Variables plotting

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

Has trending and reporting ability. Data can be dumped to a spreadsheet program and variables plotted against one another. 

A scatter plot provides a quick visual summary of where numerous data points exist in two-dimensional space. Scatter plots are commonly used to investigate positive or negative correlation between two variables plotted on the X and Y axes. 

If n(pe — pe°) + mpH > 0 then (Ox) > (Red) and the oxidant is the dominant species, and vice versa. Hence a plot of pe versus pH with (Ox) = (Red) has slope m/n and intercept pe, and for points above the line the oxidant is dominant and for points below the reductant is dominant, pe is taken as the dependent variable, plotted on the ordinate, because pH is often controlled by processes in addition to redox reactions and is therefore more properly the independent variable. 

Concentration residual vs. other pustulated variable plot [(c — c) vs. x]... 

Selected Variable Plot (Model and Variable Diagnostic) The variables that have been induded in the model should be examined to see if they are reasonable given knowledge about the chemistry of the samples and measurements. Figure 5.68 displays the calibration spectra with vertical lines indicating the variables selected for the models for components A and B. 

Concentredion Residuals vs. Other Postulated Variable Plot (Model and Sample Diagnostic) The concentration residuals in this example are acceptable, and so rt is assumed that all relevant sources of variation are being modeled, No other postulated parameters are hypothesized. 

Selected Variable Plot (Model and Variable Diagnostic) Figure 5.79 shows the three selected variables for the first derivative preprocessed data. 

Concentration Residual vs. Other Postulated Variable Plot (Model Diagnostic) For this example, no plots were generated because no variables were identified that are correlated with the concentration residuals. 

Single-variable plots of Sy(X) or S(X) such as those shown in Fig. 5.1 do not yet convey a geometrical picture of the multivariate entropy function in higher dimensions. Figure 5.2 shows a more complete 3-dimensional SUX view of the S(U,X) surface for a general extensive variable X. 

A series of laboratory experiments with a pure substance (shown in Figure 2-2) will result in data for pressure, temperature, and volume. A similar series of experiments with a two-component system will result in data for additional variables. The composition of the overall mixture, the composition of the equilibrium liquid, and the composition of the equilibrium gas are all important. Therefore, in addition to plotting combinations of temperature, pressure, and volume, additional graphs with these variables plotted against composition are possible. 

Figure 19. Reduced-variable plot of He (23S) + He differential cross sections, obtained with charge-exchange He source.

Equation 4.47a is illustrated in Fig. 4.2 for the protonation of hematite (a-Ec203) suspended in acidic perchlorate solution.24 The variable plotted along the x-axis is obtained by factoring ka on the right side of Eq. 4.47a and making use of Eq. 4.31 ... 

If the data when plotted on a graph fall on a smooth curve of complex shape then the present techniques are not applicable. In practice, therefore, we should only use the present statistical treatment when the pairs of observations on the two variables plotted on a graph give something that looks approximately a straight line. 

Define the dry-bulb temperature, wet-bulb temperature, and humid volume of humid air. Given values of any two of the variables plotted on the psychrometric chart (dry-buib and wet-bulb temperatures, absolute and relative humidity, dew point, humid volume), determine the remaining variable values and the specific enthalpy of the humid air. Use the psychrometric chart to carry out material and energy balance calculations on a heating, cooling, humidification, or dehumidification process involving air and water at 1 atm. 

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. 

The variability plot has become the most widely used reference standard for comparing variable domains. It has the advantage that it may be used to compare selected populations of light or of heavy chains, such as all human VHI chains, etc. It may be continuously kept up to date as more sequences accumulate. Most striking has been the retention of the three CDR as the numbers of chains sequenced have increased. Other parameters and correlations of other properties relate closely to findings by variability plots. The upper limit for variability depends upon n2 when n is the total number of sequences considered since there are 20 amino acids the upper limit becomes 20 x 20 = 400 if more than 20 sequences are available. 

FlG. 8. Variability plot for 67 cytochromes c. From Kabat et al. (1976b). 

Construct reduced variables plot of data using the generalized at vs. T data... 

Figure 5.1 Comparison of dimensionless canopy velocity perturbation from the theory of Finnigan and Belcher, 2004 [189] (solid line) with the no-canopy, solution of Hunt, Liebovich and Richards, 1988 [287] (dotted line). Variable plotted is where Usc = jU0. Note the solution [287] is only valid to z = —d + z0- Profiles are plotted at a series of x/L values between X/L = -2 (upwind trough) and X/L = 2 (downwind trough) on one of a series of sinusoidal ridges. The units of Z are (m) and the vertical range is from 2hi > Z > Lc.

Figure 7.1 shows other variables plotted versus reflux ratio at a constant distillate rate. A plot of B the bottoms rate expressed as a fraction of the feed, is, of course, constant at 0.5 when the overhead rate is 50 kmol/h. The condenser and reboiler... 

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