Synthetic Data

In many cases, the methods used to solve identification problems are based on an iterative minimization of some performance criterion measuring the dissimilarity between the experimental and the synthetic data (generated by the current estimate of the direct model). In our case, direct quantitative comparison of two Bscan images at the pixels level is a very difficult task and involves the solution of a very difficult optimization problem, which can be also ill-behaved. Moreover, it would lead to a tremendous amount of computational burden. Segmented Bscan images may be used as concentrated representations of the useful... 

Compound 40 has not yet been synthesized. However, there is a large body of synthetic data for nucleophilic substitution reactions with derivatives of 41 [synthesized from aliphatic and aromatic aldehydes, pyridine, and trimethylsilyl triflate (92S577)]. All of these experimental results reveal that the exclusive preference of pathway b is the most important feature of 41 (and also presumably of 40). 

In order to make this exercise as useful and as interesting as possible, we will take steps to insure that our synthetic data are suitably realistic. We will include difficult spectral interferences, and we will add levels of noise and other artifacts that might be encountered in a typical, industrial application. 

Our synthetic data simulate spectra that are measured at 100 discrete wavelengths. But, we only have 15 spectra in our training set. Thus, before we can perform ILS on our data, we must first condense our training set data to no... 

CLS, defined, 50, 71 extracted, 94 imbedded, 94 in synthetic data, 45 PRESS, 167 real, 94... 

A y,-value was then simulated for every a ,-value in Table 4.5. This new, synthetic data set had statistical properties identical ( , Sxx), or very similar Sxy, Syy, Sres) to thosc of the measured set, the difference residing in... 

Figure 3.9. Demonstration of ruggedness. Ten series of data points were simulated that all are statistically similar to those given in Table 4.5. (See program SIMILAR.) A quadratic parabola was fitted to each set and plotted. The width of the resulting band shows in what ar-range the regression is reliable, higher where the band is narrow, and lower where it is wide. The bars depict the data spread for the ten statistically similar synthetic data sets.

In the alkylation of benzene with (dichloroalkyl)chlorosilanes in the presence of aluminum chloride catalyst, the reactivity of (dichloroalkyl)silanes increases as the spacer length between the C—Cl and silicon and as the number of chloro-groups on the silicon of (dichloroalkyl)chlorosilanes decreases as similarly observed in the alkylation with (cD-chloroalkyl)silanes. The alkylation of benzene derivatives with other (dichloroalkyl)chlorosilanes in the presence of aluminum chloride gave the corresponding diphenylated products in moderate yields.Those synthetic data are summarized in Table XI. 

The synthetic data have been derived from a theoretical one-compartment model with the following settings of the parameters ... 

The synthetic data have been obtained by adding random noise with standard deviation of about 0.4 )0.g 1 to the theoretical plasma concentrations. As can be seen, the agreement between the estimated and the computed values is fair. Estimates tend to deteriorate rapidly, however, with increasing experimental error. This phenomenon is intrinsic to compartmental models, the solution of which always involves exponential functions. 

Both the determination of the effective number of scatterers and the associated rescaling of variances are still in progress within BUSTER. The value of n at the moment is fixed by the user at input preparation time for charge density studies, variances are also kept fixed and set equal to the observational c2. An approximate optimal n can be determined empirically by means of several test runs on synthetic data, monitoring the rms deviation of the final density from the reference model density (see below). This is of course only feasible when using synthetic data, for which the perfect answer is known. We plan to overcome this limitation in the future by means of cross-validation methods. 

A minimum of 2 factors are required. The synthetic data did NOT demonstrate the advantage of a single linear wavelength over a multiple wavelength model, it merely illustrated the fact that a single linear factor is not sufficient to model non-linear data. We could stop here, but, for the sake of completeness-... 

The synthetic data did NOT demonstrate the advantage of a single linear wavelength over a multiple wavelength [sic] model... (Richard Kramer)... 

As expected by all responders, and by your hosts as well, when two-factor models (either PCR or PLS) were computed, the fit of the model to the synthetic data was perfect. Table 33-1 presents a summary of the numerical results obtained, for one-factor calibration models. 

Figure 33-1 PLS loadings from the synthetic data used to test the fit of models to nonlinearity, (see Colour Plate 2)...

The absorbance spectrum in Figure 54-1 is made from synthetic data, but mimics the behavior of real data in that both are represented by data points collected at discrete and (usually) uniform intervals. Therefore the calculation of a derivative from actual data is really the computation of finite differences, usually between adjacent data points. We will now remove the quotation marks from around the term, and simply call all the finite-difference approximations a derivative. As we shall see, however, often data points that are more widely spread are used. If the data points are sufficiently close together, then the approximation to the true derivative can be quite good. Nevertheless, a true derivative can never be measured when real data is involved. 

We find that in all four cases, the coefficient of the linear term is 0.5. In Anscombe s original paper, this is all he did, and obtained the same result, but this was by design the synthetic data he generated was designed and intended to give this result for all the data sets. The fact that we obtained the same coefficient (for X) using the polynomial demonstrates that the quadratic term was indeed uncorrelated to the linear term. 

Figure 67-l(a) An illustration of the method of measuring the amount of nonlinearity showing hypothetical synthetic data to which each of the functions are fit. 

For this purpose, then, it would suffice to replace the original data with a set of synthetic data with the necessary properties. What are those properties The key properties comprise the number of data values, the range of the data values and their distribution. 

The range of the synthetic data we want to generate should be such that the A-values have the same range as the original data. The reason for this is obvious when we apply the empirically derived quadratic function (found from the regression) to the data, to compute the T-values, those should fall on the same line, and in the same relationship to the X as the original data did. 

By dividing each eigenvector component by the smallest of them, we find that the components of the fourth eigenvector [0.1826, 0.3651, 0.5477, 0.7303]T associated with the smallest eigenvalue (here zero) are in proportion of (1, 2, 3, 4) which is precisely the source composition used to produce the synthetic data (Table 9.3). The capability of this formalism to invert the data to produce relative source concentrations is therefore established. It is left to the reader to show that correct source mineralogical compositions can be retrieved using the procedure outlined in Section 9.2.2. <=... 

Figure 6.8 shows clustering results for a synthetic data example in two dimensions with three groups. The plot symbols indicate the result of k-means clustering for k 2 (left), k 3 (middle), and k 4 (right). Here it is obvious that the choice k = 3 gave the best result since it directly corresponds to the visually evident data groups. 

The example used in Figure 6.9 is analyzed by model-based clustering this synthetic data set contains two elliptical and one spherical group of objects. The different types of cluster models are evaluated for a number of clusters varying from 2 to 6. Figure 6.16 shows the evaluation of the various models (denoted by... 

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