refimprove date June 2007 Cleanup date September 2010 Wiktionary stochasticStochastic from the Greek ..., sporadic and categorically NOT intermittent. A stochastic process is one whose behavior is non ... which is analyzable in terms of probability deserves the name of stochastic process . Mathematical theory The use of the term stochastic to mean based on the theory of probability has been traced ... , specifically in probability theory , the field of stochastic process es has been a major area of research. A stochastic matrix is a matrix mathematics matrix that has non negative real number real entries that sum to one in each column. Artificial intelligence In artificial intelligence , stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing , stochastic neural network s, stochastic optimization , genetic algorithms , and genetic programming . A problem itself may be stochastic as well, as in planning under uncertainty. Natural science An example of a stochastic process in the natural world is pressure in a gas as modeled by the Wiener process ... will exhibit stochastic characteristics, such as filling the container, exerting equal pressure ... generally considered forms of stochastic simulation can be arguably traced back to the earliest ... of random numbers which had been previously used for statistical sampling. Biology Stochastic resonance In biological systems, introducing stochastic noise has been found to help improve the signal strength ... lend themselves to stochastic analysis. Gene expression , for example, is a stochastic process due ... to a Promoter biology promoter resulting from Brownian motion . Medicine Stochastic effect, or chance ... of an effect increases with dose. Cancer is a stochastic effect. Stochastic theory of hematopoiesis Geomorphology meander Stochastic theory of meander formation Creativity Simonton 2003, Psych Bulletin argues that creativity in science of scientists is a constrained stochastic behaviour ... more details
Doubly stochastic may refer to Doubly stochastic model Doubly stochastic matrix disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
otheruses4 journal named after the subject matter the article regarding the models themselves stochastic processes Infobox journal title Stochastic Models cover discipline Stochastic calculus Stochastic models formernames Communications in Statistics. Stochastic Models editor Peter Taylor publisher Taylor & Francis country abbreviation Stoch. Model. history 1985 present frequency Quarterly openaccess impact 0.449 impact year 2010 website http www.tandf.co.uk journals LSTM link1 http www.tandfonline.com toc lstm20 current link1 name Online access link2 http www.tandfonline.com loi lstm20 link2 name Online archive ISSN 1532 6349 eISSN 1532 4214 LCCN 00212884 OCLC 48483352 JSTOR CODEN SMTOBE Stochastic Models is a peer review peer reviewed scientific journal that publishes papers on stochastic process stochastic models . It is published by Taylor & Francis . It was established in 1985 under the title Communications in Statistics. Stochastic Models and obtained its current name in 2001. According to the Journal Citation Reports , the journal has a 2010 impact factor of 0.449. ref name WoS cite book year 2011 chapter Stochastic Models title 2010 Journal Citation Reports publisher Thomson Reuters edition Science accessdate 2011 11 30 work Web of Science postscript . ref The founding editor in chief was Marcel Neuts Marcel F. Neuts , ref cite doi 10.1109 90.298435 ref the current editor is Peter Taylor University of Melbourne . References Reflist External links Official website http www.tandf.co.uk journals LSTM Category Taylor & Francis academic journals Category Publications established in 1985 Category Quarterly journals Category Mathematics journals Category English language journals ... more details
Primary sources date January 2012 relies entirely on Collani and Wurzburg affiliates Stochastic thinking may be looked upon as the opposite of causal thinking , however, the term stochastic thinking is rather ambiguous, because the meaning of stochastics is not clear. It can be looked upon as a branch of mathematics, or as a cocktail of statistical ideas and probabilistic ideas , ref Andreas Eichler, Maria Gabriella Ottaviani, Floriane Wozniak and Dave Pratt, Introduction on Stochastic Thinking , Proceedings of CERME 6, January 28th February 1st 2009, Lyon France INRP 2010, http www.inrp.fr publications ... . Here stochastic thinking is explained in the sense of Bernoulli Stochastics. ref Elart von ... solving by stochastic thinking Stochastic thinking for problem solving proceeds in three steps Stochastic thinking as basis for making decisions starts with observing an effect or problem which ... the Promising Alternative. ref The second step in stochastic thinking consists of identifying ... the relation between past and future which are to be changed. The third step of stochastic thinking is to verify that the system changes are effective. The main difference between stochastic thinking and the prevailing causal thinking is the focus Stochastic thinking focus on improving the whole, while causal thinking focus on improving parts. Stochastic thinking means to think in sets and structures ... of the occurrence of problems. Effect of stochastic thinking Stochastic thinking focus ... words stochastic thinking results in a continual examination and improvement of the whole to prevent the recurrence of problems. Thus, stochastic thinking results in proactive strategies in contrast ... by a Bernoulli space which represents the basis for stochastic thinking. The Bernoulli space shows ... Stochastic thinking is oriented towards long term effects by means of continual improvement of the system ..., http www.stochastikon.com Categories Category Stochastic processes ... more details
