Parallel coordinate plot

Figure 14. Parallel coordinate plots of the dihedral angles versus the dihedral sequence that display the extent of the sampling of the conformational phase space of Ala-ciiPro-Tyr, in 2 ns free MD simulations (left hand plots) and in 2 ns biased MD simulations (right hand plots). The top plots display all the conformations sampled during a MD trajectory starting from the bl conformation, and the bottom plots, the ones sampled during a simulation starting from the al conformation (see Figure 3).

In some cases, it was deemed impossible to optimize the molecule to reach all desired properties. In particular, the concept of competing objectives is well established in multicriteria optimization case studies. Competing objectives are those where an improvement in one objective results in the deterioration in the other. Parallel coordinate plots, a simple tool that can be used to identify competing objectives, were found to be a quick and useful aid. Figure 8.17 illustrates the case of two competing objectives, solubility and in vitro enzyme potency the crossed lines clearly identify the objectives are in competition. 

FIGURE 8.17 Parallel coordinate plots allow detection of competing objectives in optimization processes. This is an extreme example, where solnhdity and in vitro potency ( Primary XC50 ) are competing. When solnhdity increases, potency decreases and vice versa. This results in the characteristic erossed-hne plot shown here. 

FIGURE 9A (a) Parallel coordinates plot of Project A compounds six most important experimental properties (normalized fiom 0 to 1) and weighted desirability score. Higher scores are depicted in green, (b) Parallel coordinates plot of the six compounds having a weighted desirability score above 0.8. The actual candidate of Project A is highlighted in red and has a score of 0.83. For color details, please see color plate section. 

FIGURE 9.4 (a) Parallel coordinates plot of Project A compounds six most important... 

Chernoff Faces, Andrews Curves, Star Diagrams, and Parallel Coordinate Plots... 

Parallel coordinates is two-dimensional graphical method for representing multiple dimensional space. In the example shown in Figure 8.13, a point in seven-dimensional space is represented by the coordinates (xi, X2, X3, X4, X5, Xg, X7). Since we cannot visualise space of more than three dimensions, the value of each coordinate is plotted on vertical parallel axes. The points are then joined by straight lines. 

Each row in the database is then plotted as a line on the parallel coordinates chart. The result will initially appear very confused with a large number of lines superimposed. The next step is to add the HI/LO constraints on each vertical axis. If a line violates any constraint on any axis then the whole line is deleted. The lines remaining will each represent an occasion in the past when all the process conditions satisfied the constraints. The final step is to choose the line for which the value on the cost axis is the lowest. Since this axis is the MVC cost function, the line with the lowest value will represent the operation that the MVC would select. 

While there is yet to be developed an entirely satisfactory solution to this problem, some ideas have been applied successfully. One is the use of a radar plot. This is similar to parallel coordinates except that the axes are arranged radially. Only a limited number of CVs and MVs are practicable - perhaps a maximum of around 12, so only the more important variables are included. 

FIG. 16-1 Isotherms (left) and isosteres (right). Isosteres plotted using these coordinates are nearly straight parallel hnes, with deviations caused by the dependence of the isosteric heat of adsorption on temperature and loading. 

The laboratory studies utilized small-scale (1-5-L) reactors. These are satisfactoiy because the reaction rates observed are independent of reac tor size. Several reac tors are operated in parallel on the waste, each at a different BSRT When steady state is reached after several weeks, data on the biomass level (X) in the system and the untreated waste level in the effluent (usually in terms of BOD or COD) are collected. These data can be plotted for equation forms that will yield linear plots on rec tangular coordinates. From the intercepts and the slope or the hnes, it is possible to determine values of the four pseudo constants. Table 25-42 presents some available data from the literature on these pseudo constants. Figure 25-53 illustrates the procedure for their determination from the laboratory studies discussed previously. 

When cr0 is plotted vs. AE, a linear relationship is obtained (Figure 3). Poly-acenes show a similar linear relationship, which may be parallel to that of the isonitrile complexes investigated and which is displaced toward lower values of o-o- A comparison between the isonitrile complexes and polyacenes demonstrates most clearly the wide ranges of activation energies, AE, and of the constant, o-o, which may be achieved merely in one series of coordination complexes as compared to polyacenes. 

Historically, however, it has been much more common for experimentalists to introduce a new variable into Figure 1, changing either the temperature of one or more samples of fixed composition, or the electrolyte concentration in a series of samples of fixed amphiphile—oil—water ratio. The former constitutes a temperature scan the latter experiment is widely known as a salinity scan. When the temperature of an amphiphile—oil—water system is varied, the phase diagram can be plotted as a triangular prism (because temperature is an intensive or field variable). When a fourth component (eg, NaCl) is added at constant temperature, tetrahedral coordinates, are appropriate (conjugate phases have different salinities, and the planes of different tietriangles are no longer parallel). 

In systems involving three components, composition is plotted on a triangular section (Figure 6.2). Pure components are represented at the comers and the grid lines show the amount of each component. All of the lines parallel to AB are lines on which the %C is constant. Those nearest C have the greatest amount of C. To represent temperature, a third dimension is needed. Figure 6.3 is a sketch of a three-dimensional ternary diagram in which temperature is the vertical coordinate. 

The most convenient way to plot a projection of an w-dimensional system in an m-dimensional linear projective subspace is to use multiple orthogonal projection, in which the directions of projection rays are parallel to the normal vectors (qi, q>,. .., qn-m) defining the projective subspace. Under this projection, a point is first projected in the direction of qi, then of q/, and so on. A convenient set of canonical coordinates describing the projective subspace is given by... 

Fig. 4-5 Ramachandran plot for alanylalanine, showing the fully allowed regions (double-hatched) and partially allowed regions (single-hatched) of and ip angles (see Fig. 4-4). The coordinates for the parallel and antiparallel /3 structures (/3p and /3a, respectively) and for the left-handed and right-handed a helices (aL and aR, respectively) are indicated.

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