XES data

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Worth noting is the fact that Barkers method does not require experimental yf values. Thus the correlating parameters Ot, b, and so on, can be ev uated from a P-X data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The worlds store of X T.E data has been compiled by Gmehling et al. (Vapor-Liquid Lquilibiium Data Collection, Chemistiy Data Series, vol. I, parts 1-8, DECHEMA, Frankfurt am Main, 1979-1990). 

As we will soon see, the nature of the work makes it extremely convenient to organize our data into matrices. (If you are not familiar with data matrices, please see the explanation of matrices in Appendix A before continuing.) In particular, it is useful to organize the dependent and independent variables into separate matrices. In the case of spectroscopy, if we measure the absorbance spectra of a number of samples of known composition, we assemble all of these spectra into one matrix which we will call the absorbance matrix. We also assemble all of the concentration values for the sample s components into a separate matrix called the concentration matrix. For those who are keeping score, the absorbance matrix contains the independent variables (also known as the x-data or the x-block), and the concentration matrix contains the dependent variables (also called the y-data or the y-block). 

Optional pretreatments can be applied, in any combination, to either the spectra (the x-data), the concentrations (the y-data) or both. 

Variance (cont) of prediction, 167 Variance scaling, 100, 174 Vectors basis, 94 Weighting of data, 100 Whole spectrum method, 71 x-block data, 7 x-data, 7 XE, 94 y-block data, 7 y-data, 7... 

Figure 3. Tg of PVC vs. content of VC (V) data measured by means of deviation from Flory-Huggins isotherm (5) (X) data measured thermomechanically (IS) (O) data obtained from limiting conversion (4).

Let us assume that we have collected a set of calibration data (X, Y), where the matrix X (nxp) contains the p > 1 predictor variables (columns) measured for each of n samples (rows). The data matrix Y (nxq) contains the q variables which depend on the X-data. The general model in calibration reads... 

As an extension of perceptron-like networks MLF networks can be used for non-linear classification tasks. They can however also be used to model complex non-linear relationships between two related series of data, descriptor or independent variables (X matrix) and their associated predictor or dependent variables (Y matrix). Used as such they are an alternative for other numerical non-linear methods. Each row of the X-data table corresponds to an input or descriptor pattern. The corresponding row in the Y matrix is the associated desired output or solution pattern. A detailed description can be found in Refs. [9,10,12-18]. 

Reciprocal Space Velocity xlO4 X (data) xi (model) x2 (data) x2 (model)... 

The projection of the X data vectors onto the first eigenvector produces the first latent variable or pseudomeasurement set, Zx. Of all possible directions, this eigenvector explains the greatest amount of variation in X. The second eigenvector explains the largest amount of variability after removal of the first effect, and so forth. The pseudomeasurements are called the scores, Z, and are computed as the inner products of the true measurements with the matrix of loadings, a ... 

PLS was originally proposed by Herman Wold (Wold, 1982 Wold et al., 1984) to address situations involving a modest number of observations, highly collinear variables, and data with noise in both the X- and Y-data sets. It is therefore designed to analyze the variations between two data sets, X, Y). Although PLS is similar to PCA in that they both model the A -data variance, the resulting X space model in PLS is a rotated version of the PCA model. The rotation is defined so that the scores of X data maximize the covariance of X to predict the Y-data. 

Hammett then designated the ionisation, in water at 25°, of m- and p-substituted benzoic acids as his standard reference reaction. He chose this reaction because reasonably precise aqueous ionisation constant, x, data were already available in the literature for quite a range of differently m- and p-substituted benzoic acids. Knowing XH and Kx for a variety of differently X-substituted benzoic acids, it is then possible to define a quantity, [Pg.362]

PLS should have, in principle, rejected a portion of the non-linear variance resulting in a better, although not completely exact, fit to the data with just 1 factor. The PLS does tend to reject (exclude) those portions of the x-data which do not correlate linearly to the y-block. (Richard Kramer)... 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

Note for this example that covar(X, Y) represents the covariance of (X, Y), stdev(X) is the standard deviation of the X data, and stdev(T) is the standard deviation of the Y data. For the MathCad program ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521), the sldcvfW) is represented by the variable symbol Sr, which can be thought of as the set of many possible standard deviations for a set of data X. 

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

There is considerable literature precedent for this reaction. In particular, Fotsch and Chamberlin (10) have reported that open chain y,8, 8,e and 6, -epoxy ketones and esters undergo cyclization in the presence of acids to form the corresponding dioxacarbenium ions. In addition, molecular orbital calculations were conducted to determine the heats of formation of the intermediates IX and X. Data from these calculations are given in Table 2. These calculations suggest that 1,6-attack (X) is... 

Figure 1.12 Seebeck coefficient of the oxide LaNil Coi- c03 as a function of the composition, x. [Data adapted from R. Robert, L. Becker, M. Trottmann, A. Reller, and A. Weidenkraft, J. Solid State Chem., 179, 3893-3899, (2006).]...

Figure 15.5 Activation enthalpy values of reaction 15.16 for different X. Data from [327]...

Additionally to the x-data, a property y may be known for each object (Figure 2.3). The property can be a continuous number, such as the concentration of a compound, or a chemical/physical/biological property, but may also be a discrete number that encodes a class membership of the objects. The properties are usually the interesting facts of the objects, but often they cannot be determined directly or only with high cost on the other hand, the x-data are often easily available. Methods from... 

The correlation coefficients can be arranged in a matrix like the covariances. The resulting correlation matrix (R, with l s in the main diagonal) is for autoscaled x-data identical to C. 

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