Statistical/probabilistic models estimation methods

Stochastic (Probabilistic) Models. One of the most significant advances in exposure estimation in the past 15 to 20 years has been the application of probabilistic statistical methods to many types of data analyses (Duan and Mage 1997 Finley and Paustenbach 1994 Morgan and Henrion 1990 US ERA 1995, 1997, 2000a). Stochastic or probabilistic techniques can help quantify variability and uncertainty in model inputs and outputs, can be used to better characterize the possible range of exposures for a particular scenario when measured data are minimal, and can be employed to better understand the uncertainty inherent in estimates developed from many different types of sources, whether quantitative or qualitative. 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

Bayesian statistics are applicable to analyzing uncertainty in all phases of a risk assessment. Bayesian or probabilistic induction provides a quantitative way to estimate the plausibility of a proposed causality model (Howson and Urbach 1989), including the causal (conceptual) models central to chemical risk assessment (Newman and Evans 2002). Bayesian inductive methods quantify the plausibility of a conceptual model based on existing data and can accommodate a process of data augmentation (or pooling) until sufficient belief (or disbelief) has been accumulated about the proposed cause-effect model. Once a plausible conceptual model is defined, Bayesian methods can quantify uncertainties in parameter estimation or model predictions (predictive inferences). Relevant methods can be found in numerous textbooks, e.g., Carlin and Louis (2000) and Gelman et al. (1997). 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

Beck JL, Katafygiotis LS (1998) Updating models and their uncertainties. I Bayesian statistical framework. J Eng Mech (ASCE) 124(4) 455 61 Beck JL, Yuen KV (2004) Model selection using response measurements Bayesian probabilistic approach. J Eng Mech (ASCE) 130(2) 192-203 Ching J, Chen YC (2007) Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection and model averaging. J Eng Mech (ASCE) 133(7) 816-832 Durovic ZM, Kovacevic BD (1999) Robust estimation with unknown noise statistics. IEEE Trans Automat Control 44(6) 1292-1296... 

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