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| Stochastic simulation is a simulation that operates with variables that can change with certain [[probability]]. [[Stochastic]] means that particular factors (values) are variable or [[random]].<ref name="sim-pro-ek">DLOUHÝ, M.; FÁBRY, J.; KUNCOVÁ, M.. Simulace pro ekonomy. Praha : VŠE, 2005.</ref>
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| With stochastic model we create a projection which is based on a set of [[random]] values. Outputs are recorded and the projection is repeated with a new set of [[random]] (variable) values. Previous steps are repeated until reasonable amout of data is gathered (thousandfold, millionfold, ..). In the end, the [[distribution]] of the outputs shows the most probable estimates as well as a frame of expectations (fringe values dividing those we still can expect from the ones we should not).<ref name="sim-pro-ek" />
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| ==Stochastic==
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| [[Stochastic]] means "pertaining to conjecture"; from Greek stokhastikos "able to guess, conjecturing"; from stokhazesthai "guess"; from stokhos "a guess, aim, target, mark". The sense of "randomly determined" was first recorded in 1934, from German Stochastik. <ref>stochastic. (n.d.). Online Etymology Dictionary. Retrieved January 23, 2014, from Dictionary.com website: http://dictionary.reference.com/browse/stochastic</ref>
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| ==Discrete-event simulation==
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| In order to determine the next event in a stochastic simulation, the rates of all possible changes to the state of the model are computed, and then ordered in an array. Next, the cumulative sum of the array is taken, and the final cell contains the number R, where R is the total event rate. This cumulative array is now a discrete cumulative distribution, and can be used to choose the next event by picking a random number z~U(0,R) and choosing the first event, such that z is less than the rate associated with that event.
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| ===Probability distributions===
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| A probability distribution is used to describe the potential outcome of a random variable.
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| Limits the outcomes where the variable can only take on discrete values. <ref name="ASM">Rachev, Svetlozar T. Stoyanov, Stoyan V. Fabozzi, Frank J., "Chapter 1 Concepts of Probability" in Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization : The Ideal Risk, Uncertainty, and Performance Measures, Hoboken, NJ, USA: Wiley, 2008</ref>
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| ====Bernoulli Distribution====
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| ''Main article [[Bernoulli distribution]]''
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| A random variable X is [[Bernoulli distribution|Bernoulli-distributed]] with parameter ''p'' if it has only two possible outcomes, usually encoded as 1 (success or defaul) or 0 (failure or survival).<ref name="ASM" />
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| Example: Toss of coin <ref name="bernoulli">Bernoulli Distribution, The University of Chicago - Department of Statistics, [online] available at http://galton.uchicago.edu/~eichler/stat22000/Handouts/l12.pdf</ref>
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| Define
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| X = 1 if head comes up and
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| X = 0 if tail comes up
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| Both realizations are equally likely:
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| P(X = 1) = P(X = 0) = 1/2
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| Of course, the two outcomes may not be equally likely (e.g. success of medical treatment).<ref name="bernoulli" />
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| ====Binomial Distribution====
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| ''Main article [[Binomial distribution]]''
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| A [[Binomial distribution|binomial distributed]] random variable Y with parameters ''n'' and ''p'' is obtained as the sum of ''n'' independent and identically [[Bernoulli distribution|Bernoulli-distributed]] [[random]] variables X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub><ref name="ASM" />
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| Example: A coin is tossed three times. Find the probability of getting exactly two heads.
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| This problem can be solved by looking that the sample space. There are three ways to get two heads.
