Stochastic simulation algorithm

Combined strategic and operational model 2 Genetic algorithm and simulation model to analyze stochastic effects... 

We can, however, analyze these problems within the framework of the stochastic formulation by looking for an exact solution, or by using the probability generating functions, or the stochastic simulation algorithm. 

For linear systems, the differential equation for the jth cumulant function is linear and it involves terms up to the jth cumulant. The same procedure will be followed subsequently with other models to obtain analogous differential equations, which will be solved numerically if analytical solutions are not tractable. Historically, numerical methods were used to construct solutions to the master equations, but these solutions have pitfalls that include the need to approximate higher-order moments as a product of lower moments, and convergence issues [383]. What was needed was a general method that would solve this sort of problem, and that came with the stochastic simulation algorithm. 

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

For most macroscopic dynamic systems, the neglect of correlations and fluctuations is a legitimate approximation [383]. For these cases the deterministic and stochastic approaches are essentially equivalent, and one is free to use whichever approach turns out to be more convenient or efficient. If an analytical solution is required, then the deterministic approach will always be much easier than the stochastic approach. For systems that are driven to conditions of instability, correlations and fluctuations will give rise to transitions between nonequihbrium steady states and the usual deterministic approach is incapable of accurately describing the time behavior. On the other hand, the stochastic simulation algorithm is directly applicable to these studies. 

As previously, initial conditions for the compartmental model and the enzymatic reaction were set to tiq = [100 50], and so = 100, eo = 50, and cq = 0, respectively. Figures 9.31 and 9.32 show the deterministic prediction, a typical run, and the average and confidence corridor for 100 runs from the stochastic simulation algorithm for the compartmental system and the enzyme reaction, respectively. Figures 9.33 and 9.34 show the coefficient of variation for the number of particles in compartment 1 and for the substrate particles, respectively. 

By dealing with the evolution of constraints (i.e., Ramachandran basins) rather than the backbone torsional coordinates themselves, the dynamics are judiciously simplified [31]. The algorithm consists of a stochastic simulation of the coarsely resolved dynamics, simplified to the level of time-evolving Ramachandran basin assignments. An operational premise is that steric restrictions imposed by the side chains on the backbone may be subsumed into the basin-hopping dynamics. The side chain constraints define regions in the Ramachandran map that can be explored in order to obtain an optimized pattern of nonbonded interactions. 

Kastner J., Solomon J. and Fraser S. (2002). Modeling a hox gene network in silico using a stochastic simulation algorithm. Developmental Biology. 246, pp 122-131. 

In a series of papers, Gillespie makes an elegant argument for the use of stochastic simulation in chemical kinetic modeling [7, 8] and provides the following simulation algorithm [7, p.2345]. 

If not is the total probability for reaction, is the probability that a reaction has not occurred during time interval r, which leads directly to Equation 4.94 for choosing the time of the next reaction. We wlU derive this fact in Chapter 8 when we develop the residence-time distribution for a CSTR. Shah, Ramkrishna and Borwanker call this time the interval of quiescence, and us.e it to develop a stochastic. simulation algorithm for particulate. system.dyii.amic.s rather than. c.bem.ical. kinetics.114]. 

To arrive at a true optimal subset of variables (wavelengths) for a given data set, consideration of all possible combinations should in principle be used but it is computationally prohibitive. Since each variable can either appear, or not, in the equation and since this is true with every variable, there are 2"-possible equations (subsets) altogether. For spectral data containing 500 variables, this means 2 possibilities. For this type of problems, i.e. for search of an optimal solution out of the millions possible, the stochastic search heuristics, such as Genetic Algorithms or Simulated Annealing, are the most powerful tools [14,15]. 

Fast measurements also require fast and robust parameter extraction. Biischel et al. (2011) compared the usability of stochastic algorithms, evolution, simulated annealing, and particle filter for robust parameter extraction from impedance spectra and concluded that particle filter delivered the most reliable results. 

Abstract Results of stochastic simulations of micellization kinetics are presented. The algorithm used was derived from the general Monte Carlo method introduced by D.T. Gillespie (1976, J. Phys. Chem. 22 403-434) and applied to micelle formation according to a mechanism that allows association and dissociation among n-mers of whatever aggregation number. With a careful choice of thermodynamic and... 

Asmussen, S. and Glynn, P. (2007) Stochastic Simulation Algorithms and Analysis, Springer, Berlin. 

Fuz Stochastic simulation, Neural network, genetic algorithm... 

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