Reduction, data

The following dataset possibilities exist (for studies with multiple concentrations per duration) with respect to probit analysis (applicable CK IH studies cited for each)  

The computer can order the printer or the screen to furnish spectra in table or graph form, supposing a normalization to 100 of the base peak intensity or of a fraction of this intensity if the detection of weak peaks is sought. It can also order the production of time-dependent data (total ion current, temperatures, voltages). 

In addition, it can also print a chromatogram, a variation with time of the intensities registered at a few given masses. It can suggest elementary compositions for the peaks. 

This chapter is concerned with the pre-treatment of data and so far we have discussed the nature of data, the properties of the distribution of data, and means by which data may be scaled. All of these matters are important, in so far as they dictate what can be done with data, but perhaps the most important is to answer the question what information does the data contain . It is most unlikely that any given data set will contain as many pieces of information as it does variables. That is to say, most data sets suffer from a degree of redvmdancy and this section describes ways by which redundantgr may be identified and, to some extent at least, eliminated. This stage in data analysis is called data reduction in which selected variables are removed from a data set. It should not be confused with dimension 

Another fairly obvious test to apply to the variables in a data set is to identify those parameters which have constant, or nearly constant, values. Such a situation may arise because a property has been poorly chosen in the first place, but may also happen when structural changes in the compounds in the set lead to compensating changes in physicochemical properties. Some data analysis packages have a built-in facility for the identification of such ill-conditioned variables. At this stage in data reduction it is also a good idea to examine the distribution of each of the variables in the set so as to identify outliers or variables which have become indicators , as discussed in Section 3.3. Values of the distribution parameters may also be used as decision criteria when choosing which of a pair of correlated variables to retain. 

This introduces the correlation matrix. Having removed ill-conditioned variables from the data set, a correlation matrix is constructed by calculation of the correlation coefficient between each pair of variables in the 

Having identified pairs of correlated variables, two problems remain in deciding which one of a pair to eliminate. First, is the correlation real , in other words has the high correlation coeSicient arisen due to a true correlation between the variables or is it caused by some point and cluster effect (see Section 6.2) due to an outlier. The best, and perhaps simplest, way to test the correlation is to plot the two variables against one another, effects due to outliers will then be apparent. It is also worth considering whether the two parameters are likely to be correlated with one another. If one is electronic and the other steric then there is no reason to expect a correlation, although one may exist, of course. On the other hand, maximum width and molecular weight may well be correlated for a set of molecules with similar overall shape. 

The second problem, having decided that a correlation is real, concerns the choice of which descriptor to eliminate. One approach to this problem is to delete those features which have the highest number of correlations with other features. This results in a data matrix in which the maximum number of parameters has been retained but in which the inter-parameter correlations are kept low. Another way in which this can be described is to 

As mentioned, EXAFS is defined by the variation of the mass absorption coefficient  

Equation 7.5 describes the energy-dependent EXAFS oscillations that can be interpreted using the EXAFS equation (as given in Equation 7.1 or Equation 7.2). X(k) EXAFS as a function of electron wave vector. 

EXAFS analysis is performed by a number of data treatment steps  

Add the experimental spectra and individual contributions of the detectors (for, e.g., fluorescence). 

Subtract a background function post-edge to obtain % E.  

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The accuracy of our calculations is strongly dependent on the accuracy of the experimental data used to obtain the necessary parameters. While we cannot make any general quantitative statement about the accuracy of our calculations for multicomponent vapor-liquid equilibria, our experience leads us to believe that the calculated results for ternary or quarternary mixtures have an accuracy only slightly less than that of the binary data upon which the calculations are based. For multicomponent liquid-liquid equilibria, the accuracy of prediction is dependent not only upon the accuracy of the binary data, but also on the method used to obtain binary parameters. While there are always exceptions, in typical cases the technique used for binary-data reduction is of some, but not major, importance for vapor-liquid equilibria. However, for liquid-liquid equilibria, the method of data reduction plays a crucial role, as discussed in Chapters 4 and 6. 

Figure 1 compares data reduction using the modified UNIQUAC equation with that using the original UNIQUAC equation. The data are those of Boublikova and Lu (1969) for ethanol and n-octane. The dashed line indicates results obtained with the original equation (q = q for ethanol) and the continuous line shows results obtained with the modified equation. The original equation predicts a liquid-liquid miscibility gap, contrary to experiment. The modified UNIQUAC equation, however, represents the alcohol/n-octane system with good accuracy. 

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

The continuous line in Figure 16 shows results from fitting a single tie line in addition to the binary data. Only slight improvement is obtained in prediction of the two-phase region more important, however, prediction of solute distribution is improved. Incorporation of the single ternary tie line into the method of data reduction produces only a small loss of accuracy in the representation of VLE for the two binary systems. 

Two further examples of type I ternary systems are shown in Figure 19 which presents calculated and observed selectivities. For successful extraction, selectivity is often a more important index than the distribution coefficient. Calculations are shown for the case where binary data alone are used and where binary data are used together with a single ternary tie line. It is evident that calculated selectivities are substantially improved by including limited ternary tie-line data in data reduction. 

