Parameter estimation robustness

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

Another way is the robust parameter estimation on the basis of median statistics (see Sect. 4.1.2 Danzer [1989] Danzer and Currie [1998]). For this, all possible slopes between all the calibration points bij = (yj — yi)/(xj — X ) for j > i are calculated. After arranging the b j according to increasing values, the average slope can be estimated as the median by... 

Occasionally it is convenient to refer to the p function in (11.21), but generally the form (11.22) is used in robust M-estimation. The use of the t(r form is due to Hampel s concept of the influence function (Hampel et al., 1986). According to the IF concept, the value of it represents the effect of the residuals on the parameter estimation. If iff is unbounded, it means that an outlier has an infinite effect on the estimation. Thus, the most important requirement for robustness is that iff must be bounded and should have a small value when the residual is large. In fact, the value of the iff function corresponds to the gross error sensitivity (Hampel etal., 1986), which measures the worst (approximate) influence that a small amount of contamination of fixed size can have on the value of the estimator. 

We will follow the guidance of Albert Einstein to make everything as simple as possible, but not simpler. The reader will find practical formulae to compute results like the correlation matrix, but will also be reminded that there exist other possibilities of parameter estimation, like the robust or nonparametric estimation of a correlation. 

Model validation is a process that involves establishing the predictive power of a model during the study design as well as in the data analysis stages. The predictive power is estimated through simulation that considers distributions of PK, PD, and study-design variables. A robust study design will provide accurate and precise model-parameter estimations that are insensitive to model assumptions. 

L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. Applied interval analysis with examples in parameter and state estimation robust control and robotics. Springer-Verlag, Londres, 1991. 

Approaches based on parameter estimation assume that the faults lead to detectable changes of physical system parameters. Therefore, FD can be pursued by comparing the estimates of the system parameters with the nominal values obtained in healthy conditions. The operative procedure, originally established in [23], requires an accurate model of the process (including a reliable nominal estimate of the model parameters) and the determination of the relationship between model parameters and physical parameters. Then, an online estimation of the process parameters is performed on the basis of available measures. This approach, of course, might reveal ineffective when the parameter estimation technique requires solution to a nonlinear optimization problem. In such cases, reduced-order or simplified mathematical models may be used, at the expense of accuracy and robustness. Moreover, fault isolation could be difficult to achieve, since model parameters cannot always be converted back into corresponding physical parameters, and thus the influence of each physical parameters on the residuals could not be easily determined. 

Despite highly developed computer technologies and numerical methods, the application of new-generation rate-based models requires a high computational effort, which is often related to numerical difficulties. This is a reason for the relatively limited application of modeling methods described above to industrial problems. Therefore, a further study in this field - as well as in the area of model parameter estimation - is required in order to bridge a gap and to provide process engineers with reliable, consistent, robust and user-friendly simulation tools for reactive absorption operations. 

Robust parameter estimates are then obtained following Equation 6.11 to Equation 6.13 as... 

Integral PFR data cannot be used directly in this way, since one has a differential equation that describes the conversion profile along the catalyst bed (eq 2). For simple cases this can be integrated analytically, yielding an implicit expression in the observed variable (eq S). Sometimes the independent variable W/Ff is now used as observed variable and its SSR minimized [9], but this interchange of dependent and independent variable destroys the error properties and the parameter error limits are not correct. Often the parameter estimates correspond well [9] and can be used as starting guesses for more robust minimization to determine the real error bounds. 

Consider now robustness. If the estimators A are computed from independent response variables then, as noted in Section 1, the estimators have equal variances and are usually at least approximately normal. Thus the usual assumptions, that estimators are normally distributed with equal variances, are approximately valid and we say that there is inherent robustness to these assumptions. However, the notion of robust methods of analysis for orthogonal saturated designs refers to something more. When making inferences about any effect A, all of the other effects At (k i) are regarded as nuisance parameters and robust means that the inference procedures work well, even when several of the effects ft are large in absolute value. Lenth s method is robust because the pseudo standard error is based on the median absolute estimate and hence is not affected by a few large absolute effect estimates. The method would still be robust even if one used the initial estimate 6 of op, rather than the adaptive estimator 6L, for the same reason. 

Basu, A., Paliwal, K. K., 1989. Robust M-Estimates and Generalized M-estimates for Autoregressive Parameter Estimation. TENCON 89, 4th IEEE region 10 Int. conf., Bombay. Biegler, L. T., Grossmann, I. E., 2004. Retrospective on optimization. Comput. Chem. Eng., 28, 1169. 

State estimation has been proposed as a way to improve our ability to predict hydrate formation in subsea pipelines. PF and MHE, state-of-the-art state estimation methods, have been reviewed and tested with a simple example case study with satisfactory results. Strategies based on both MHE and PF are being tested at present. The ultimate aim is to develop an efficient observer by relying on the robustness and the optimisation-based approach of MHE to provide initial guesses on the one hand, and the speed of PF on the other hand to solve the state and parameter estimation problem. 

Some commercial RTO systems either back-calculate model parameters from available process measurements or use some form of least-squares parameter estimation.P The back-calculation approach has been shown to be less robust to errors in the data and model than the least-squares approach and, as a result, is less desirable in RTO applications. The least-squares model updating problem can be solved by a number of nonlinear programming solvers, and the estimated parameters used for model-based optimization. 

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