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| HHH, '''HHT, HTH, THH''', TTH, THT, HTT, TTT
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| The answer is 3/8 (= 0.375). <ref>http://www.elderlab.yorku.ca/~aaron/Stats2022/BinomialDistribution.htm</ref>
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| ====Poisson Distribution====
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| ''Main article [[Poisson Distribution]]
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| The [[Poisson Distribution]] depends on only one parameter, λ, and can be interpreted as an approximation to the [[binomial distribution]] when the parameter ''p'' is a small number. A [[Poisson Distribution|poisson-distributed]] [[random]] variable is usually used to describe the [[random]] number of events occuring over a certain time interval.<ref name="ASM" />
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| Typical exemple problem: If 3% of the electric bulbs manufactured by a company are defective find the probability that in a sample of 100 bulbs exactly 5 bulbs are defective. ( Given e-0.25= 0.7788 ) <ref>http://ncalculators.com/math-worksheets/poisson-distribution-example.htm</ref>
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| ===Methods===
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| ====Direct and first reaction methods====
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| Published by [[Dan Gillespie]] in 1977, and is a linear search on the cumulative array. See [[Gillespie algorithm]].
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| Gillespie’s Stochastic Simulation Algorithm (SSA) is essentially an exact procedure for numerically simulating the time evolution of a well-stirred chemically reacting system by taking proper account of the randomness inherent in such a system. <ref name="ssa">Stephen Gilmore, An Introduction to Stochastic Simulation - Stochastic Simulation Algorithms, University of Edinburgh, [onlina] available at http://www.doc.ic.ac.uk/~jb/conferences/pasta2006/slides/stochastic-simulation-introduction.pdf</ref>
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| It is rigorously based on the same microphysical premise that underlies the chemical master equation and gives a more realistic representation of a system’s evolution than the deterministic reaction rate equation (RRE) represented mathematically by ODEs. <ref name="ssa" />
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| As with the chemical master equation, the SSA converges, in the limit of large numbers of reactants, to the same solution as the law of mass action.
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| ====Next reaction method====
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| Published 2000. This is an improvement over the first reaction method where the unused reaction times are reused. To make the sampling of reactions more efficient, an indexed priority queue is used to store the reaction times. On the other hand, to make the recomputation of propensities more efficient, a dependency graph is used. This dependency graph tells which reaction propensities to update after a particular reaction has fired.
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| ====Optimised and sorting direct methods====
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| Published 2004 and 2005. These methods sort the cumulative array to reduce the average search depth of the algorithm. The former runs a presimulation to estimate the firing frequency of reactions, whereas the latter sorts the cumulative array on-the-fly.
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| ====Logarithmic direct method====
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| Published in 2006. This is a binary search on the cumulative array, thus reducing the worst-case time complexity of reaction sampling to O(log M).
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| ====Partial-propensity methods====
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| Published in 2009, 2010, and 2011 (Ramaswamy 2009, 2010, 2011). Use factored-out, partial reaction propensities to reduce the computational cost to scale with the number of species in the network, rather than the (larger) number of reactions. Four variants exist:
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| * PDM, the partial-propensity direct method. Has a computational cost that scales linearly with the number of different species in the reaction network, independent of the coupling class of the network (Ramaswamy 2009).
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| * SPDM, the sorting partial-propensity direct method. Uses dynamic bubble sort to reduce the pre-factor of the computational cost in multi-scale reaction networks where the reaction rates span several orders of magnitude (Ramaswamy 2009).
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| * PSSA-CR, the partial-propensity SSA with composition-rejection sampling. Reduces the computational cost to constant time (i.e., independent of network size) for weakly coupled networks (Ramaswamy 2010) using composition-rejection sampling (Slepoy 2008).
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| * dPDM, the delay partial-propensity direct method. Extends PDM to reaction networks that incur time delays (Ramaswamy 2011) by providing a partial-propensity variant of the delay-SSA method (Bratsun 2005, Cai 2007).
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| The use of partial-propensity methods is limited to elementary chemical reactions, i.e., reactions with at most two different reactants. Every non-elementary chemical reaction can be equivalently decomposed into a set of elementary ones, at the expense of a linear (in the order of the reaction) increase in network size.
