Parameter estimation, Monte Carlo

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

If the Worst Case analysis determines that not all circuits will pass a specified performance parameter, the Monte Carlo analysis may be used to estimate what percentage of the circuits will pass. 

Global SA is based on simulations where results are conditioned on uncertainty distributions across all parameters. Uncertainty is quantitatively defined for all parameters (models) through the use of appropriate distribution models (28) or using distributions from prior reports or models. The latter method, which does not require an assumed model parameter probability distribution function, may include use of fuzzy set theory (29) or the use of bootstrapped estimates from previous estimations. Monte Carlo methods are required to simulate from the uncertainty distributions at the intertrial level. This usually requires one set of simulations with a large number of replicates. The number of trial replicates is discussed in Chapter 33, where this number may need to be further increased for the global SA. 

Characterizing the overall uncertainties associated with the PBPK model estimates is also an important component of the PBPK model evaluation and application. This includes characterizing the uncertainties in model outputs resulting from the uncertainty in the PBPK model parameters. Traditionally, Monte Carlo has been employed for performing uncertainty analysis of PBPK models (39, 40). Some of the recent techniques that have been applied for the uncertainty analysis of PBPK models include the stochastic response surface method (SRSM) (38, 41) and the high-dimensional model reduction (HDMR) technique (42). 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

We have presented applications of a parameter estimation technique based on Monte Carlo simulation to problems in polymer science involving sequence distribution data. In comparison to approaches involving analytic functions, Monte Carlo simulation often leads to a simpler solution of a model particularly when the process being modelled involves a prominent stochastic coit onent. 

MD simulations in expHcit solvents are stiU beyond the scope of the current computational power for screening of a large number of molecules. However, mining powerful quantum chemical parameters to predict log P via this approach remains a challenging task. QikProp [42] is based on a study [3] which used Monte Carlo simulations to calculate 11 parameters, including solute-solvent energies, solute dipole moment, number of solute-solvent interactions at different cutoff values, number of H-bond donors and acceptors (HBDN and HBAQ and some of their variations. These parameters made it possible to estimate a number of free energies of solvation of chemicals in hexadecane, octanol, water as well as octanol-water distribution coefficients. The equation calculated for the octanol-water coefficient is ... 

The third method used to interpret the level of risk associated with chlorpy-rifos use is Monte Carlo simulation. This method provides a range of exposure estimates for the evaluation of the uncertainty in a risk estimate based on ranges of input variables. The first step in performing a Monte Carlo simulation is determination of a model to describe the dose. This model describes the relationship between the input parameters and dose, and a specific model is presented here for one group of workers. 

The uncertainty of the fitted values of these two parameters has been estimated objectively by means of a Monte-Carlo simulation model. The data points on each curve in Figure 5 are the mean of 100 calculated points and each point is the "best-fit" of the parameter to a simulated measurement in a simulated indoor environment in which allowance is made for fluctuations of the parameters. 

As mentioned above, for the simulation in dimethylformamide (DMF) of the same reaction [53], the parameters for the substrate were not changed from the parametrization in water. For DMF the parameters were adopted from the OPLS parametrization of the pure liquid. The transferability was tested in part by performing a Monte Carlo simulation for CT plus 128 DMF molecules and evaluating the heat of solution for the chloride ion. The obtained value compares favorably with the experimental estimate. It is important to remark here that when potentials are used to simulate different solutions to the ones used in the parametrization process, they no longer are "effective" potentials. This fact becomes more evident in the simulation of solutions of small ions with localized charge that polarizes the neighboring solvent molecules. In this case it is convenient to consider the n-body corrections. 

From this dynamic interpretation of the Monte Carlo averaging we can obtain a formal estimate of the number of steps Mq that have to be omitted at the beginning of the averaging. Usually, the order parameter ijf is the slowest relaxing quantity and then... 

Combustion of aluminum particle as fuel, and oxygen, air, or steam as oxidant provides an attractive propulsion strategy. In addition to hydrocarbon fuel combustion, research is focussed on determining the particle size and distribution and other relevant parameters for effectively combusting aluminum/oxygen and aluminum/steam in a laboratory-scale atmospheric dump combustor by John Foote at Engineering Research and Consulting, Inc. (Chapter 8). A Monte-Carlo numerical scheme was utilized to estimate the radiant heat loss rates from the combustion products, based on the measured radiation intensities and combustion temperatures. These results provide some of the basic information needed for realistic aluminum combustor development for underwater propulsion. 

In this method, each assessment factor is considered uncertain and characterized as a random variable with a lognormal distribution with a GM and a GSD. Propagation of the uncertainty can then be evaluated using Monte Carlo simulation (a repeated random sampling from the distribution of values for each of the parameters in a calculation to derive a distribution of estimates in the population), yielding a distribution of the overall assessment factor. This method requires characterization of the distribution of each assessment factor and of possible correlations between them. As a first approach, it can be assumed that all factors are independent, which in fact is not correct. 

Eurthermore, uncertainties in the exposure assessment should also be taken into account. However, no generally, internationally accepted principles for addressing these uncertainties have been developed. For predicted exposure estimates, an uncertainty analysis involving the determination of the uncertainty in the model output value, based on the collective uncertainty of the model input parameters, can be performed. The usual approach for assessing this uncertainty is the Monte Carlo simulation. This method starts with an analysis of the probability distribution of each of the variables in the uncertainty analysis. In the simulation, one random value from each distribution curve is drawn to produce an output value. This process is repeated many times to produce a complete distribution curve for the output parameter. 

Monte Carlo method, 210, 21 propagation, 210, 28] Gauss-Newton method, 210, 11 Marquardt method, 210, 16 Nelder-Mead simplex method, 210, 18 performance methods, 210, 9 sample analysis, 210, 29 steepest descent method, 210, 15) simultaneous [free energy of site-specific DNA-protein interactions, 210, 471 for model testing, 210, 463 for parameter estimation, 210, 463 separate analysis of individual experiments, 210, 475 for testing linear extrapolation model for protein unfolding, 210, 465. 

Selected entries from Methods in Enzymology [vol, page(s)] Generation, 240, 122-123 confidence limits, 240, 129-130 discrete variance profile, 240, 124-126, 128-129, 131-133, 146, 149 error response, 240, 125-126, 149-150 Monte Carlo validation, 240, 139, 141, 146, 148-149 parameter estimation, 240, 126-129 radioimmunoassay, 240, 122-123, 125-127, 131-139 standard errors of mean, 240, 135 unknown sample evaluation, 240, 130-131 zero concentration response, 240, 138, 150. 

In classical statistics, the most important type of criterion for judging estimators is a high probability that a parameter estimate will be close to the actual value of the parameter estimated. To implement the classical approach, it is necessary to quantify the closeness of an estimate to a parameter. One may rely on indices of absolute, relative, or squared error. Mean squared error (MSB) has often been used by statisticians, perhaps usually because of mathematical convenience. However, if estimators are evaluated using Monte Carlo simulation it is easy to use whatever criterion seems most reasonable in a given situation. 

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