Stochastic computing is a collection of techniques that represent continuous values by streams of random ... the similarity in their names, stochastic computing is distinct from the study of randomized algorithm ... to compute math p times q math . Stochastic computing performs this operation using probability instead ..., stochastic computing represents numbers as streams of random bits and reconstructs numbers by calculating ... of reconstruction, devices that perform these operations are sometimes called stochastic averaging processors. In modern terms, stochastic computing can be viewed as an interpretation of calculations ... Image RASCEL stochastic computer 1969.png thumb alt A photograph of the RASCEL stochastic computer. The RASCEL stochastic computer, circa 1969 Stochastic computing was first introduced in a pioneering ... journal last1 Poppelbaum first1 W. last2 Afuso first2 C. last3 Esch first3 J. title Stochastic computing ... cite journal last Gaines first B. title Stochastic Computing journal AFIPS SJCC year 1967 volume 30 ... stochastic computation. A host ref cite book last1 Mars first1 P. last2 Poppelbaum first2 W. title Stochastic and deterministic averaging processors year 1981 ref of these machines were constructed ... computer based on a regular array of stochastic computing element logic year 1969 location University ... and 1970s, stochastic computing ultimately failed to compete with more traditional digital logic, for reasons outlined below. The first and last International Symposium on Stochastic Computing ref cite conference title Proceedings of the first International Symposium on Stochastic Computing ... in the area dwindled over the next few years. Although stochastic computing declined as a general method ... Systems Science title Stochastic Computing Systems last Gaines first B. R. editor last Tou editor ... Computing Machines, Proceedings IEEE, NAPA title A stochastic neural architecture that exploits ... has turned towards stochastic decoding, which applies stochastic computing to the decoding of error ... more details
Stochastic calculus is a branch of mathematics that operates on stochastic process es. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly. The best known stochastic process to which stochastic calculus is applied is the Wiener process named in honor of Norbert Wiener , which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates. The main flavours of stochastic calculus are the It calculus and its variational relative the Malliavin calculus . For technical reasons the It integral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation particularly in engineering disciplines. The Stratonovich integral can readily be expressed in terms of the It integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and does therefore not require It s lemma . This enables problems to be expressed in a co ordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than R sup n sup . The dominated ... integral is central to the study of stochastic calculus. The integral math int H ,dX math is defined ... integral. Applications A very important application of stochastic calculus is in quantitative finance , in which asset prices are often assumed to follow stochastic differential equations . In the Black ... date August 2011 References Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application ... http arxiv.org PS cache arxiv pdf 0712 0712.3908v2.pdf Preprint Category Stochastic calculus Category ... more details
lead missing date March 2012 Stochastic programming is a framework for modeling Optimization mathematics optimization problems that involve uncertainty . Whereas deterministic optimization problems are formulated ... and Optimization mathematics optimal in some sense. Stochastic programming mathematical model ... Lectures on stochastic programming Modeling and theory series MPS SIAM Series on Optimization volume ... be taken in response to each random outcome. Stochastic programming has applications in a broad ... and William T. Ziemba eds. . Applications of Stochastic Programming . MPS SIAM Book Series on Optimization 5, 2005. ref ref Applications of stochastic programming are described at the following website, http stoprog.org Stochastic Programming Community . ref Biological Applications Stochastic dynamic ... staged, rather than two staged. Economic Applications Stochastic dynamic programming is a useful ... Howitt, R., Msangi, S., Reynaud, A and K. Knapp. 2002. Using Polynomial Approximations to Solve Stochastic .... Solvers FortSP solver for stochastic programming problems See also Portal Computer science Stochastic ... V. Louveaux. Introduction to Stochastic Programming . Springer Verlag, New York, 1997. cite book last1 Kall first1 Peter last2 Wallace first2 Stein W. title Stochastic programming series Wiley Interscience ... G. Ch. Pflug Optimization of Stochastic Models. The Interface between Simulation and Optimization . Kluwer, Dordrecht, 1996. Andras Prekopa . Stochastic Programming. Kluwer Academic Publishers, Dordrecht, 1995. Andrzej Ruszczynski and Alexander Shapiro eds. . Stochastic Programming . Handbooks in Operations ... last2 Dentcheva first2 Darinka last3 Ruszczy ski first3 Andrzej title Lectures on stochastic ... of Stochastic Programming . MPS SIAM Book Series on Optimization 5, 2005. External links http stoprog.org Stochastic Programming Community Home Page DEFAULTSORT Stochastic Programming Category Stochastic optimization Category Stochastic algorithms Category Mathematical optimization Category Operations ... more details
