A3 Gram Schmidt OrthogonalizationAn Example

We now discuss the details of orthogonalization by using the example of the ECE model for chronocoulometry in eq. (4.5). Direct use of this equation led to slow convergence using steepest descent or Marquardt-

Levenberg algorithms, and indications of partial parameter correlation were obtained from correlation matrices.

A model can be considered a linear combination of functions that may contain nonlinear parameters. Gram-Schmidt orthogonalization [3] leads to an equivalent model from the explicit model for a single potential step chronocoulometric response for the ECE reaction. The orthogonalized model is a linear combination of orthogonal and normalized functions derived from the explicit ECE model.

As mentioned previously, the correlation between parameters can be thought of as lack of orthogonality between the parameter axes in the error surface plot. Such correlations can be expressed as the scalar products of appropriate pairs of unit vectors representing parameter axes in the parameter space. The scalar product of two axis vectors will be unity for full correlation and close to zero for uncorrelated parameters. As discussed already, a matrix containing correlation coefficients for all pairs of parameters can be constructed from the results of a regression analysis.

A representative correlation matrix for eq. (4.5) regressed onto a set of simulated data with 0.3% noise (Table 4.4) shows the correlation between parameters bi and b2 at -0.981, suggesting partial correlation and significant nonorthogonality. As in the previous example, this method did not cause serious errors in final parameter values, but the convergence times using a program with a Marquardt algorithm were quite long. Matrix inversion errors were sometimes encountered. The analysis providing the matrix in Table 4.4 required 27 iterations for convergence.

The object of orthogonalization is to transform the nonorthogonal model in eq. (4.5) to the form of eq. (4.6). In the new model, the hk(t}) are orthonormal functions, and the Bk are a new set of four adjustable parameters.

The starting basis functions are chosen from the overall form of eq. (4.5):

In developing the orthonormal basis function set, the next step is to accept one of the starting functions without a change. Hence, the first orthogonal

Table 4.4 Correlation Matrix for the Fit of Simulated Data onto the Nonorthogonal Model in Eq. (4.5)

0.834

1 b3

function was made identical to fx(t). Normalization of gi was then performed. The normalized function h\ took the form h,{t) = gl(i)/M (4.8)

where N\ is a normalization factor such that

Solving for N{ gave the first function of the orthonormal basis set:

According to Gram-Schmidt rules [3], the second function of the orthogonal basis set is a linear combination of the second function from the original model and the first orthonormal function. That is, g2(t)=f2(t)-a2Mt)- (4-11)

Here, the second term on the right-hand side of eq. (4.11) removes the component of f2(t) in the direction of /](()■ Imposing the orthogonality requirement of a zero scalar product of basis functions,

produced the a21 constant:

/= i where (t) is the average value of t. Substitution for a2i in eq. (4.11) gives g2(t) = t-(t). (4.14)

Finally, normalization of g2 yields

The last orthogonal function takes the form g3(t) = /3(0 - a32h2(t) - a3Mt) (4.16)

The coefficients a32 and aiX are found from the requirement of orthogonality between g3 and both h2 and hx. A complication arises here because the function /3 contains the nonlinear parameter k, represented here as B„. Thus, a32 and a31 are functions of B0 and must be evaluated each time B0 changes during the iterative optimization of the parameters by the regression analysis. Also, the normalization factor N3 depends on B0 and has to be reevaluated whenever the latter changed. To simplify the regression calculations the following expression was derived:

Thus, the orthogonal model for the ECE reaction in chronocoulometry is given by eq. (4.6) with huh2, and h3 from eqs. (4.10), (4.15), and (4.18). The parameters to be used for regression analyses are B0 = k (the same as b0 in the nonorthogonal model of eq. (4.5), Bu B2, and B3.

A detailed description of the applications of this orthogonal model can be found in the original literature [2], Here, we briefly discuss its performance on the same set of data that gave rise to Table 4.4 using the nonorthogonal model. The orthogonal model converged in 4 iterations, compared to 27 for the nonorthogonal model. We see from Table 4.5 that the offdiagonal matrix elements are very small indeed. Orthogonalization of the model has effectively removed correlations between parameters and dramatically improves the rate of convergence.

A.4. Fourier Deconvolution

This method can improve the resolution of data before nonlinear regression analysis. It has been used successfully for pretreatment of spectroscopic data composed of a collection of severely overlapped peaks [4]. In practice, Fourier deconvolution increases the resolution of a spectrum by decreasing the width of the component peaks, while keeping their peak positions and fractional areas the same. The fractional area is the area of the component peak divided by the total area of all the component peaks. The user of the software specifies a resolution enhancement factor proportional to the desired amount of decrease in the half-width of the component peaks.

Fourier deconvolution increases the resolution of a spectrum, as shown

Table 4.5 Correlation Matrix for the Fit of Simulated Data onto the Orthogonal Model in Eq. (4.6)

Bo Bi B2

Figure 4.2 Illustration of a Fourier deconvolution for enhancing the resolution of raw data consisting of two severely overlapped peaks.

in the conceptual illustration in Figure 4.2. Programs for this procedure are available in the literature [5], in software provided by many infrared spectrometer manufacturers, and can be constructed by using Mathcad or Matlab (see the end of Chapter 2 for sources of these mathematics software packages) with their built-in fast Fourier transform capabilities.

Fourier deconvolution has been used extensively in the analysis of infrared spectra of proteins. In this application, the deconvolution aids in identifying the number of underlying peaks in the spectra. If the main interest of the researcher is to obtain peak positions and areas, nonlinear regression analysis of the Fourier deconvoluted spectrum should give identical results as the analysis of raw spectral data. We shall encounter further examples of the use of Fourier deconvolution in Chapter 7.

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