Parameter estimation identifiability

Model The level of sophistication needs to be identified. Prehm-inary usage of the model should identify the uniqueness of parameter estimates and conclusions to be drawn. 

Identifying the minimum number of specific measurements containing the most information such that the model parameters are uniquely estimated requires that the model and parameter estimates be known in advance. Repeated unit tests and model building exercises will ultimately lead to the appropriate measurements. However, for the first unit test in absence of a model, the identification of the minimum number of measurements is not possible. 

In what follows we briefly review some of the previous attempts to analyze the available spectra of plutonium (6). In addition, we estimate energy level parameters that identify at least the gross features characteristic of the spectra of plutonium in various valence states in the lower energy range where in most cases, several isolated absorption bands can be discerned. The method used was based on our interpretation of trivalent actinide and lanthanide spectra, and the generalized model referred to earlier in the discussion of free-ion spectra. 

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. 

Model parameters are usually determined from expterimental data. In doing this, sensitivity analysis is valuable in identifying the experimental conditions that are best for the estimation of a particular model parameter. In advanced software packages for parameter estimation, such as SIMUSOLV, sensitivity analysis is provided. The resulting iterative procedure for determining model parameter values is shown in Fig. 2.39. 

The second case study corresponds to an existing pyrolysis reactor also located at the Orica Botany Site in Sydney, Australia. This example demonstrates the usefulness of simplified mass and energy balances in data reconciliation. Both linear and nonlinear reconciliation techniques are used, as well as the strategy for joint parameter estimation and data reconciliation. Furthermore, the use of sequential processing of information for identifying inconsistencies in the operation of the furnace is discussed. 

In the second example, that of an industrial pyrolysis reactor, simplified material and energy balances were used to analyze the performance of the process. In this example, linear and nonlinear reconciliation techniques were used. A strategy for joint parameter estimation and data reconciliation was implemented for the evaluation of the overall heat transfer coefficient. The usefulness of sequential processing of the information for identifying inconsistencies in the operation of the furnace was further demonstrated. 

Therefore, a flexible method to evaluate physical and chemical system parameters is still needed (2, 3). The model identification technique presented in this study allows flexibility in model formulation and inclusion of the available experimental measurements to identify the model. The parameter estimation scheme finds the optimal set of parameters by minimizing the sum of the differences between model predictions and experimental observations. Since some experimental data are more reliable than others, it is advantageous to assign higher weights to the dependable data. 

The parameter estimation approach is important in judging the reliability and accuracy of the model. If the confidence intervals for a set of estimated parameters are given and their magnitude is equal to that of the parameters, the reliability one would place in the model s prediction would be low. However, if the parameters are identified with high precision (i.e., small confidence intervals) one would tend to trust the model s predictions. The nonlinear optimization approach to parameter estimation allows the confidence interval for the estimated parameter to be approximated. It is thereby possible to evaluate if a parameter is identifiable from a particular set of measurements and with how much reliability. 

Using the concentration and obscuration measurements allow all of the kinetic parameters of interest to be identified. The simulated data is shown in Figures 1-2. The parameter estimation results corresponding to these measurements are given in Table 1. These results indicate that these measurements may provide enough process information to allow identification, even in the presence of measurement noise. This hypothesis is investigated experimentally in the next section. 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This "brute force" approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations... 

All reduced form parameters are estimable directly by using least squares, so the reduced form is identified in all cases. Now, 71 = n /n2. On is the residual variance in the euqation (yi - yiy2) = Sj, so G must be estimable (identified) if 71 is. Now, with a bit of manipulation, we find that con - ron = -On/A. Therefore, with On and Yi "known" (identified), the only remaining unknown is y2. which is therefore identified. With 71 and y2 in hand, P may be deduced from n2. With y2 and P in hand, ct22 is the residual variance in the equation (y2 - Px -YiVi) 2, which is dfrectly estimable, therefore, identified. 

Consider all alternatives for estimation in terms of reliability, accuracy, time required, and cost efficiency. Develop predictive models that allow for in silico screening, rather than necessitating prior synthesis of compound. Analyze literature for both pharmacokinetic and toxicokinetic parameter estimation, to identify models that already exist or ones that could be suitably modified for the parameter of interest... 

Process identification and parameter estimation has been applied in water quality and wastewater treatment systems (7-9). The overall oxygen transfer coefficient can be determined on-line. The hydraulic dispersion has been identified by manipulation of the influent flow rate or the return sludge flow rate (9). 

Usually, a mathematical model simulates a process behavior, in what can be termed a forward problem. The inverse problem is, given the experimental measurements of behavior, what is the structure A difficult problem, but an important one for the sciences. The inverse problem may be partitioned into the following stages hypothesis formulation, i.e., model specification, definition of the experiments, identifiability, parameter estimation, experiment, and analysis and model checking. Typically, from measured data, nonparametric indices are evaluated in order to reveal the basic features and mechanisms of the underlying processes. Then, based on this information, several structures are assayed for candidate parametric models. Nevertheless, in this book we look only into various aspects of the forward problem given the structure and the parameter values, how does the system behave ... 

Experimental arcs in the spectrum are not always ideal semicircles, and this complicates parameter estimation. Nevertheless, there are still basic rules for estimating the initial values [8, 9], The key is to identify the region of the spectrum in which one element dominates and then estimate the value of the element in this region. For example, the resistor s impedance dominates the spectrum at a low frequency, while the impedance of a capacitor approaches zero at a high frequency and infinity at a low frequency also, individual resistors can be recognized based on the horizontal regions in a Bode plot. 

I expect that SA of stochastic and multiscale models will be important in traditional tasks such as the identification of rate-determining steps and parameter estimation. I propose that SA will also be a key tool in controlling errors in information passing between scales. For example, within a multiscale framework, one could identify what features of a coarse-level model are affected from a finer scale model and need higher-level theory to improve accuracy of the overall multiscale simulation. Next a brief overview of SA for deterministic systems is given followed by recent work on SA of stochastic and multiscale systems. 

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. 

In our group we have used SA in lattice 2D and 3D KMC in order to identify key parameters for parameter estimation from experimental data (see corresponding section below). Finite difference approximations of NSC were employed (Raimondeau et al., 2003 Snyder and Ylachos, 2004). Drews et al. (2003a) motivated by extraction of parameters for Cu electrodeposition, obtained an expression for the sensitivity coefficient, analogous to Eq. (9), that minimizes the effect of noise on the NSC assuming that the variance of the stochastic correction is unaffected by the perturbation. 

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