Methods of data reduction

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. 

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. 

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. 

Another useful method of data reduction is the Dixon plot, where l/v is plotted against [I], the inhibitor concentration, at a fixed [D]. This allows for the determination of without the need to determine the absolute concentration of [D]—a great advantage in cases in which the substrate is a polynucleotide or a protein, as is often the case in chemotherapy. 

There have been a number of methods of data reduction proposed, some of which are briefly described here. 

This method combines the advantages of simplicity, direct control of frequency, and minimum reliance on mathematical modeling assumptions. It differs from other embodiments of the direct-force approach (4), (5) in the means for reacting the forces applied to the sample and in the techniques for measuring the imposed force and the resultant shear deformation. Described in this article are the test configuration, the principle of operation, the method of data reduction, typical measurements, consistency checks, and an application of measured data. 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

Fourier Transform Method. Another method of data reduction is to take a fast Fourier transform (FFT) of the wave (10). As indicated in Figure 7, the Fourier transform of a damped sine wave with a single frequency is a single maximum in the frequency domain at the frequency of the oscillation. The amplitude (H) of the transformed data as a function of angular frequency ([Pg.344]

Discussion. The four methods of data reduction were used to analyze the raw data of the same TBA specimen during a slow (0.25°C/min) temperature scan (Figure 8). A comparison of the spectra indicates that they all gave similar results over the range of period (0.3 to 1.8 sec) and logarithmic decrement (0.01 to 1.08) encountered in the experiment. (The automated torsion pendulum has been used to reduce data with a range of 0.1 to 15 sec. for the period, and 0.001 to 4.0 for the... 

An alternative method of data reduction was reported early in the history of gas chromatography by Hoare and Purnell (12-15 see refs. 16,17 for recent applications), who considered the dependence oTffie specific retention volume on the solute saturation vapor pressure pA. Thus, taking the view [now recognized to be naive (18) see later], that the observed mole fraction-based solute activity coefficient" can be decomposed into "athermal" and "thermal" components (19-22) v v-... 

The techniques of infrared emission, ultraviolet and visible absorption and emission, and time-of-flight mass spectrometry have also been utilized and will be discussed along with a general description of the shock tube method and various methods of data reduction and refinement. 

The purpose of this section is to provide an overview of the several technologies that are available to measure the dynamic gait variables listed earlier, including stride and temporal parameters, kinematics, kinetics, and dynamic EMG. Methods of data reduction will be described in a following section. 

These classification methods use different principles and rules for learning and prediction of class membership, but wiU usually produce a comparable result. Some comparisons of the methods have been given (i.e., Kotsiantis, 2007 Rani et al., 2006). Although the modem methods such as SVM have demonstrated very good performance, the drawback is that the model becomes an incomprehensible black-box that removes the explanatory information provided by, for example, a logistic regression model. However, classification performance usually outweighs the need for a comprehensible model. PCA has been used for classification based on bioimpedance measurements. Technically, PCA is not a method for classification but rather a method of data reduction, more suitable as a parameterization step before the classification analysis. 

The experimental use of flame theory is a simpler problem because the measured profiles allow the replacement of the coupled differential equations with a pointwise set of algebraic equations. For example, to determine the rate of appearance (or disappearance) of a particular species at various points (i.e., at various temperatures in the flame) requires a knowledge of the first and second derivatives of the composition of a species and the temperature and gas velocity at the point in question. These are experimental quantities available from the flame structure measurements (see, e.g.. Fig. 8). The usual method of data reduction involves two steps (1) the calculation of the species flux [Eq. (2)] (i.e., the amount of the species passing through a unit area per unit time)—this takes the effect of diffusion into account quantitatively (2) from this flux curve the rate of species production is obtained by differentiation [Eq. (3)]. 

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