Sometimes, a rather poor fit of the model or difficulty in convergence may be encountered if the model contains a large number of parameters and some of them are partially correlated. One example involves fitting an absorbance spectrum to a model containing five overlapped peaks. From Section 3.B.3, we know that we could use a model consisting of five Gaussian peaks with 15 parameters, consisting of peak widths, positions, and heights for each peak. However, a regression analysis that attempts to find all parameters at once will often fail to converge properly.
The solution to this problem is to obtain reliable estimates of the peak positions, such as by Fourier deconvolution or derivative spectra , then to fix the peak positions in a preliminary regression analysis. Once values of the 10 parameters describing widths and heights of the peaks are found in this preliminary analysis, they can be used as a starting point for the final nonlinear regression to find the best values of all the parameters. This technique will be discussed in more detail in Chapter 7.
Table 4.8 Program Statements Useful to Keep Parameters Positive
IF ftj < 0 then bi = 1 (or some other realistic positive number) y(calc) = regression model
y(calc) = regression model
Note: "IF" statements need to be placed in the routine that computes y(calc).
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