Levenberg-Marquardt nonlinear procedure

The procedure was tested on simulated time domain MRS data where the model data consisted of metabolite peaks at 3.2, 3.0 and 2.0 ppm representing choline, creatine and IV-acetylaspartate (NAA) respectively, with corresponding values of Ak of 1.0, 1.0 and 3.0 units.89 White noise of specified standard deviation, crt, was then added. The Levenberg-Marquardt method requires suitable initial values for each of the nine parameters being fitted. The initial values of the three frequencies were taken as their known values. An exponentially decaying curve with a constant offset parameter was fitted, using a nonlinear least-squares fit, to the envelope of the free induction decay, Mv(t), in order to obtain an initial value for T and for the amplitudes, each of which was taken to be one-third of the amplitude of the envelope. The constant offset was added to account for the presence of the noise. 

A nonlinear, multiparameter regression procedure (Levenberg-Marquardt method) was applied to estimate the kinetic parameters involved in Equations (51)-(54). The experimental concentrations of the pollutant (4-CP) and of the main intermediate species (4-CC and HQ) at different reaction times were compared with model predictions. Under the operating conditions of the experimental runs, it was found that the terms a Ci- cp(f)/ aiC4-cc(f)/ and 02CHQ(t) were much lower than 1. As a result, the final expressions employed for the regression of the kinetic parameters are the following ... 

Figure 3.8 Example of parameter redundancy in nonlinear models. Symbols were generated using the model given by Eq. (3.97). Solid line is the predicted fit using Eq. (3.98). The biexponential model predicted values cannot be distinguished from data generated using a triexponential equation. Starting values were 10,1,10, and 0.25. Model was fit using the Levenberg— Marquardt method within the NLIN procedure in SAS.

The Arrhenius form of the parameters are used, i.e., ki=Aie, Ki=Aje, and the exponent % were estimated using the nonlinear regression software package SAS (Statistical Analysis Software). The Proc Model (with Marquardt-Levenberg method) and Fit Procedures in SAS were used for this purpose. The results are shown in Table 1 and Table 2. 

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