Three-dimensional scatter plots

The results show that DE-MS alone provides evidence of the presence of the most abundant components in samples. On account of the relatively greater difficulty in the interpretation of DE-MS mass spectra, the use of multivariate analysis by principal component analysis (PCA) of DE-MS mass spectral data was used to rapidly differentiate triterpene resinous materials and to compare reference samples with archaeological ones. This method classifies the spectra and indicates the level of similarity of the samples. The output is a two- or three-dimensional scatter plot in which the geometric distances among the various points, representing the samples, reflect the differences in the distribution of ion peaks in the mass spectra, which in turn point to differences in chemical composition of... 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

The aim of the graphics software, (tMod, threeD), is to enable the user to rapidly examine a large number of two- and three-dimensional scatter plots. At present the program is capable of handling up to 120 variables with up to 200 observations for each. 

Figure 8.39 shows a three-dimensional scatter plot of the first three PC scores obtained from a PCA analysis of 987 calibration spectra that were collected for a specific on-line analyzer calibration project. In this case, cluster analysis was done using the first six PCs (all of which cannot be displayed in the plot ) in order to select a subset of 100 of these samples for calibration. The three-dimensional score plot shows that the selected samples are well distributed among the calibration samples, at least when the first three PCs are considered. 

Figure 8.39 Three-dimensional scatter plot of the first three PCA scores obtained from a set of original calibration data. The calibration samples selected by the cluster analysis method are marked with an x. ...

For bivariate data the simple scatter plot of variate x against variate y is popular and there are several ways in which this can be extended to accommodate fmther variables. Figure 9 iUustrates an example of a three-dimensional scatter plot. The data used are from Table 11, representing the results of the... 

Figure 16 A three-dimensional scatter plot of the polymer spectra projected on to the first three principal components...

Figure 7.20 Three-dimensional scatter plot of the first three principal components analysis. Subjects that are removed from the bulk of the data represent influential observations that may be influencing the parameter estimates overall. ...

For bivariate data the simple scatter plot of variable y against variable a is popular and there are several ways in which this can be extended to accommodate further variables. Figure 1.12 illustrates an example of a three-dimensional scatter plot. The data used are from Table 1.11, representing the results of the analysis of nine alloys for four elements. The concentration of three analytes, zinc, tin, and iron, are displayed. It is immediately apparent from the illustration that the samples fall into one of two groups, with one sample lying between the groups. This pattern in the data is more readily seen in the graphical display than from the tabulated data. 

This style of representation is limited to three variables and even then the diagrams can become confusing, particularly for a lot of points. One method for graphically representing multivariate data ascribes each variable to some characteristic of a cartoon face. These Chernoff faces have been used extensively in the social sciences and adaptations have appeared in the analytical chemistry literature. Figure 1.13 illustrates the use of Chernoff faces to represent the data from Table 1.11. The size of the forehead is proportional to tin concentration, the lower face to zinc level, mouth to nickel, and nose to iron concentration. As with the three-dimensional scatter plot, two groups can... 

If we had a data set with three variables it would still be possible to visualize the whole information by a three-dimensional scatter plot, in which the coordinates of each object are the values of the variables. But what to do if there are more than three variables What we need therefore is a technique permitting the visualization by simple bi- or tri-dimensional scatter plots of the majority of the information contained in a highly dimensional data set. This technique is Principal Component Analysis (PCA), one of the simplest and most used methods of multivariate analysis. PCA is very important especially in the preliminary steps of an elaboration, when one wants to perform an exploratory analysis in order to have an overview of the data. 

Just like structural keys, pharmacophore keys can be readily extended to account for multiple conformations. Three-point pharmacophore keys also lend themselves to visualization in the form of a three-dimensional scatter plot (see Section 5.4). A number of people have followed up Sheridan s work, most notably the groups at Chemical Design, Rhone-Poulenc, and Abbott. ... 

