Two Column Plots

Two column plots are more effective if the two stationary phases are very different from each other. If they are of opposite polarity, then the elution order of the analytes in a given sample should be very different (opposite) on the two phases. For example, in the GC separation of a mixture of n-heptane, tetrahydrofuran, 2-butanone, and n-propanol, the elution order on a polar phase like Carbowax is in the order just listed on a nonpolar phase like SE-30, the order is the exact opposite Neither separation follows the boiling point order. 

Figure 4-8. Plot of the first two column vectors of the loadings matrlK of PCA,...

Figure 3.4 Schematic representation of the steps involved in obtaining a two-dimensional NMR spectrum. (A) Many FIDs are recorded with incremented values of the evolution time and stored. (B) Each of the FIDs is subjected to Fourier transformation to give a corresponding number of spectra. The data are transposed in such a manner that the spectra are arranged behind one another so that each peak is seen to undergo a sinusoidal modulation with A second series of Fourier transformations is carried out across these columns of peaks to produce the two-dimensional plot shown in (C).

Fig. 31.3. (a,b) Reproduction of distances D and angular distances 0 in a score plot (a = 1) or loading plot (p = 1) in the common factor-space (c,d) Unipolar axis through the representation of a row or column and through the origin 0 of space. Reproduction of the data X is obtained by perpendicular projection of the column- or row-pattern upon the unipolar axis (a + P = 1). (e,0 Bipolar axis through the representation of two rows or two columns. Reproduction of differences (contrasts) in the data X is obtained by perpendicular projection of the column- or row-pattern upon the bipolar axis (a + P = 1). 

Figure 32.8 shows the biplot constructed from the first two columns of the scores matrix S and from the loadings matrix L (Table 32.11). This biplot corresponds with the exponents a = 1 and p = 1 in the definition of scores and loadings (eq. (39.41)). It is meant to reconstruct distances between rows and between columns. The rows and columns are represented by circles and squares respectively. Circles are connected in the order of the consecutive time intervals. The horizontal and vertical axes of this biplot are in the direction of the first and second latent vectors which account respectively for 86 and 13% of the interaction between rows and columns. Only 1% of the interaction is in the direction perpendicular to the plane of the plot. The origin of the frame of coordinates is indicated... 

A dissimilarity plot is then obtained by plotting the dissimilarity values, dj, as a function of the retention time i. Initially, each p 2 matrix Y, consists of two columns the reference spectrum, which is the mean (average) spectram (normalised to unit length) of matrix X, and the spectrum at the /th retention time. The spectrum with the highest dissimilarity value is the least correlated with the mean spectrum, and it is the first spectrum selected, x, . Then, the mean spectrum is replaced by x, as reference in matrices Y, (Y, = [x j x,]), and a second dissimilarity plot is obtained by applying eq. (34.14). The spectrum most dissimilar with x, is selected (x 2) and added to matrix Y,-. Therefore, for the determination of the third dissimilarity plot Y, contains three columns [x, x 2 /]> wo reference spectra and the spectmm at the /th retention time. 

Initially, A = 0.5 mols, B0 = 0 and D0 = 0.05. The reaction is run at constant pressure and temperature. Given the data of the first two columns beteen -the rate and the fractional conversion, confirm that the assumed rate equation is correct. Also check if the plot of rate against concentration has the peak that is characteristic of many autocatalytic reactions. 

Response to impulse input of tracer is shown in the first two columns. Find the principal RTD functions and prepare plots of E(tr) and F(tr). Identify the times between which a specified fraction of the tracer has left the vessel. 

Tracer impulse input data are given in the first two columns. Find various response functions and make plots of F(t) and A(tr). 

The last two columns show Tpftr and TcSTRi so one can simply read Ca(j) and T(t) fi om Table 5-1, and, since this is the same problem worked previously, the previous graphs can be plotted simply from this spreadsheet (see Figure 5-10). 

Principal components analysis can be best understood using a simple m o-variable example. With only two variables it is possible to plot the row space without the need to reduce the number of variables. Although this docs not fully present the utilit> of PCA. it is a good demonstration of how it functions. A two-dimensional plot of the row space of an example data set is shown in Figure 4.23. The data matrix consists of two columns, representing the two measurements, and 40 rows, representing the samples. Each row of the matrix is represented as a point (O) on the graph. 

Figure 4.2. The row space plot of a matrix with two columns.

If you followed the quest of the previous paragraphs, realize that these referenced Maxwell enthalpy charts and Table 1.10 are near directly and totally governed by temperature alone. Observe how Fig. 1.5 lays all the enthalpies to display two curves plotted as enthalpy vs. temperature. Both curves start at 0°F and end at 1000°F. With the two pressure curves as shown in Fig. 1.5, one can determine any enthalpy value, gas or liquid. You simply need one temperature. Pressure-based interpolation may then be made linearly between the two temperature intercept points of these two curves as shown in Fig. 1.5. Please note that the dashed temperature lines are the same as the column temperatures given in Table 1.10. Thus, Table 1.10 may be used just as if one were using the curve types of Fig. 1.5 to derive enthalpy values. Table 1.10 is proposed as an improved, easier-to-read resource as compared to a curve-plotted chart. The table gives an advanced get-ahead step, giving you the curve points to read to make your interpolation. 