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| ==Continuous simulation ==
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| While in discrete [[state space]] it is cleary distinguished between particular states (values) in continuous space it is not possible due to certain continuity. The system usually change over time, variables of the model, then change continuously as well. Continuous simulation thereby simulates the system over time, given [[differential equation|differential equations]] determining the rates of change of state variables. <ref>Crespo-Márquez, A., R. R. Usano and R. D. Aznar, 1993, "Continuous and Discrete Simulation in a
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| Production Planning System. A Comparative Study"</ref>
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| Exemple of continuous system is ''the predator/prey model''<ref>Louis G. Birta, Gilbert Arbez (2007). Modelling and Simulation, p. 255. Springer.</ref> or cart-pole balancing <ref>http://anji.sourceforge.net/polebalance.htm</ref>
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| ===Probability distributions===
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| ====The Normal Distribution====
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| ''Main article [[Normal Distribution]]''
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| The [[random]] variable X is said to be [[normal distribution|normally distributed]] with parameters μ and σ, abbreviated by X ∈ N(μ, σ<sup>2</sup>), if the density of the [[random]] variable is given by the formula <ref name="ASM" />
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| <math>
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| f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }.
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| </math>
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| x ∈ R. <ref name="ASM" />
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| Many things actually are [[normal distribution|normally distributed]], or very close to it. For example, height and intelligence are approximately [[normal distribution|normally distributed]]; measurement errors also often have a [[normal distribution]].<ref>University of Notre Dame, Normal Distribution, [online] available at http://www3.nd.edu/~rwilliam/stats1/x21.pdf</ref>
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| ====Exponential Distribution====
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| ''Main article [[Exponential distribution]]''
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| [[Exponential distribution]] describes the time between events in a [[Poisson process]], i.e. a process in which events occur continuously and independently at a constant average rate.
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| The [[exponential distribution]] is popular, for example, in [[queuing theory]] when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect. <ref name="ASM" />
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| ====Student's t-distribution====
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| ''Main article [[Student's t-distribution]]''
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| [[Student's t-distribution]] are used in finance as probabilistic models of assets returns. The [[density function]] of the t-distribution is given by the following equation:<ref name="ASM" />
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| <math>f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\!</math>
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| where <math>\nu</math> is the number of ''[[degrees of freedom (statistics)|degrees of freedom]]'' and <math>\Gamma</math> is the [[gamma function]].
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| For large values of ''n'', the [[Student's t-distribution|t-distribution]] doesn't significantly differ from a standard [[normal distribution]]. Usually, for values ''n'' > 30, the [[Student's t-distribution|t-distribution]] is considered as equal to the standard [[normal distribution]].
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| ====Other distributions====
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| * [[Extreme Value Distribution]]
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| * Generalized [[Extreme Value Distribution]]
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| ===Methods===
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| ====τ leaping method====
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| Since the SSA method keeps track of each trasition, it would be impractical to implement for certain applications due to high time complexity. Gallespie proposed a solution in [[approximation procedure]], the [[Tau-leaping|tau-leaping method]] which decreases computational time with the minimal loss of accuracy. <ref>D.T. Gillespie, A General Method for Numerically Simulating the stochastic time evolution of coupled chemical reactions, The Journal of Computational Physics, 22 (1976), 403–434.</ref>
