Non-linear least-squares optimisation

If the reaction order (n) with respect to component A is known in advance, the reaction model in Equation 8.14 can be integrated. Assuming the reaction is first order in component A ( n= 1), the rate constant, k, can be determined by the non-linear least-squares optimisation indicated in Equation 8.16 ... 

The aim of the multivariate evaluation methods is to fit a reaction model to the measured reaction spectrum on the basis of the Beer-Lambert law and thus identify the kinetic parameters of the model. The general task can be described by the non-linear least-squares optimisation described in Equation 8.20 ... 

The authors state that the iteration method used converges to an accurate value of Px- In fact, the iteration method does not converge and cannot be used to find P. Other methods must be used such as non-linear least squares optimisation, but a good fit can hardly be expected between the model and the experimental data over the whole range of sulphate concentrations. 

Later, in Chapter 4.4, General Optimisation, we discuss non-linear least-squares methods where the sum of squares is minimised directly. What is meant with that statement is, that ssq is calculated for different sets of parameters p and the changes of ssq as a function of the changes in p are used to direct the parameter vector towards the minimum. 

Figure A-31 Solubility of NiO in basic solutions as a function of initial molality of NaOH. Symbols refer to experimental data [80TRE/LEB2], the lines have been obtained from non-linear least squares fits to the experimental data in order to optimise the stability constants for Ni(OH)j ...

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

In general, the error e tic-q-i+j, 0) is a non-linear function of the parameter vector 0. Therefore, the above problem is a well-known nonlinear least squares problem (NLSP) that may be solved by various optimisation algorithms such as the Levenberg-Marquardt algorithm [2], the quasi-Newton method or the Gauss-Newton (GN) algorithm [3]. 

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