Fig. 1.4. Plots of the actual, Epacked and Eopen, and the fictitious, E, electric field strengths in the packed and the open segments against X for Case I with a0pen/ Opacked = 3.1. At the top, the discontinuity in the electric field strength at the interface of the two column segments is illustrated. Rest of the conditions as in Figure 1.2.

Qualitative analysis is enhanced if data are acquired on more than one system. For example, in GC it is fairly common and easy to run a sample on each of two columns that are chosen to be widely different in their polarities. The results can be plotted as net retention volumes or as Kovats index values on either linear or log scales as shown in Figure 6.5. In either case, straight lines result for homologous series, thus aiding qualitative identifications. The principle is simple the more data, the more reliable the analysis. 

One of die simplest plots is that of die score of one PC against die odier. Figure 4.10 illustrates die PC plot of the first two PCs obtained from case study 1, corresponding to plotting a graph of die first two columns of Table 4.3. The horizontal axis represents the scores for die first PC and the vertical axis diose for die second PC. This picture can be interpreted as follows ... 

The selection of test items by such a design can be accomplished as follows For each constituent of the reaction system, two principal property axes should be considered. The columns of a two-level fractional factorial design matrix contain an equal number of minus and plus signs. If we let the columns pairwise define the selection of test systems, four combinations of signs are possible [(—),(—)], [(-), (+)]. [(+), (—)]. and [( + ), (+)]. These combinations of signs correspond to different quadrants in the score plots. Hence we can use the sign combinations of two columns to define from which quadrant in the score plot a test item should... 

The extractablllty of coals with pyridine after this treatment Is shown In Figure 3. The left column Indicates the extract-ability of the untreated coal. The two columns on the right-hand side show the Influence of the hydrogen fluoride. As a comparison, the Influence of aluminum chloride has been plotted, too. It can be seen that the extractablllty of high volatile coal decreased, probably due to the condensing effects of the catalysts on the coal molecules. 

If more than two columns (or rows) of data are selected for plotting. Excel uses the leftmost column or uppermost row as the independent variable (plotted on the X Axis) and the remaining rows or columns as the dependent variables (plotted on the Y Axis). Figure 5-3 illustrates one column of x data and two columns of y data to be selected for a chart. If the data series are non-adjacent. 

FIGURE 16.10. Two-dimensional plot showing factors. Data in column 1 are plotted against data in column 2. The result is approximately a straight line, which, since this line is only one-dimensional, can be considered as just one factor. 

Carry out a least-squares analysis of the experimental data provided in the first two columns of Table 8-1 and plotted in Figure 8-9. 

Figure 16-8 Data conversion and plots for Example 16-10. (a) The data are used to calculate the two columns In [A] and 1/[A]. (b) Test for zero-order kinetics a plot of [A] versus time. The nonlinearity of this plot shows that the reaction does not follow zero-order kinetics, (c) Test for first-order kinetics a plot of In [A] versus time. The observation that this plot gives a straight line indicates that the reaction follows first-order kinetics, (d) Test for second-order kinetics a plot of 1/[A] versus time. If the reaction had followed second-order kinetics, this plot would have resulted in a straight line and the plot in part (c) would not.

Figures 4-7 and 4-8, which show VjF, plotted against the integrals in (J) and (K), were prepared by graphical integration, with Fig. 4-5 used for the relations of X, to 7s6 nd jsg. The values of the integrals are given in the last two columns of Table 4-7. We see that in contrast to Fig. 4-6, based on S2, the data do not fall on a straight line as required by Eqs. (J) and (K). Also, the points for various reactants ratios are not in agreement. We conclude that a second-order mechanism based on either 85 or Sg as the reactive sulfur species does not agree with the experimental data.

Figure 2.24. The concept of orthogonality as shown by retention plots of two sets of columns for a variety of different analytes. (A) Since the log k data of the two columns (C8 and C18) are well correlated for most analytes, these two columns are expected to yield similar elution profiles. (B) The selectivity differences of a C18 and a polar-embedded phase (amide) column lead to very scattered correlation of their respective retention data. Methods using a C18 and a polar-embedded column are therefore termed orthogonal and expected to yield very dissimilar profiles. Diagram courtesy of Supelco, Inc.

When this is run, the plot shown in Figure 5.5 is produced. With virtually no programming other than a simple SQL statement, a plot of two columns of data from a table can be produced using R. Of course, once this data is read into an R dataframe, many other complex statistical operations can also be performed. Some of these are discussed in Chapter 12. 

The derivative -dCi./Jt is determined by calculating and plotting (-ACf lAt) as a function of time, t, and then using the equal-area ditferentiation technique (Appendix A.2) to determine -dC ldO as a function of C. First, we calculate the ratio t-AC Ar) from the first two columns of Table E5-1.2 the result is written in the third column. Nexl we use Table E5-1.2 to plot the third column as a function of the... 

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