Non-linear parameter estimation

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

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). 

Non-linear parameter estimation is far from a trivial task, even though it is greatly simplified by the availability of user-friendly program packages such as a) SIMUSOLV (Steiner et al., 1986), b) ESL, c) a set of BASIC programs (supplied with the book of Nash and Walker-Smith, 1987) or d) by mathematical software (MATLAB). ISIM itself does not supply these advanced features, but ISIM programs can easily be translated into other more powerful languages. 

Non-linear parameter estimation methods can also be used to estimate ka and kt. Two methods are recommended. Both use an organism-water two-compartment model (i.e., Equation (29)). The first method is the BIOFAC method (Blau and Algin 1978). This method is based on the condition that the chemical concentration in the water is constant throughout the duration of the experiment, such that Equation 29 can be integrated to give... 

In summary, sensitivity analysis, which has heretofore been considered a very large and costly calculation, has been reduced to the point where it may be done as a matter of routine, and even incorporated into other computations that rely on sensitivity values, as for example non-linear parameter estimation. 

Herzberg, T. (1983) MODFTT - A General Computer Program for Non-Linear Parameter Estimation. Inst. Chem. Eng., University of Trondheim, Norway. 

Fig. 15 Objective functions for non-linear parameter estimations in multiresponse situations...

Ferraris and Donati [73] reevaluated the data of Nielsen et al. using both the Temkin equation Eq. (112) modified by Dyson and Simon and Eq. (83) used by Nielsen et al. They used the same thermodynamic data as Dyson and Simon. A non-linear parameter estimation program was used to calculate the constants in the two equations minimizing the sum of squares defined by ... 

Using UCB condensation data base [7] for both pure and steam/air mixture available, the unknown parameters of the correlations have been estimated by using Marquardt-Levenberg non-linear parameter estimation method [8] which provides quicker convergence than alternative methods. Results of present correlations are given in Table I. 

Marquadt, D. W., An algorithm for least-squares estimation of non-linear parameters, J. Soc. Indust. Appl. Math., 11, 431, 1963. 

Marquardt, D.W., "An Algorithm for Least Squares Estimation of Non-Linear Parameters", Journ. Soc. Ind. Appl. Math., 1963,... 

The term a gives the slope of the left-hand ascending side of the curve and (a - b) that of the right-hand descending side. The non-linear parameter jS, which must be estimated by a stepwise iteration procedure, relates to the volume ratio of the aqueous and lipid phases in the system. Setting jS = 1 and b 2a produces the original McFarland model. Kubiny s bilinear model can be derived from kinetically controlled model systems as well as from equilibrium models, indicating that it is valid under diffusion control as well as under equilibrium or pseudo-equilibrium conditions. For many data sets, the bilinear function aptly fits the experimental observations. Difficulties in calculations may arise from unbalanced data sets, which often occur in environ-... 

D.W. Marquardt An algorithm for least square estimation of non-linear parameters, Soc.Ind.Appl.Math.J.11(1963)431-441 /33/ U. Hoffmann and H. Hofmann Einfuhrung in die Optimierung Verlag Chemie, Weinheim 1971... 

Step 2 Non-linear regression (Levenberg-Marquardt method) of the correct solution (eq. 10) holding tp and Pp constant at 0. In order to fit the data by a non-linear regression, estimates of the parameters b and have to be supplied. The estimate for the parameter b is computed by means of eq. 9 assuming an... 

Table 1 Relaxation spectrum and estimated values of the non-linear parameters in the XPP, PTT-XPP and mLeonov models for mLLDPE Exact 0201 at ISO C.

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

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