Fitting and Plotting

Many problems in engineering and sciences require the use of graphic tools to fit experimental data and plot them. If we start with plotting, the basic function is plot. We show different plotting routines in the examples. We first define the vectors  

Hint We use after the line if we do not want to see on the command window the result of the computation. Finally, the use of the function plot requires the vectors and we can add as options the symbols and line types for each of the series between 

A brief summary of the plotting options can be seen in the command window by using help plot . 

We can add labels, legend, and title directly in the command window (or in a script)  

A hint In the command window, we can use the arrows up to bring back the previous operations and reuse the information so that we do not have to retype them. 

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Here x ll angle between the in-plane component of the scattering vector q and the magnetic field/f. I q is the background term. Below the transition temperature, one expects to see harmonics. The coefficients measure the strength of -fold harmonics. Each of the yields an independent degree of bond-orientational order. The temperature dependence of the first seven members of the set Cf, is obtained from such a fit and plotted in Fig. 17. The results explicitly show that as the temperature decreases the sample smoothly develops first C5 order, then C12 order, then C,g order and so on. This transition evolves smoothly and... 

Data Fitting and Plotting with Known Functional Forms... 

This section will consider least squares data fitting and plotting of data using known or assumed functional forms. In order to get a feel for the use of the nlstsqO code, a simple example is first shown in Listing 7.10. This is a very simple example where data is generated by lines 5 through 9 from the equation ... 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

Demonstration Fit up a dashpot and spring model (Fig. 19.8) and hang it from a support. Hang a weight on the lower end of the combination and, using a ruler to measure extension, plot the creep out on the blackboard. Remove weight and plot out the reverse creep. 

In Figure 7, the resistance to mass transfer term (the (C) term from the Van Deemter curve fit) is plotted against the reciprocal of the diffusivity for both solutes. It is seen that the expected linear curves are realized and there is a small, but significant, intercept for both solutes. This shows that there is a small but, nevertheless, significant contribution from the resistance to mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in Figures 5, 6 and 7 support the Van Deemter equation extremely well. 

We established the coordinate system and plotted the curves using the Z-voltage of the piezoelectric ceramic as the X axis and the lateral voltage of the detector with four quadrants (V/s, Vii and Vn - Vi ) as the Y axis, as shown in Fig. 6. We fit linearly the varieties of the lateral voltages in the detector with four quadrants to the varieties of the... 

Figure 3.9. Demonstration of ruggedness. Ten series of data points were simulated that all are statistically similar to those given in Table 4.5. (See program SIMILAR.) A quadratic parabola was fitted to each set and plotted. The width of the resulting band shows in what ar-range the regression is reliable, higher where the band is narrow, and lower where it is wide. The bars depict the data spread for the ten statistically similar synthetic data sets.

Recall that the unlmodal first order and second order cases had good parameter accuracy and fit, and this is easily understood since curves (a) and (b) in Figure 6 have the same shape as those for Equation 11 (in Figure 5). However, curve (c) has a distinctly different shape indicating that Equation 11 cannot adequately describe the deactivation characteristics of this distribution. Figure 7 shows plots for the two other F(x) functions introduced earlier (Equations 14 and 15), which appear to represent the situation much better. The rate Equations 16 and 17 were fit to the same simulated polymerization rate curves with both showing much better 6 accuracy and large improvements in... 

In the trajectory study of Cl-—CHjCl complex formation by Cl" + CH3C1 association, the number of complexes with a lifetime t, i.e. N(t), was evaluated for different Cl" + CH3C1 initial conditions.36,37 The resulting plots of N(t) are highly nonexponential and plots of N(t)/N(0) were fit with the biexponential function... 

Using the actual dimensions of commercial steel pipe from Appendix F, plot the pipe wall thickness versus the pipe diameter for both Schedule 40 and Schedule 80 pipe, and fit the plot with a straight line by linear regression analysis. Rearrange your equation for the line in a form consistent with the given equation for the schedule number as a function of wall thickness and diameter ... 

The data are tabulated and plotted. All three lines are good linear fits, but... 

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