Stochastic parameter

Deterministic vs. stochastic an optimization problem can be based on deterministic parameters assuming certain input data or reflect uncertainty including random variables in the model in value chain management deterministic parameters are the basic assumptions extended models also model specifically uncertain market parameters such as demand and prices as stochastic parameters based on historic distributions in chemical commodities, this approach has some limitations since prices and demand are not normally distributed but depend on many factors such as crude oil prices (also later fig. 37). 

The pitting potential is a stochastic parameter. The potential de-CTeases as the area inaeases, because the chance of there being a weak spot for pit initiation is greater on a large surface than on a small surface. 

Compensating slack variables accounting for shortfall and/or surplus in production are introduced in the stochastic constraints with the following results (i) inequality constraints are replaced with equality constraints (ii) numerical feasibility of the stochastic constraints can be ensured for all events and (iii) penalties for feasibility violations can be added to the objective function. Since a probability can be assigned to each realization of the stochastic parameter vector (i.e., to each scenario), the probability of feasible operation can be measured. In this... 

Snodgrass and Kitanidis [61] also used a probabilistic approach combining Bayesian theory and geostatistical techniques. In their method, the source function to be estimated is discretized into components that are assigned a known stochastic structure with unknown stochastic parameters. The method incor-... 

Total exposure is defined from the combination of contact, exposure and uptake scenarios for each route of entry, and dose measures are calculated. These dose measures contain concentration estimates and short- and long-term average doses in terms of milligram of chemical per day per kilogram of body weight. The program allows for stochastic parameters and each parameter can attain a normal, log-normal or uniform distribution, or an empirical distribution defined by data. Exposure and dose distributions reflect stochastic parameters and these distributions can be depicted and percentiles can be quantified. 

The program provides sensitivity analyses for each stochastic parameter, in which mean exposures or doses as a function of the value of a selected stochastic parameter are depicted and analyzed. 

Lee, J. H., and Cooley, B. L., Optimal feedback control strategies for state-space systems with stochastic parameters, IEEE Trans. AC, in press (1998). 

The third group of models succeeds in accounting for some thermodynamic properties in the stable region of water (0 C starting point for describing the short-time region by properly defining a continuous stochastic variable that could be made to depend suitably on the stochastic parameters of Eq. (4.1). 

We consider a periodic change of the stochastic parameter K(t) as an external transformation to know how good the adiabatic invariant is. Since Eo(t) is the value of the energy at time t determined by the conservation of p(Eo(f)>f) = ll( o(0), 0). the deviation and the variance of the adiabatic invariance are simply related to those of the Hamiltonian as... 

Select using stochastic parameter sensitivities 8 D-optimal design... 

Specification of the source of data for model, material and geometrical parameters, as well as stochastic parameters. Specification of the results of the spatial assignment. 

Hierarchical Bayesian models are commonly used in many biological apphcations because they incorporate both physical and statistical models with uncertainty. These models are used in pathway analysis because of their ability to manage multiple data sources, uncertain, physical models with stochastic parameters, and expert opinion. A summary of these models is given in WiMe (2004). In brief, all unknowns are treated as if they are random and evaluated probabilistically as follows ... 

The two-site model was applied to obtain stochastic parameters for MgCl2/intemal donor/TiCU solid catalyst component used in combination with A1(C2H5)3 and external donor for propylene polymerization. The type and the amount of internal donor were varied. With respect to the fraction for asymmetric site, the two-site model enabled us to conclude that new kinds of active centers are generated in specific cases where external donor is believed to be replacing weaker internal donor during polymerization. 

In the area of isotactic polypropylene, there have been several publications (4 6) where the two-site model stochastic parameters are successfully utilized to explain the effect of external donors that are used in combination with solid catalyst con onent. However, few studies employing the two-site model have been made of the effect of internal donor on catalyst components. 

The theory and the parameters of the two-site model have been presented elsewhere in detail (3), but is briefly reviewed here for convenience. The model is composed of three parameters these are stochastic parameters describing the role of the first and the second sites, and the fraction of the polymers obtained from the first site. More precisely, the first parameter a is the probability of the selection of d (or 1) monad in an asymmetric site the second parameter a is that of m diad in a symmetric site and the last parameter oo is the fraction of the polymers obtained from the asymmetric site. 

The design of plants against earthquakes requires, as a mle, the determination of response spectra of the object under consideration (building, vessel etc.) to the excitation caused by the movements from the earthquake. A detailed treatment is beyond the present scope. Instead a simple assessment of the loads on the support columns of a spherical tank is presented based on [24]. However, it goes beyond the treatment given there by explicitly accounting for stochastic parameters. 

It is evident that the results strongly depend on the stochastic parameter weather . 

The stochastic parameter humidity of the air , which cannot be predicted for the moment of occurrence of the fireball, has a substantial influence on the conditional probability of death, as can also be seen from Fig. 10.30. 

ABSTRACT In this paper we consider nncertainties in the distribution of random variables due to small-sample observations. Based on the maximum entropy distribution we assume the first four stochastic moments of a random variable as uncertain stochastic parameters. Their uncertainty is estimated by the bootstrap approach from the initial sample set and later considered in estimating the variation of probabilistic measures. 

Figure 5. Maximum entropy distributions of the modified soil parameter samples and obtained distributions of the uncertain stochastic parameters from the bootstrap approach. 

In our final analysis we consider the variation in the stochastic parameters. Based on a first order... 

The required derivatives canbe obtained forthe FORM approach very efficiently as reported in Most and Knabe (2009). Based on this Taylor series approximation the failure probability can be calculated quite accurate for each specific sample of the stochastic parameters close to the mean value vector po- This calculation is performed here for all 10000 bootstrap samples. The resulting histogram of the reliability index is shown in Figure 6 using log-normally distributed

maximum entropy distributions. 

Based on the assumption of almost normally distributed stochastic parameters p the variance of the failure probability can be directly estimated from the covariance matrix of these parameters... 

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