Parameter estimation multiple-response

This parameter-estimation technique has also been extended to the multiple-response case (D3). Just as was seen in the multiple-response... 

SA of SODEs describing chemically reacting systems was introduced early on, in the case of white noise added to an ODE (Dacol and Rabitz, 1984). In addition to expected values (time or ensemble average quantities), SA of variances or other correlation functions, or even the entire pdf, may also be of interest. In other words, in stochastic or multiscale systems one may also be interested in identifying model parameters that mostly affect the variance of different responses. In many experimental systems, the noise is due to multiple sources as a result, comparison with model-based SA for parameter estimation needs identification of the sources of experimental noise for meaningful conclusions. 

The sequential planning for the systems C02-Diphenyl and C02-n-Propylbenzene is shown in Tables 2 and 3, respectively. In Figure 1 is depicted the Hunter-Reiner and Ferraris-Forzatti deviations for the C02-n-Propylbenzene system. It can be noted that the selected points for both methodologies are (essentially) the same. This can be explained due to the very small estimated variances of the responses and, hence the Ferraris-Forzatti methodology reduces to the Hunter-Reiner, as can be verified by Equations (5) to (7). This occurred because of, for the systems studied here, the parameters were estimated with small uncertainties. So, the additional effort of the Ferraris-Forzatti methodology is not justified for these systems (the same result was obtained for the C02-Diphenyl), However, the same can not be true when considering multiple responses. 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

In the last twenty years, various non-deterministic methods have been developed to deal with optimum design under environmental uncertainties. These methods can be classified into two main branches, namely reliability-based methods and robust-based methods. The reliability methods, based on the known probabiUty distribution of the random parameters, estimate the probability distribution of the system s response, and are predominantly used for risk analysis by computing the probability of system failure. However, variation is not minimized in reliability approaches (Siddall, 1984) because they concentrate on rare events at the tail of the probability distribution (Doltsinis and Kang, 2004). The robust design methods are commonly based on multiobjective minimization problems. The are commonly indicated as Multiple Objective Robust Optimization (MORO) and find a set of optimal solutions that optimise a performance index in terms of mean value and, at the same time, minimize its resulting dispersion due to input parameters uncertainty. The final solution is less sensitive to the parameters variation but eventually maintains feasibility with regards probabilistic constraints. This is achieved by the optimization of the design vector in order to make the performance minimally sensitive to the various causes of variation. 

The higher-level parameters in the multiple regression model enable quantification of how chemicals influence each other in relation to the measured response. Suppose that fil 2 (the estimated function parameter for the first-level interaction term between chemical 1 and chemical 2) in a regression model... 

The true model parameters (ft, ftj(...) are partial derivatives of the response function / and cannot be measured directly. It is, however, possible to otain estimates, ft, bV], bVl, of these parameters by multiple regression methods in which the polynomial model is fitted to known experimental results obtained by varying the settings of xr. These variations will then define an experimental design and are conveniently displayed as a design matrix, D, in which the rows describe the settings in the individual experiments and the columns describe the variations of the experimental variables over the series of experiments. 

Changing the exit gas pressure also gave three multiple steady state responses in which the same set of operating parameters produced different reactor profiles. Finally, a rough estimate for the location of the bifurcation points was given for the coal moisture, steam feed rate, and exit gas pressure transient response runs. 

These parameters can be estimated by multiple linear regression. This method is described below. By this procedure, the polynomial model is fitted to known experimental results so that the deviations between the observed responses and the corresponding responses calculated from the model are as small as possible. How these calculations are done and how the experiments should be laid out to obtain good estimates of the model parameters is treated in detail in the chapters that follow. 

To be of any practical value, a response surface model should give a satisfactory description of the variation of y in the experimental domain. This means that the model error R(x) should be negligible, compared to the experimental error. By multiple linear regression, least squares estimates of the model parameters would minimize the model error. Model fitting by least squares multiple linear regression is described in the next section. 

From this conclusion follows, that a factorial design can be used to fit a response surface model to account for main effects and interaction effects. In the concluding section of this chapter is discussed how the properties of the model matrix X influence the quality of the estimated parameters in multiple regression. It is shown that factorial design have optimum qualities. 

Figure 10.67. Parameters from multiplicative effect in GEMANOVA model. The estimated response at specified levels of the four factors equal the product of the corresponding effects plus the muscle term, which varies between 31 and 33.

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