARDIA [27,28] technique which yields clusters of pure variables (columns), for a certain pure factor. 

The spatial temperature distribution established under steady-state conditions is the result both of thermal conduction in the fluid and in the matrix material and of convective flow. Figure 2. 9.10, top row, shows temperature maps representing this combined effect in a random-site percolation cluster. The convection rolls distorted by the flow obstacles in the model object are represented by the velocity maps in Figure 2.9.10. All experimental data (left column) were recorded with the NMR methods described above, and compare well with the simulated data obtained with the aid of the FLUENT 5.5.1 [40] software package (right-hand column). Details both of the experimental set-up and the numerical simulations can be found in Ref. [8], The spatial resolution is limited by the same restrictions associated with spin... 

Scolecite gave the opportunity to relate the electron density features of Si-O-Si and Si-O-AI bonds to the atomic environment and to the bonding geometry. After the multipolar density refinement against Ag Ka high resolution X-ray diffraction data, a kappa refinement was carried out to derive the atomic net charges in this compound. Several least-squares fit have been tested. The hat matrix method which is presented in this paper, has been particularly efficient in the estimation of reliable atomic net charges in scolecite. 

At this point it is important to note that the flow model (a hydrologic cycle model) can be absent from the overall model. In this case the user has to input to the solute module [i.e., equation (1)] the temporal (t) and spatial (x,y,z) resolution of both the flow (i.e., soil moisture) velocity (v) and the soil moisture content (0) of the soil matrix. This approach is employed by Enfield et al. (12) and other researchers. If the flow (moisture) module is not absent from the model formulation (e.g., 14). then the users are concerned with input parameters, that may be frequently difficult to obtain. The approach to be undertaken depends on site specificity and available monitoring data. 

Second, the resolution achieved in a 2-D experiment, particularly in the carbon domain is nowhere near as good as that in a 1-D spectrum. You might remember that we recommended a typical data matrix size of 2 k (proton) x 256 (carbon). There are two persuasive reasons for limiting the size of the data matrix you acquire - the time taken to acquire it and the shear size of the thing when you have acquired it This data is generally artificially enhanced by linear prediction and zero-filling, but even so, this is at best equivalent to 2 k in the carbon domain. This is in stark contrast to the 32 or even 64 k of data points that a 1-D 13C would typically be acquired into. For this reason, it is quite possible to encounter molecules with carbons that have very close chemical shifts which do not resolve in the 2-D spectra but will resolve in the 1-D spectrum. So the 1-D experiment still has its place. 

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