Estimation errors deterministic

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... 

By again neglecting statistical errors due to and the deterministic errors at time t + 2At could be estimated by using... 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

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. 

The assumptions that need to be met in order to obtain parameter estimates with smaller variances are that both W(k) and v(k) be independrat white-noise sequences with a zero mean. When the system vector is deterministic (not randomly affected by system error) and invariant with respect to independent variable (e.g. potential, wavelength, time, etc.), the simplification W(k)=0 and F=I, where I is the identity matrix, are possible in Eqn.(l). 

Prior to model estimation the question that it will be used to answer and the specific manner in which it will be used should be explicitly stated. Using a model to answer a question is the act of simulation. There are two types of simulation deterministic and stochastic. In a deterministic simulation, the statistical model is ignored and no error is introduced into the model—the results are error-free. For example, given data from single-dose administration of a drug it may be of interest to predict the typical concentration-time profile at steady-state under a repeated dose administration regimen. A deterministic simulation would be useful in this case. 

Steric similarities can be estimated using procedures based on the matching of atoms or of superimposed volumes. For example, Dean and co-workers have discussed the matching of pairs of atoms, one from a target structure and one from a database structure, so as to minimize the sum of the squared distance errors both deterministic (106) and nondeterministic (107-110) algorithms have been reported, with run times of a few CPU seconds for a pair of structures. Meyer and Richards (111) measure the similarity of a pair of molecules by the extent to which their volumes overlap, and describe an efficient algorithm to obtain good, but not necessarily optimal, overlaps for pairs of structurally related molecules. 

The improved performance of the multiscale approach is due to the ability of orthonormal wavelets to approximately decorrelate most stochastic processes, and compress deterministic features in a small number of large wavelet coefficients. These properties permit representation of the prior probability distribution of the variables at each scale as a Gaussian or exponential function for stochastic and deterministic signals, respectively. Consequently, computationally expensive non-parametric methods need not be used for estimating the probability distribution of the coefficients at each scale. If the probability distribution of the contaminating errors and the prior can be represented as a Gaussian, the multiscale Bayesian approach provides... 

Simulation techniques vary based on the degree of randomness desired. If one wants the typical subject only, a deterministic solution is achieved by solving the structural model for a given input at defined sampling times. Error (intra- and interindividual) partitions are not necessary. Similarly, the extremes of a certain parameter can be simulated by fixing the estimate of that parameter to its upper or lower bound and proceeding with a deterministic solution. More informed simulations will, of course, specify the degree of stochasticity required to answer the questions posed. For... 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

This problem was solved using the Point Estimate technique with the following set of values (after (Kelly 1993)) m=100 t, Wb = 66 t, cos Stt rad/s, cob = 7T rad/s, Vs = 0.02, Vb = 0.1. For the peak ground acceleration Ag a mean value equal to 0.25g was employed. The results for the standard deviation of the base and structural displacements are 0.0676 m and = 0.0135 m. The solution required only six deterministic calculations of the displacement responses. Finally, the standard deviations of these displacements without regard to parameter uncertainties and using their mean values were found to be 0.0614 m and 0.0123 m, respectively. For both cases the error in omitting the uncertainties in model parameters is 10.2%. 

Structural reliability methods have been applied to estimate runway-overrun probabilities and were shown to be suitable for this purpose. In particular, the calculated sensitivities and FORM importance measures support the interpretation and further development of the model. With the help of the FORM importance measures, one can simplify the model by modeling the quantities of minor importance deterministically. We have found that the parameters Temperature , Pressure , Reverser deployment , Spoiler deployment and End breaking may be modeled deterministically without inducing a large error. This does, however, not imply that the influence of these quantities is small. It can only be concluded that the uncertainty associated with the quantity is not significant. The parameter sensitivities were in this paper calculated based on the samples obtained with subset simulation. These parameter sensitivities describe the effect of a change of the mean or standard deviation on the probability of failure. We presented the parameter sensitivities in the form of elasticities, which are typically easier to interpret. From the results it can be seen that the variables, which are of main importance according to the FORM importance measures, are... 

Estimation procedures for show one special feature. Compared to estimation problems in o er areas of engineering we know much less what the accurate physical relation is. Often we cannot use straightforward deterministic modeling, as our knowl dge of is insufficient. We also have experimental errors. On the other hand, we know much more about than we do in the areas for which purely statistical model-builcting procedures have been developed. What we really need is a new approach that incorporates into statistical estimating procedures the considerable knowledge of the thermodynamic constraints... 

In this paper the estimation problem is solved from a point of view which is essentially different from the stochastic approach of the conventional Kalman filter. The time-variant state estimation problem is re-phrased into a time-invariant parameter estimation problem, and least-squaxes techniques, which is derived under deterministic framwork, are then used. The advantage of using least-squaxes approach is that it does not require a pHori knowledge of the noise statistics, the initial values of the estimated state, and its corresponding error covariance. Close relations are found between the Kalman filter and the least-squaxes Alter. Finally, a numerical example is provided to illustrate the feasibility of the proposed method. 

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