Mathematical stochastic uncertainty

Kousa et al. [20] classified exposure models as statistical, mathematical and mathematical-stochastic models. Statistical models are based on the historical data and capture the past statistical trend of pollutants [21]. The mathematical modelling, also called deterministic modelling, involves application of emission inventories, combined with air quality and population activity modelling. The stochastic approach attempts to include a treatment of the inherent uncertainties of the model [22],... 

From a mathematical perspective (see Equation 4.1), CA simply represents the weighted harmonic mean of the individual ECx values, with the weights just being the fractions / , of the components in the mixture. This has important consequences for the statistical uncertainty of the CA-predicted joint toxicity. As the statistical uncertainty of the CA-predicted ECx is a result of averaging the uncertainties of the single substance ECx values, the stochastic uncertainty of the CA prediction is always smaller than the highest uncertainty found in all individual ECx values. Perhaps contrary to intuition, the consideration of mixtures actually reduces the overall stochastic uncertainty, which is a result of the increased number of input data. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

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... 

Fora recent survey of reactive and stochastic chemical batch scheduling approaches, the reader is referred to Floudas and Lin [2], For a survey of the different types of probabilistic mathematical models that explicitly take uncertainties into account, see Sahinidis [12]. For detailed information about stochastic programming and its applications, the reader is referred to the books of Birge and Louveaux [9], Ruszczynski and Shapiro [10], or Wallace and Ziemba [26]. 

The theory behind every measurement method can be generalised by Eq (1) [1]. Some quantity (or quantities, measurands) is measured, which has a specific relationship to the sought quantity. The measurand can be regarded to be a stochastic variable associated with an uncertainty, which implies that the sought quantity is also a random variable. The mathematical relationship depends on the physical model, that is, the model of the physical phenomenon of interest, for example temperature, pressure, and volume flow. The physical model always includes limitations, which implies that the measurement method has restrictions that is, it will only function in a certain measuring range and according to the assumption of the model. 

Uncertainty25 Uncertainty can be incorporated into strategic analysis in different ways (cf. Kallrath and Maindl 2006, p. 39). From a mathematical point of view, models are either deterministic (D) or stochastic (S). The latter is here defined as explicitly considering a probabilistic repre-... 

For the most frequently used low-dose models, the multi-stage and one-hit, there is an inherent mathematical uncertainty in the extrapolation from high to low doses that arises from the limited number of data points and the limited number of animals tested at each dose (Crump et al., 1976). The statistical term confidence limits is used to describe the degree of confidence that the estimated response from a particular dose is not likely to differ by more than a specified amount from the response that would be predicted by the model if much more data were available. EPA and other agencies generally use the 95 percent upper confidence limit (UCL) of the dose-response data to estimate stochastic responses at low doses. 

There are various ways to classify mathematical models (5). First, according to the nature of the process, they can be classified as deterministic or stochastic. The former refers to a process in which each variable or parameter acquires a certain specific value or sets of values according to the operating conditions. In the latter, an element of uncertainty enters we cannot specify a certain value to a variable, but only a most probable one. Transport-based models are deterministic residence time distribution models in well-stirred tanks are stochastic. 

Problems of uncertainty and inaccuracy can be addressed by using statistical and stochastic methods that have been described before [29,30]. The fuzzy logic approach provides a mathematical framework for representation and calculation of inaccurate data in AI methods [31,32]. Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the values between exactly true and exactly false. 

Many systems of interest to industrial engineers involve randomness or unpredictability for example, in the arrival of jobs requiring processing, or in machine breakdowns. In attempting to understand these systems for the purpose of design or control, a mathematical model is needed. Often, randomness and uncertainty must be captured explicitly in the model in order to represent the system reasonably faithfully. Such a model is called a stochastic model. The value of these models is that they enable us to predict the performance of a new system or the effect of a change in an existing system. 

The aforementioned methods can be applied to evaluate the reliability of engineering systems subjected to stochastic input with a given mathematical model. On the other hand, if a parametric model of the underlying system is available and the probability density function of these parameters is obtained by Bayesian methods, the uncertain parameter vector can be augmented to include the model parameters and the uncertain input components. Then, robust reliability analysis can proceed for stochastic excitation with an uncertain mathematical model. This allows for more realistic reliability evaluation in practice so that the modeling error and other types of uncertainty of the mathematical model can be taken into account. 

Contrary to t3q>ically deterministic mathematical models available in the literamre, real-world apphcations are usually surrounded with uncertainty. The two main approaches for dealing with uncertainty are stochastic programming and robust optimization. For developing stochastic programming models, probability distributions of uncertain parameters should be known in advance. However, in many practical situations, there is no information or enough information for obtaining probability distribution of uncertain parameters. Robust optimization models are viable answers to these situations via providing solutions that are always... 

Vahdani B, Naderi-Beni M (2014) A mathematical programming model for recycling network design under uncertainty an interval-stochastic robust optimization model. Int J Adv Manuf Technol 73 1057-1071... 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

The finite element method (FEM) has become the dominant computational method in structural engineering. In general, the input parameters in the standard FEM assume deterministic values. In earthquake engineering, at least the excitation is often random. However, considerable uncertainties might be involved not only in the excitation of a structure but also in its material and geometric properties. A rational treatment of these uncertainties needs a mathematical concept similar to that underlying the standard FEM. Thus, FEM as a numerical method for solving boundary value problems has to be extended to stochastic boundary value problems. The extension of the FEM to stochastic boundary value problems is called stochastic finite element method (SEEM). 

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