Column-standardization

Column-standardization is the most widely used transformation. It is performed by division of each element of a column-centered table by its corresponding column-standard deviation (i.e. the square root of the column-variance)  

In the context of data analysis we divide by n rather than by (n - 1) in the calculation of the variance. This procedure is also called autoscaling. It can be verified in Table 31.5 how these transformed data are derived from those of Table 31.4. 

Atmospheric data from Table 31.1, after column-standardization. 

In the corresponding column-standardized biplot of Fig. 31.7 we find all representations of the eight chromatographic methods more or less at the same distance from the origin of space. The circle is distorted because of the large difference between the contributions of the first and second latent variables (95 and 4%) and the choice of a = [3 = 0.5 which has been made at the outset. The combined effect is an apparent dilation of the vertical axis. 

The distances between compounds in Fig. 31.7 are not notably affected by the transformation in comparison with the previous Fig. 31.6. This biplot allows more easily to perceive the correlations between measurements. Three clusters are now put in evidence, namely (1) DMSO and DMF, (2) ethanol and propanol, (3) octanol, dioxane, THF and methylenedichloride. The line segments drawn from the origin have been added to emphasize these groupings. Unipolar axes could have been defined here in the same way as in Fig. 31.6. Bipolar axes on the column-standardized biplot, however, cannot be interpreted directly in terms of the original data in X. 

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Fig. 29.8. (a) Pattern of points in column-space S (left panel) and in row-space S" (right panel) before column-centering, (b) After column-centering, the pattern in 5 is translated such that the centroid coincides with the origin of space. Distances between points in S are conserved while those in S" are not. (c) After column-standardization, distances between points in S and 5" are changed. Points in 5" are located on a (hyper)sphere centered around the origin of space. 

A vector of column-standard deviations provides for each column of X a measure of the spread of the elements around the corresponding column-mean ... 

The geometrical interpretation of column-standard deviations is in terms of distances of the points representing the columns of Y from the origin of 5" (multi-... 

The matrix Cp contains the variances of the columns of X on the main diagonal and the covariances between the columns in the off-diagonal positions (see also Section 9.3.2.4.4). The correlation matrix Rp is derived from the column-standardized matrix Zp ... 

W = diag (1/d ) where d represents the vector of column-standard... 

In this case the column-norms dp are called column-standard deviations. The square of these numbers are the column-variances, whose sum represents the global variance in the data. Note that the column-variances are heterogeneous which means that they are very different from each other. 

Fig. 31.7. Biplot of chromatographic retention times in Table 31.2, after column-standardization of the data. Unipolar semi-axes have been drawn through all points representing methods. The particular arrangement of the methods is indicative for the presence of a strong size component in the data.

In this case it is required that the original data in X are strictly positive. The effect of the transformation appears from Table 31.6. Column-means are zero, while column-standard deviations tend to be more homogeneous than in the case of simple column-centering in Table 31.4 as can be seen by inspecting the corresponding values for Na and Cl. 

The biplot of Fig. 31.9 shows that both the centroids of the compounds and of the methods coincide with the origin (the small cross in the middle of the plot). The first two latent variables account for 83 and 14% of the inertia, respectively. Three percent of the inertia is carried by higher order latent variables. In this biplot we can only make interpretations of the bipolar axes directly in terms of the original data in X. Three prominent poles appear on this biplot DMSO, methylene-dichloride and ethylalcohol. They are called poles because they are at a large distance from the origin and from one another. They are also representative for the three clusters that have been identified already on the column-standardized biplot in Fig. 31.7. 

Look at the last column, standard form. As you go down this column, each entry is multiplied by three to get the next entry below, such as 81 x 3 = 243. Notice that as you go UP this column, you would divide each entry by three to get the previous entry above. Now let s extend the table to include an exponent of zero and some negative exponents ... 

Note that correspondence analysis cannot be used with centred data, and that some kind of column standardization has to be used when the range of variables is very different. 

Table 6-4 gives, in the first column, standard values of Gibbs energies of formation from the elements AGf° for a variety of pure solids, gases, and liquids as well as values for substances in solution at the hypothetical 1 M activity. As an example, consider the value of AGf 0 for pure liquid acetic acid, -389.1 kj mol-1. The equation for its formation from the elements is ... 