As is the case for structural keys, pharmacophore keys can be readily extended to account for multiple conformations. Additionally, because pharmacophores are two- and three-dimensional objects, they are able to capture information on molecular shape and chirality. Three-point pharmacophore keys also lend themselves well to visualization via three-dimensional scatter plots (see section Visualization without Dimensionality Reduction below). Sheridan s original work has been extended by a number of groups, most notably those at Chemical Design [27], Rhone-Poulenc [30], and Abbott [10]. Davies and Briant [31] have employed pharmacophore keys for similarity/diversity selection using an iterative procedure that takes into account the flexibility of the compounds and the amount of overlap between their respective keys (see section Boolean Logic). 

These are plotted in Fig. 10.6, which shows the net intensity envelope in the xy plane as a solid line and represents the horizontally and vertically polarized contributions to the resultant by the broken lines. Since 0 is symmetrical with respect to the x axis, the three-dimensional scattering pattern is generated by rotating the solid contour around the x axis. 

Figure 2. Two- and three-dimensional perspective plots of a scattering halo for the 107. HPC-L solution at 45°C.

Results of ordination analysis are visualized using two-dimensional plots of the new axes. Although sometimes each axis can be viewed independently, or three axes can be viewed on a three-dimensional plot, typically a two-dimensional scatter plot is most informative. Interpretation of ordination scatter plots is quite intuitive, but we provide a few guidelines. Each PCA and COA principal component (PC) is decor-related (orthogonal), and thus each axis represents an independent or different trend... 

In the pharmacokinetic literature, three types of plots are most commonly seen scatter plots, histograms, and bar charts. Good plotting practices for each of these graph types will be presented. Beyond these plots are others, like QQ plots, dot plots, and three-dimensional surface plots. It is beyond the scope of this chapter to cover every conceivable plot, but many of the good plotting practices that will be put forth should carry over into other plot types as well. 

The plot of scattered field amplitude is shown in the upper part of Fig. 5.1 lb. It is the so-called sine-integral function. The scattering intensity is shown in the lower part of the figure. Integrating over the y and z co-ordinates we obtain the three-dimensional scattering amplitude F(q)= pFA AyA and intensity 7(q)= p"F2(A,AyA,)2. 

Pharmacophore plots are three-dimensional scatter diagrams that represent the three-point pharmacophores exhibited by a particular structure. The axes represent the distances between the three centers, and each point represents a distinct pharmacophore, labeled by a symbol that indicates the three centers, and color-coded according to whether it contains 1, 2, or 3 identical centers. A typical pharmacophore plot generated using Chemical Design s Chem-X suite is shown in Figure 6. 

The evolution of a wavepacket representing the H -b H2 scattering reaction for a particular set of inifial conditions is plotted on Figure 2 as a series of snapshots. To display the three-dimensional (3D) wavepacket on a two-dimensional (2D) plot, the reduced density... 

The central purpose of a graph is to present, summarize, and/or highlight trends in data or sets of data. Graphs of various types (e.g., scatter plots, contour plots, two- and three-dimensional line graphs, and bar graphs) are used for different purposes thus, authors must match their purpose with the appropriate type of graph. 

FIGURE 3.17 Three-dimensional plot of scattering intensity as a function of scattering angle and elution volume for a broad molecular weight distribution (MWD) polystyrene (PS) (NITS standard reference 706). (Courtesy of Wyatt Technology Corporation, Santa Barbara, CA 93117 wyatt wyatt.com. With permission.)... 

Fig. 4.3. Different methods of plotting one set of two-dimensional (forward scatter vs. side scatter) data. A Two separate histograms, a dot plot, a three-dimensional plot, and contour plots according to two different plotting algorithms. B Four additional contour plotting algorithms for the same data.

So far, we have discussed LEED patterns entirely in terms of two-dimensional lattices. However, the third dimension is also involved, because electrons can be scattered from deeper layers. For each layer spacing there are combinations of electron energies and angles of diffraction for which a Bragg condition in the vertical direction is satisfied. In such situations one observes very intense spots. Plots of the spot intensity as a function of the electron energy are called I-V plots these contain, in combination with the LEED pattern itself, three-dimensional... 

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