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| Instead of taking incremental steps in time, keeping track of X(t) at each time step as in the SSA method, the [[Tau-leaping|tau-leaping method]] leaps from one subinterval to the next, approximating how many transitions take place during a given subinterval. It is assumed that the value of the leap, τ, is small enough that there is no significant change in the value of the transition rates along the subinterval [t, t + τ]. This condition is known as the leap condition. The [[Tau-leaping|tau-leaping method]] thus has the advantage of simulating many transitions in one leap while not losing significant accuracy, resulting in a speed up in computational time.<ref>H.T. Banks, Anna Broido, Brandi Canter, Kaitlyn Gayvert,Shuhua Hu, Michele Joyner, Kathryn Link, Simulation Algorithms for Continuous Time Markov Chain Models, [online] available at http://www.ncsu.edu/crsc/reports/ftp/pdf/crsc-tr11-17.pdf</ref>
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| ==Combined simulation==
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| It is often possible to model one and the same system by use of completely different world views. [[Discrete event simulation]] of a problem as well as [[continuous event simulation]] of it (continuous simulation with the discrete events that disrupt the continuous flow) may lead eventually to the same answers. Sometimes however, the techniques can answer different questions about a system. If we necessarily need to answer all the questions, or if we don't know what purposes is the model going to be used for, it is convenient to apply combined continuous/discrete [[methodology]]. <ref name="cellier">Francois E. Cellier, Combined Continuous/Discrete Simulation Applications, Techniques, and Tools</ref>
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| ==Monte Carlo Simulation==
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| Monte Carlo is an estimation procedure. The main idea is that if it is necessary to know the average value of some [[random]] variable and its [[distribution]] can not be stated, and if it is possible to take samples from the [[distribution]], we can estimate it by taking the samples, independenty, and averaging them. If there are sufficiently enough samples, then the law of large numbers says the average must be close to the tru value. The cenrtal limit theorem says that the average has a [[Gaussian distribution]] around the true value.<ref name="mc">Cosma Rohilla Shalizi, Monte Carlo, and Other Kinds of Stochastic Simulation, [online] available at http://vserver1.cscs.lsa.umich.edu/~crshalizi/notebooks/monte-carlo.html </ref>
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| Simple example: We need to measure area of a shape with a complicated, irregular outline. The Monte Carlo approach is to draw a square around the shape and measure the square. Then we throw darts into the square, as uniformly as possible. The fraction of darts falling on the shape gives the ratio of the area of the shape to the area of the square. In fact, it is possible to cast almost any integral problem, or any averaging problem, into this form. It is necessary to have a good way to tell if you're inside the outline, and a good way to figure out how many darts to throw. Last but not least, we need to throw the darts uniformly, i.e., a good [[random number generator]].<ref name="mc"/>
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| ===Application===
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| There are wide possibilities for use of Monte Carlo Method <ref name="sim-pro-ek" />:
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| * Statistic experiment using generation of random variables (e.g. dice)
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| * [[sampling method]]
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| * [[Mathematics]] (e.g. numerical integraion, multiple integrals)
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| * [[Reliability Engineering]]
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| * [[Project Mnagement]] (SixSigma)
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| * [[Experimental particle physics]]
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| * [[simulation|Simulations]]
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| * [[Risk Measurement]]/[[Risk Mangement]] (e.g. Portfolio value estimation)
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| * [[Economy]] (e.g. finding the best fitting Demand)
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| * [[Process Simulation]]
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| * Operation Research
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| ==Random Number Generators==
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| ''main article: [[Random Number Generation]]''
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| For [[simulation]] experiments (including Monte Carlo) it is necessary to generate [[random]] numbers (as values of variables). The problem is, that the computer is highly [[determinism|deterministic]] machine - basicaly, behind each process there is always an algorithm, [[determinism|deterministic]] computation changing inputs to outuputs, therefore it is not easy to generate uniformly spread [[random]] numbers over a defined interval or set. <ref name="sim-pro-ek" />
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| [[Random Number Generation|Random number generator]] is a device capable of producing a sequence of numbers which can not be "easily" indentified with [[determinism|deterministic]] properties. This sequence is then called ''Sequence of stochastic numbers''.<ref name="donald">Donald E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms - chapitre 3 : Random Numbers (Addison-Wesley, Boston, 1998).</ref>