Fig. 7 Typical reversed-pbase separation of amino acids. Precolumn derivatization of a standard amino acid mixture was achieved employing FMOC. Resolution was achieved by gradient elution with acetonitrile, methanol, and acetate buffer (pH 4.2) on a C,8 column. Standard three-letter abbreviations for amino acids are used also, CySO H = cysteic acid. (From Ref. 164. Copyright 1983 Elsevier Science.)...

Fig. 8 Separation of standard mixture employing precolumn derivatization with AQC. Gradient elution with acetonitrile and acetate buffer (pH 5.0) was employed with a C18 column. Standard three-letter abbreviations for amino acids were used also, CA = cysteic acid, AMQ = hydrolyzed excess reagent, and nle = norleucine. Data was supplied by Stephen D. Smith, Ross Products Division of Abbott Laboratories, Columbus, OH.

The best way to follow column changes is by way of column standard plate counts. For discussion purposes, we will use the four-standard mixture of acetophenone, nitrobenzene, benzene, and toluene described in the discussion on efficiency factor (Chapter 4). Our column will be a Ci8 reverse-phase column run in 70% acetonitrile/water at 254 nm. In an initial run, we obtain four peaks whose interpeak a s double between each pair. After we discuss reverse phase, we will see how these killers affect normal phase columns. 

It s better to pick a time convenient for you than to have to do this process on an emergency basis in the middle of a critical separation. I would have a tested column ready as a replacement. Replace the dirty column after washing out the buffer, cap it, and, then, wash the old column off-line when you have more time. You never seem to wash everything off the column. After you ve used a column for a while, you often will find a brown or black residue at the column head under the column frit on opening even a freshly washed column. Don t worry about it if the column standards run correctly. 

When the baseline is stable, inject 15/iL of column standards. (Lab note Inj ect by overfilling the syringe, point the needle up, pull the barrel back until you can see the meniscus, tap out visible bubbles in the liquid, push the plunger to the 15- L mark, wipe outside the barrel with a lab wipe with a pulling motion. Insert into injector. Load the injector loop slowly, and leave the needle in place.) Thrn the injection handle quickly. Remove the injection needle, and flush three times with solvent. 

Figure 8.3. ID analysis of B1O3" in drinking water by ion chromatography with ICP-MS detection study of polyatomic ions produced at mass 79 and 81 from a 300 p,g g-1 sulfate matrix a) and a 100 p,g g-1 phosphate matrix (b). Std denotes the peak produced by a 100 p.L post-column standard injection of BrOj". Reprinted from [395] with permission Copyright 1999 American Chemical Society.

Fig. 13 shows an example of how important it is to avoid contamination, and its influence on the final result of the analysis. The analytical procedure was a pre-concentration of aluminum by adsorption of its complex with an organic reagent (chrome azurol S) onto a polymeric material (polyethylene powder) packed into a column. Standard solutions were prepared and the procedure carried out after adopting stepwise precautions to avoid contamination. It can be seen that, only after adopting all steps, was the contamination controlled. The graph also shows that the lower the aluminum content in the sample the higher is the contribution of the contamination sources. 

Column Standardization After equilibrating the HPLC system at a flow rate of 0.8 mL/min for at least 1 h, inject 50 il. of the Standard Solution five times, allowing 15 min between injections. Record the retention times of the various components in the Standard Solution. Retention times for each component should agree within 2 s. Insert the average retention time along with the molecular weight of each component into the calibration table of the molecular weight distribution software. 

Procedure Inject 50 p.L of the Sample Preparation, following the same conditions and procedure as described under Column Standardization. Using the Molecular Weight Distribution software of the data-reduction system, generate a molecular weight distribution curve of the sample. There is no measurable peak above a molecular weight of 22,000. Monomers... 

Supercritical fractionation of a liquid lipid feed material is usually carried out in a packed column. Standard columns are not available commercially and have to be custom built either in-house or by manufacturers of extraction units. Lab-scale and pilot-scale supercritical columns, 0.6-13.6 m high with internal diameters of 14.3-68 mm are available in research labs around the world and have been used for the processing of deodorizer distillates (56, 57, 86-90), vegetable and fish oils (91-105), cocoa butter, and milkfat (106-109). A schematic diagram of a typical SCCO2 fractionation column (2.8 m, 2.54 cm o.d.), which was designed and built... 

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