MATLAB Examples

This section presents three examples that show how to implement various forms of regression analysis in MATLAB. The topics considered are linear regression, nonlinear regression, and system identification. AH examples are based on real data obtained from experiments. Appropriate MATLAB code, as well as the final results, is provided so that the reader can modify these examples to fit their particular needs. 

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The capability of dragging results from one cell to others is a very useful property of Excel and becomes even more powerful in combination with the dollar operator ( ) correctly applied within the cell reference. Referring to the previous Matlab example, if the scalar element foi,2 (cell F3) is to be subtracted from matrix A (A3 C4) in Excel, putting the dollar operator ( ) in front of the column and row reference of the source cell containing the scalar foi,2 ( F 3), prevents "dragging-over" of the source cell F3 in both column and row direction. 

Symmetric matrices are square matrices that are identical to their transpose. They are invariant to an inflection at their main diagonal, i.e. invariant to the interchange of row and column index. In the former Matlab example both the 2x2 matrix Y1 and the 3x3 matrix Y2 are symmetric. 

For the time being let us assume that we know all the individual concentrations of four mixtures of three chemical components forming matrix C. Let us also suppose that we know the molar absorptivities of all three components at six wavelengths, matrix A. From those two matrices one can construct a multivariate measurement, matrix Y. In this or a similar way, most "experimental" data matrices used in later chapters will be simulated. A simple Matlab example ... 

In order to keep the spreadsheet reasonably small, the dimensions are much smaller than those in the Matlab examples. It is still a consecutive reaction scheme the spectra were recorded at 11 times and at 6 wavelengths. 

MATLAB Example 4.1 function to give sorted eigenvectors and eigenvalues... 

MATLAB Example 4.2 program to perform principal component... 

MATLAB Example 4.3 Principal component analysis using the SVD... 

MATLAB Example 4.6 function for calculating Malinowski s RE, IND, AND REV FUNCTIONS... 

Modeling and visualization of a reaction A —B requires only a few lines of MATLAB code (see MATLAB Example 7.1), including a plot of the concentration profiles, as seen in Figure 7.1. Of course this task can equally well be performed in Excel. 

Solutions for the integration of ODEs such as those given in Equation 7.5 are not always readily available. For nonspecialists, it is difficult to determine whether there is an explicit solution at all. MATLAB s symbolic toolbox provides a very convenient means of producing the results and also of testing for explicit solutions of ordinary differential equations, e.g., for the reaction 2A — B, as seen in MATLAB Example 7.2. (Note that MATLAB s symbolic toolbox demands lowercase characters for species names.)... 

In MATLAB Example 7.3a and 7.3b we give the code for a simplex optimization of the first-order kinetic example discussed above. Refer to the MATLAB manuals for details on the simplex function fminsearch. Note that all three parameters k, G. and % are fitted. The minimal ssq is reached at k = 0.048 s sA,. = 106.9 M cnr1, and b,x = 400.6 M 1cm 1. 

MATLAB Example 7.3b employs the function that calculates ssq (and also C). It is repeatedly used by the simplex routine called in MATLAB Example 7.3a. In Ligure 7.2 we have already seen a plot of the experimental data together with their fit. 

Figure 7.15 shows the calculated corresponding concentration profiles using the rate constants = 10" s 1 and k2 = 1 M s 1 for initial concentrations [A]0 = 1 M and [B](i = 0M. We used MATLAB s Runge-Kutta-type ODE-solver oae45. In MATLAB Example 7.6b, the function is given that generates the differential equations. It is repeatedly called by the ODE-solver in MATLAB Example 7.6a. 

FIGURE 2.2 Code Block 2—a Matlab example highlighting common machine precision issues encountered. 

In languages such as Java and C, it is not possible to use a variable without declaring a type for it. So, the Matlab example in Figure 2.4 would not run properly in those languages. In this code, there is an error on line 11 in Code Block 4, a... 

In Section 18.5> we will discuss matrix algebra in more dentil. If you do not have an adequate background in matrix algebra, you may want to read Section 18.5 before smdying the MATLAB examples that deal with matrices. 

Burden et al. [5] give an algorithm involving the Crank-Nicolson method for solving Equation 9.99, while Constantinides et al. [9] has MATLAB examples involving the nonhomogeneous case of Equation 9.99. 

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