Curve Fitting with MATLAB

In Secdon 14.8, we discussed the concept of curve fitting. MATLAB offers a variety of curve-fitting options. We will use Example 14.11 to show how you can also use MATLAB to obtain an equation that closely fits a set of data points. For Example 14.11 (Revisited), we will use the 

Find the equation that best fits the following set of data points in Table 15.13. 

In Secdon 14.8, plots of data points revealed that the teladonship between y and x is qua-dradc (second order polynomial). To obtain the coefficients of the second order polynomial that best fits the given data, we will type the following sequence of commands. The MATLAB Command Window for Example 14.11 (Revisited) is shown in F jire 15.27. 

Upon execution of the polyfit command, MATLAB will return the following coefficients, Cq = 1, Cl = —3, and C2 = 2, which leads to the equation j/ = - 3x + 2. 

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In this chapter, you will learn how to perform curve fitting with MATLAB in a manner suitable for those with little or no programming experience. You will also see how to automate your entire curvefitting workflow, including ... 

To use nonlinear regression, you minimize Eq. (E.3) with respect to the unknown parameters. Polynomial and multiple regression do this too (behind the scenes), but for nonlinear curve fits it is necessary to use functions such as Solver in Excel and fminsearch in MATLAB. This is demonstrated using the same example given above for multiple regression. 

A polynomial fnnction is the first approach suggested by most of the curve-fitting tools available (e.g.. Excel, graphing calculators, MATLAB). Usually, it is a univariate (single-variable) polynomial function with constant parameters given by... 

Figure 12.4 shows the experimental versus simulated data obtained from the MATLAB program during reaction at 50°C. It is observed that the simulated curves adequately fit with the experimental data. An increasing trend in the rate of methyl ester formation with reaction temperature found in Figure 12.5 confirms that the reaction is favoured at higher temperatures, in line with those reported in the literature (Noureddini and Zhu, 1997 Kusdiana and Saka, 2001 Zhou et al, 2003 Schumacher, 2005 Vicente et al, 2005, 2006 Issariyakul and Dalai, 2010). 

NOTE We will show how to proceed with curve-fitting process showing both the old and new look of the MATLAB Curve Fitting Toolbox. In general. Fig. 5.2q will be reserved for the old look and Fig. 5.2b for the new look whenever there is a difference worth mentioning. 

Second, define the dependent (y) and independent (x) variables. Click the Data button (shown as a framed button in Fig. 5.15). Figure 5.16 shows the data window that defines thex andy variables. With the new look of MATLAB Curve Fitting Toolbox, you define the dependent variable (y), the independent variable (x), the model main category, and the method t e from the drop-down lists (shown as framed boxes in Fig. 5.15bL... 

Based on the aforementioned results, the model with the estimated parameters is definitely a misfit, as also shown in Fig. 5.26. The same terrible situation will occur with the new look MATLAB Curve Fitting Toolbox (see Fig. 5.25bl. 

Finally, from the File menu in the main window of the Curve Fitting Toolbox, you may choose Generate Code from the drop-down list to create the M-file, which when executed will create a plot similar to the plot in the main Curve Fitting Toolbox, using the data that you provide as input. You can use this function with the same data you used with the MATLAB Curve Fitting Toolbox or different data sets. You may want to edit the function to customize the code. 

We do not design our own algorithm here but use the fin Insearch. m function supplied by Matlab. It is based on the original Nelder, Mead simplex algorithm. As an example, we re-analyse our exponential decay data Data Decay. m (see p. 106], this time fitting both parameters, the rate constant and the amplitude. Compare the results with those from the linearisation of the exponential curve, followed by a linear least-squares fit, as performed in Linearisation of Non-Linear Problems, (p.127). 

Analysis of variance (ANOVA) analyses were performed using the general statistical package StatView 5.01 (SAS Institute, Cary, NC, USA). The ANOVAs were calculated as repeated-measures ANOVAs with wells as within factor for phase 1 and with plates as within factor for subsequent phases. Specialized statistics, such as comparison of fits of different calibration curves, were calculated in MATLAB 5.1 (MathWorks, Natick, MA, USA) using custom routines. 

To do the integration, volume data were obtained from the NIST program at 0.1 bar intervals from 1 to 10 bars, and at 10 bar intervals from 10 to 1000 bars. Integration was done numerically in matlab , after fitting the curve with a spline function. If you try this, don t forget to get V, RT/P, and P in compatible units. The easiest way is to change volumes in cm moU to Jbar mol by multiplying by 0.1. 

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