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| The algorithms typically rely on pseudo random numbers, computer generated numbers mimicking true random numbers, to generate a realization, one possible outcome of a proces. <ref name="hellander">Andreas hellander, Stochastic Simulation and Monte Carlo Methods, [online] available at http://www.it.uu.se/edu/course/homepage/bervet2/MCkompendium/mc.pdf</ref>
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| Methods for obtaining [[random]] numbers exist for a long time and are used in many different fields (like [[gaming]]). However, these number suffer from certain bias. Currently the best methods, expected to produce truly random sequences are natural methods that take advantage of the random nature of [[quantum phenomena]].<ref name="donald" />
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| ==See also==
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| *[[Gillespie algorithm]]
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| *[[Network simulation]]
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| *[[Network traffic simulation]]
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| *[[Simulation language]]
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| *[[Queueing theory]]
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| *[[Discretization]]
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| ==References==
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| {{reflist}}
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| * (Slepoy 2008): {{cite journal| title=A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks | journal= Journal of Chemical Physics| volume= 128| issue=20 | pages= 205101 | year=2008| doi = 10.1063/1.2919546| pmid=18513044| last1=Slepoy| first1=A| last2=Thompson| first2=AP| last3=Plimpton| first3=SJ }}
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| * (Bratsun 2005): {{cite journal| author=D. Bratsun, D. Volfson, J. Hasty, L. Tsimring, | title=Delay-induced stochastic oscillations in gene regulation | journal= PNAS| volume= 102| issue=41 | pages= 14593–8 | year=2005| doi = 10.1073/pnas.0503858102| pmid=16199522| pmc=1253555 }}
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| * (Cai 2007): {{cite journal| author=X. Cai, | title=Exact stochastic simulation of coupled chemical reactions with delays | journal= J. Chem. Phys.| volume= 126| pages= 124108 | year=2007| doi = 10.1063/1.2710253| issue=12| pmid=17411109 }}
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| *{{cite book| last=Hartmann| first= A.K.|year=2009|title=Practical Guide to Computer Simulations|publisher=World Scientific|isbn=978-981-283-415-7|url=http://www.worldscibooks.com/physics/6988.html}}
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 17.7. Stochastic Simulation of Chemical Reaction Networks | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=946}}
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| * (Ramaswamy 2009): {{cite journal| author=R. Ramaswamy, N. Gonzalez-Segredo, I. F. Sbalzarini, | title=A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks | journal= J. Chem. Phys.| volume= 130| pages= 244104 | year=2009| doi = 10.1063/1.3154624| issue=24| pmid=19566139 }}
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| * (Ramaswamy 2010): {{cite journal| author=R. Ramaswamy, I. F. Sbalzarini, | title=A partial-propensity variant of the composition-rejection stochastic simulation algorithm for chemical reaction networks | journal= J. Chem. Phys.| volume= 132| pages= 044102 | year=2010| doi = 10.1063/1.3297948| issue=4| pmid=20113014 }}
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| * (Ramaswamy 2011): {{cite journal| author=R. Ramaswamy, I. F. Sbalzarini, | title=A partial-propensity formulation of the stochastic simulation algorithm for chemical reaction networks with delays | journal= J. Chem. Phys.| volume= 134| pages= 014106 | year=2011| doi = 10.1063/1.3521496| pmid=21218996| issue=1 }}
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| == External links ==
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| ;Software
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| * [http://www.resassure.com ResAssure] - Stochastic reservoir simulation software - solves fully implicit, dynamic three-phase fluid flow equations for every geological realisation.
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| * [http://cain.sourceforge.net/ Cain] - Stochastic simulation of chemical kinetics. Direct, next reaction, tau-leaping, hybrid, etc.
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| * [http://mosaic.mpi-cbg.de/?q=downloads/stochastic_chemical_net pSSAlib] - C++ implementations of all partial-propensity methods.
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| * [http://stochpy.sourceforge.net StochPy] - Stochastic modelling in Python
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| * [http://steps.sourceforge.net/STEPS/Home.html STEPS] - STochastic Engine for Pathway Simulation using swig to create Python interface to C/C++ code
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| [[Category:Stochastic processes]]
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| [[Category:Stochastic simulation]]
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