Full stochastic model

Malaysia). Investment in Malaysia manages to reduce risk over that in Indonesia due to the lower volatility of natural gas prices in Malaysia. Figure 12.13 compares the risk curves and shows values of VaR and OV. Table 12.3 compares the risk indicators more closely. The VaR reduces from 18.1% but the OV (UP) reduces 18.9% and the risk area ratio (RAR) is equal to 2.2. This means that the loss in opportunity is more than twice the gain in risk reduction. The application of the decomposition procedure rendered similar solutions to those obtained using the full stochastic model. The use of regret analysis in this case produced similar but slightly less profitable answers. 

Olsson, G. Hansson, 0. "Stochastic modeling and computer control of a full scale wastewater treatment plant" Proc. Symp. on Systems and Models in Air and Water Pollution, The Institute of Measurement and Control, London, Sep. 1976. 

The possible states in each compartment are n0i + n02 + 1. Therefore R is a 256-dimensional matrix. The initial condition for the master equation is Pio,5 (0) = 1. Figures 9.25 and 9.26 show the associated probabilities for each state as functions of time for the central and peripheral compartments, respectively. In these figures the disk area is proportional to the associated probability, the full markers are the expected values, and the solid lines the solution of the deterministic model. As already mentioned, we note that the expectation of the stochastic model follows the time profile of the deterministic system. 

Owing to this reduction to a Markov process the model can again be treated in full detail. A stochastic process that can be made Markovian by means of one additional variable is said to be Markovian of the second degree , and if more variables are needed it is Markovian of some higher degree. 

In order to analyze for process problems or design stochastic controllers one usually needs to run full scale plant tests and develop process dynamic models from the resulting data. This section discusses several aspects related to the modelling of polymer reactors for these purposes. 

In order to understand how the algorithm actually works and to construct an explicit expression for the error it is not convenient to work with the metadynamics equations (12) in their full generality. Instead, we notice that the finite temperature dynamics of the collective variables satisfies, under rather general conditions, a stochastic differential equation [54,55]. Furthermore, in real systems the quantitative behavior of metadynamics is perfectly reproduced by the Langevin equation in its strong friction limit [56]. This is due to the fact that all the relaxation times are usually much smaller than the typical diffusion time in the CV space. Hence, we model the CVs evolution with a Langevin t3rpe dynamics ... 

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