Algebraic models

There has been fierce debate (see Refs. 232, 235-237) over the usefulness of the preceding methods and the matter is far from resolved. On the one hand, the use of algebraic models such as modified DR equations imposes artificial constraints, while on the other hand, the assumption of the validity of the /-plot in the MP method is least tenable just in the relatively low region where micropore filling should occur. 

As in algebraic models, the error term accounts for the measurement error as well as for all model inadequacies. In dynamic systems we have the additional complexity that the error terms may be autocorrelated and in such cases several modifications to the objective function should be performed. Details are provided in Chapter 8. 

The objective function is a suitable measure of the overall departure of the model calculated values from the measurements. For an individual measurement the departure from the model calculated value is represented by the residual e,. For example, the i,h residual of an explicit algebraic model is... 

The choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. 

Now we tum our attention to algebraic models that can only be represented implicitly though an equation of the form,... 

As we have already mentioned in Chapter 2, assuming linearity with respect to the unknown parameters, the general algebraic model can be reduced to the following form... 

Other Nonlinear Regression Methods for Algebraic Models... 

Having the smoothed values of the state variables at each sampling point and having estimated analytically the time derivatives, n we have transformed the problem to a usual nonlinear regression problem for algebraic models. The parameter vector is obtained by minimizing the following LS objective function... 

The above parameter estimation problem can now be solved with any estimation method for algebraic models. Again, our preference is to use the Gauss-Newton method as described in Chapter 4. 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

If the equality constraint involves independent variables and parameters in an algebraic model, i.e.. it is of the form, model equations reduces the number of unknown parameters by one. 

To compensate for the errors involved in experimental data, the number of data sets should be greater than the number of coefficients p in the model. Least squares is just the application of optimization to obtain the best solution of the equations, meaning that the sum of the squares of the errors between the predicted and the experimental values of the dependent variable y for each data point x is minimized. Consider a general algebraic model that is linear in the coefficients. 

Esposito, W. R. and C. A. Floudas. Parameter Estimation in Nonlinear Algebraic Models via Global Optimization. Comput Chem Eng 22 S213-220 (1998). 

Optimizers for Stand-Alone Operation or Embedded Applications / 8.9.2 Spreadsheet Optimizers / 8.9.3 Algebraic Modeling Systems... 

Finally, an algebraic model relationship is included in order to check on the total component material balance achieved in the simulation. The last lines specify the chemical reaction rate terms and calculate the total number of moles present at any time during the reaction. 

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