Generalization to ODE Models with Nonlinear Dependence on the Parameters

2 GENERALIZATION TO ODE MODELS WITH NONLINEAR DEPENDENCE ON THE PARAMETERS 

The derivative approach described previously can be readily extended to ODE models where the unknown parameters enter in a nonlinear fashion. The exact same procedure to obtain good estimates of the time derivatives of the state variables at each sampling point, t can be followed. Thus the governing ODE 

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. 

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Generalization to ODE Models with Nonlinear Dependence on the Parameters... 

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