Density function estimation errors

The first task considered is the robust estimation of fitting parameters. Following to Peter Huber, the consideration is built at the assumption that the density function of the experimental random errors (8) can be presented in the following form ... 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

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). 

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

Vibrational frequencies measured in IR experiments can be used as a probe of the metal—ligand bond strength and hence for the variation of the electronic structure due to metal—radical interactions. Theoretical estimations of the frequencies are obtained from the molecular Hessian, which can be straightforwardly calculated after a successful geometry optimization. Pure density functionals usually give accurate vibrational frequencies due to an error cancellation resulting from the neglect of... 

Box and Draper (1965) took another major step by deriving a posterior density function p 6 Y), averaged over S, for estimating a parameter vector 6 from a full matrix Y of multiresponse observations. The errors in the observations were assumed to be normally distributed with an unknown m X m covariance matrix S. Michael Box and Norman Draper (1972) gave a corresponding function for a data matrix Y of discrete blocks of responses and applied that function to design of multiresponse experiments. 

Box and Draper (1965) derived a density function for estimating the parameter vector 6 of a multiresponse model from a full data matrix Y, subject to errors normally distributed in the manner of Eq. (4.4-3) with a full unknown covariance matrix E. With this type of data, every event u has a full set of m responses, as illustrated in Table 7.1. The predictive density function for prospective data arrays Y from n independent events, consistent with Eqs. (7.1-1) and (7.1-3), is... 

Maximum likelihood (ML) estimation can be performed if the statistics of the measurement noise Ej are known. This estimate is the value of the parameters for which the observation of the vector, yj, is the most probable. If we assume the probability density function (pdf) of to be normal, with zero mean and uniform variance, ML estimation reduces to ordinary least squares estimation. An estimate, 0, of the true yth individual parameters (pj can be obtained through optimization of some objective function, 0 (0 ). ModeP is assumed to be a natural choice if each measurement is assumed to be equally precise for all values of yj. This is usually the case in concentration-effect modeling. Considering the multiplicative log-normal error model, the observed concentration y is given by ... 

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