We consider three types of m-component liquid-liquid systems. Each system requires slightly different data reduction and different quantities of ternary data. Figure 20 shows quarternary examples of each type. 

Using the ternary tie-line data and the binary VLE data for the miscible binary pairs, the optimum binary parameters are obtained for each ternary of the type 1-2-i for i = 3. .. m. This results in multiple sets of the parameters for the 1-2 binary, since this binary occurs in each of the ternaries containing two liquid phases. To determine a single set of parameters to represent the 1-2 binary system, the values obtained from initial data reduction of each of the ternary systems are plotted with their approximate confidence ellipses. We choose a single optimum set from the intersection of the confidence ellipses. Finally, with the parameters for the 1-2 binary set at their optimum value, the parameters are adjusted for the remaining miscible binary in each ternary, i.e. the parameters for the 2-i binary system in each ternary of the type 1-2-i for i = 3. .. m. This adjustment is made, again, using the ternary tie-line data and binary VLE data. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Vapor-Liquid Equilibrium Data Reduction for Acetone(1)-Methanol(2) System (Othmer, 1928)... 

Bevington P R 1969 Data Reduction and Error Analysis for the Physical Sciences (New York McGraw Hill) pp 36-43... 

Intensive data reduction is an efficient inetl iod of managing large datasets. Generally, hasl i codes are used within chemical information processes such as molecule identification and recognition of identical atoms [9S]. 

Data reduction. The process of transforming the initial digital or analog representation of output from a spectrometer into a form that is amenable to interpretation, e.g., a bar graph, a table of masses versus intensities. 

Preprocessor. A device in a data-acquisition system that performs a significant amount of data reduction by extracting specific information from raw signal representations in advance of the main processing operation. A preprocessor can constitute the whole of a data-acquisition interface, in which case it must also perform the data-acquisition task (conversion of spectrometer signal to computer representation), or it can specialize solely in data treatment. 

Another troublesome aspect of the reactivity ratios is the fact that they must be determined and reported as a pair. It would clearly simplify things if it were possible to specify one or two general parameters for each monomer which would correctly represent its contribution to all reactivity ratios. Combined with the analogous parameters for its comonomer, the values rj and t2 could then be evaluated. This situation parallels the standard potential of electrochemical cells which we are able to describe as the sum of potential contributions from each of the electrodes that comprise the cell. With x possible electrodes, there are x(x - l)/2 possible electrode combinations. If x = 50, there are 1225 possible cells, but these can be described by only 50 electrode potentials. A dramatic data reduction is accomplished by this device. Precisely the same proliferation of combinations exists for monomer combinations. It would simplify things if a method were available for data reduction such as that used in electrochemistry. 

An approach to copolymerization has been advanced by Price and Alfrey which attempts to both combine resonance and polarity considerations and accomplish the data reduction strategy of the last paragraph. It should be conceded at the outset that the Price-Alfrey method is only semiquantitative in its success. Its greatest usefulness is probably in providing some orientation to a new system before launching an experimental investigation. 

The special appeal of this approach is that it allows the heat of mixing to be estimated in terms of a single parameter assigned to each component. This considerably simplifies the characterization of mixing, since m components (with m 6 values) can be combined into m(m - l)/2 binary mixtures, so a considerable data reduction follows from tabulating 6 s instead of AH s. Table 8.2 is a list of CED and 6 values for several common solvents, as well as estimated 6 values for several common polymers. 

Data Reduction Correlations for G and the activity coefficients are based on X T.E data taken at low to moderate pressures. The ASOG and UNIFAC group-contribution methods depend for validity on parameters evaluated from a large base of such data. The process... 

The data-reduction procedure just desciiDed provides parameters in the correlating equation for g that make the 8g residuals scatter about zero. This is usually accomphshed by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y. Equation (4-282) is solved for yjP and written for species 1 and for species 2. Adding the two equations gives... 

For the first time through a liqmd-liquid extrac tion problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabe-Thiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. 

FI Spath. Cluster-Analysis Algorithms for Data Reduction and Classification of Objects. Chichester Ellis Florwood, 1980. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Having made the comparison with experiment one may then make an assessment as to whether the simulation agrees sufficiently well to be useful in interpreting the experiment in detail. In cases where the agreement is not good, the detennination of the cause of the discrepancy is often instructive. The errors may arise from the simulation model or from the assumptions used in the experimental data reduction or both. In cases where the quantities examined agree, the simulation can be decomposed so as to isolate the principal components responsible for the observed intensities. Sometimes, then, the dynamics involved can be described by a simplified concept derived from the simulation. 

Aides (administrative, data reduction, engineering, environmental health, laboratory, sanitarian, and unspecified), assistants (administrative, fiscal, laboratory, and legal), draftspersons, laboratory workers, mechanics, project illustrators, samplers, and technicians (air pollution control, air quality monitoring station, electronic, engineering, instrument, and unspecified). 

Alternatively to (3), account for certain end and edge effects (e.g., shear-extension coupling) in the data-reduction process. 

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