Marquardt procedure

Therefore, the trial model function will in general be a nonlinear function of the independent variable, time. Various mathematical procedures are available for iterative x2 minimization of nonlinear functions. The widely used Marquardt procedure is robust and efficient. Not all the parameters in the model function need to be determined by iteration. Any kinetic model function such as Equation 3.9 consists of a mixture of linear parameters, the amplitudes of the absorbance changes, A and nonlinear parameters, the rate constants, kb For a given set of kb the linear parameters, A, can be determined without iteration (as in any linear regression) and they can, therefore, be eliminated from the parameter space in the nonlinear least-squares search. This increases reliability in determining the global minimum and reduces the required computing time considerably. 

In this way a function 4> is defined, viiich we describe as the surface above the parameter space. Marquardts procedure starts with a set of inferred initial values ... 

The differences between the target spectrum on the left-hand side and the right-hand side of the above formula are minimized through the Levenberg-Marquardt procedure by best fitting of the constant parameters t, tl,... t Ck, , (,k, 0)k-... 

Valko, P. and S. Vajda, "An Extended Marquardt -type Procedure for Fitting Error-In-Variables Models", Computers Chem. Eng., 11, 37-43 (1987). 

The most effective spectrophotometric procedures for pKa determination are based on the processing of whole absorption curves over a broad range of wavelengths, with data collected over a suitable range of pH. Most of the approaches are based on mass balance equations incorporating absorbance data (of solutions adjusted to various pH values) as dependent variables and equilibrium constants as parameters, refined by nonlinear least-squares refinement, using Gauss-Newton, Marquardt, or Simplex procedures [120-126,226],... 

Marquardt (1963), Levenberg (1944), and others have suggested that the Hessian matrix of fix) be modified on each stage of the search as needed to ensure that the modified H(x),H(x), is positive-definite and well conditioned. The procedure adds elements to the diagonal elements of H(x)... 

Valko, P., and Vadja, S. (1987). An extended Marquardt-type procedure for fitting error-in-variable models. Comput. Chem. Eng. 11, 37-43. 

In practice, initial guesses of the fitting parameters (e.g. pre-exponential factors and decay times in the case of a multi-exponential decay) are used to calculate the decay curve the latter is reconvoluted with the instrument response for comparison with the experimental curve. Then, a minimization algorithm (e.g. Marquardt method) is employed to search the parameters giving the best fit. At each step of the iteration procedure, the calculated decay is reconvoluted with the instrument response. Several softwares are commercially available. 

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

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 ... 

C.Carrot, J.Guillet, J.F.May, J.P.Puaux, Application of the Marquardt-Levenherg procedure to the determination of discrete relaxation spectra, Makromol. Chem., Theory Simul. 1 (1992), 215-231. 

To solve the equations of problem P ), the optimization algorithms used are Levenberg-Marquardt and Trust-Region procedures. These methods enable computation of the solution by using the Jacobian matrix and the Hessian matrix (or its approximation) related to the objective function E(Y) [57]. 

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. 

Fluorescence decay profiles are analysed as a sum of up to three exponentials by an iterative reconvolution procedure which has been described elsewhere (17-19) and is based on the Marquardt algoritlim. Goodness of fit is judged by inspection of the weighted residuals, autocorrelation function of the weighted residuals, reduced chi-square value and the Durbin-Watson parameter. 

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