Curve fitting 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... 

If the parameters enter the equation linearly, then the minimization problem reduces to a set of linear equations which are solved easily by Excel and MATLAB. The effectiveness of the curve fit is often reported as values of the linear correlation coefficient squared, r. The linear correlation coefficient is defined as (Press et al., 1986, p. 484) ... 

Figure E.l. Linear curve fit to data in Table E.l using MATLAB.

Figure E.4. Linear (a) and quadratic (b) curve fit to the data in Table E.2 using MATLAB.

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. 

The parameters in the adsorption isotherms were estimated from the experimental equilibrium data using MATLAB Curve Fitting Toolbox. The comparison of experimental and estimated data by Langmuir, Freundlich, Redlich-Peterson and combined Langmuir-Freundlich models for the investigated systems are presented in Figures 1 to 3 for six investigated systems. 

Once the selection of a possible kinetic model and suitable reactor model are complete (equation (8-1)), a non-linear, least square method can be adopted to determine the kinetic and adsorption parameters. This can be achieved by minimizing an objective function representing the sum of the differences between the model concentration estimates and the measured experimental concentrations. This non-linear, least square fit can be performed using the curve fit functions available in Matlab, as recommended by Ibrahim (2(X)1). 

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... 

There are many factors that can affect the reduced value of fluence rate from (1278.1 mW.mm ) at (0.02 mm) to (890.84 mW.mm ) at (0.08 mm). The amounts of light in the diffusion region are decreased and converted to scattered photons and absorbed photons because the transition photons from the stratum corneum to the epidermis were chromophore dependence and skin depth dependence. The part of fig 2b represents a fitting curve that was not linearly but an exponential relation. The correlation coefficients were 0.97 to 0.98 for all curves. Curve fitting by MATLAB software was used to estimate the transmission through epidermis and to calculate the photodynamic dose per penetration depth of skin. 

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 ... 

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. 

Figure 5.15(a) The main window for the MATLAB Curve Fitting Toolbox (old look). 

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... 

Figure 5.17 The MATLAB Curve Fitting Toolbox is ready for the next step, i.e., the fitting step, after properly defining the x and y variables.

Twelfth, click the OK button (shown as a framed button in Fig. 5.24). MATLAB will return to the main Fit Editor window as shown in Fig. 5.25a. Figure 5.25b shows the new look of the Curve Fitting Toolbox window where the user selects Custom Equation from the main category drop-down list and then enters the expression fory = f x a, b, c). 

Figure 5.25(a) The old look of the MATLAB Curve Fitting Toolbox, where it is ready to start the process of curve fitting or nonlinear regression. 

Figure 5.25(b) The new look of the MATLAB Curve Fitting Toolbox. After successfully entering the syntax error-free model (i.e., y = f(x)), MATLAB will either start the process of curve fitting or wait for the user s command if the Auto fit option is de-selected. 

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. 

Figure 5.5Qq shows the 95% confidence interval which brackets (or sandwiches) the curve. From a statistics point of view, the 95% confidence interval means that out of 100 samples being measured fory, at the given x 95 of them will have a value ofy that lies within the range ofy, (i.e., between lower f(x,) and upper / (x,)) at the given x,. For the new look of the MATLAB Curve Fitting Toolbox (see Fig. 5.25bi. click on the Tools menu, followed by the Prediction Bounds submenu, and pick up, for example, a 95% confidence interval so that the 95% confidence envelope will sandwich the curve, as shown in Fig. 5.30b. 

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. 

Figure S.34 shows the default main window of the MATLAB Surface Fitting Toolbox, which is similar to that of Fig. 5.25b (a new look for the MATLAB Curve Fitting Toolbox).

Figure 5.34 The default main window of the MATLAB Surface/Curve Fitting Toolbox.

NAN values are not a serious issue for the Surface/Curve Fitting Tool. For example, some solubility data at either 20 or 25°C are missing in Table 5.1. Nevertheless, MATLAB will take care of such a shortage in data and consider only fully populated rows. A fully populated row means all data entries are supplied for each column. 

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. 

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