Technical date September 2011 Stochastic resonance SR is a phenomenon that occurs in a threshold measurement ... non zero level of stochastic input noise thereby lowering the response threshold ref name MossReview cite journal author Moss F, Ward LM, Sannita WG title Stochastic resonance and sensory information ... resonate s at a particular noise level. Definition Stochastic resonance is observed when noise added ... ratio as a function of noise intensity shows a shape. Strictly speaking, stochastic resonance occurs ... wide band stochastic force noise . The system response is driven by the combination of the two ... switch rate induced by the sole noise the stochastic time scale . citation needed date December 2010 Thus the term stochastic resonance . Stochastic resonance was discovered and proposed for the first ... author Benzi R, Parisi G, Sutera A, Vulpiani A title Stochastic resonance in climatic ... has been applied in a wide variety of systems. Nowadays stochastic resonance is commonly invoked when ... stochastic resonance Suprathreshold stochastic resonance is a particular form of stochastic resonance ... systems where stochastic resonance occurs, suprathreshold stochastic resonance occurs not only ..., hence the qualifier, suprathreshold, in suprathreshold stochastic resonance. Neuroscience psychology and biology Main Stochastic resonance sensory neurobiology Stochastic resonance has been observed ... title Neural synchrony in stochastic resonance, attention, and consciousness journal Can J Exp Psychol ... Gammaitoni L, H nggi P, Jung P, Marchesoni F title Stochastic resonance journal Review of Modern Physics ... overview of stochastic resonance. Signal analysis A related phenomenon is dither ing applied to analog ... Gammaitoni L title Stochastic resonance and the dithering effect in threshold physical systems journal ... SR and dithering p4691 1.pdf doi 10.1103 PhysRevE.52.4691 bibcode 1995PhRvE..52.4691G ref Stochastic ... C title Measurement of weak transmittances by stochastic resonance journal Optics Letters volume ... more details
Stochastic control is a subfield of control theory which deals with the existence of uncertainty in the data ... probability distribution affects the state evolution and the observation of the controllers. Stochastic ... average cost despite the presence of these noises. ref http www.answers.com topic stochastic control theory?cat technology Definition from Answers.com ref An extremely well studied formulation in stochastic .... ref Turnovsky, Stephen, Optimal stabilization policies for stochastic linear systems The case ... 1990, 161 164. ref The field of stochastic optimal control SOC developed greatly since the 1970s .... Major mathematical developments were by W. Fleming and R. Rishel, Deterministic and Stochastic Optimal ... 2006 . These techniques were applied by J. L. Stein in Stochastic Optimal Control and the US ... stochastic control theory is concerned with models in which random system disturbances are allowed. The controller knows the state of the system at each instant of time. For stochastic systems there are many ... in the stochastic differential equation is usually wealth or net worth. Determinants of the change in wealth are usually the stochastic returns to assets and the interest rate. The maximization, say of the expected logarithm of net worth at a terminal date T, is subject to stochastic processes on the components ... 2012 is that the application of Stochastic Optimal Control SOC is very helpful in understanding and predicting ... leverage balances risk against expected growth. The environment is stochastic the capital gain, productivity of capital and interest rate are stochastic variables, and for an insurance company, such as AIG, the claims are also stochastic. He associates the housing price bubble with the growth ... Time Finance, Blackwell 1990 W. Fleming and R. Rishel, Deterministic and Stochastic Optimal ... 2006 J. L. Stein in Stochastic Optimal Control and the US Financial Crisis, Springer Science 2012 . Reflist See also Stochastic process Control theory math stub Category Control theory Category Stochastic ... more details
In probability theory , stochastic drift is the change of the average value of a stochastic process stochastic random process . A related term is the drift rate which is the rate at which the average changes. This is in contrast to the random fluctuations about this average value. For example, the process which counts the number of heads in a series of math n math coin toss es has a drift rate of 1 2 per toss. Stochastic drifts in population studies Longitudinal studies of secular events are frequently conceptualized as consisting of a trend component fitted by a polynomial , a cyclical component often fitted by an analysis based on autocorrelation s or on a Fourier series , and a random component stochastic drift to be removed. In the course of the time series analysis , identification of cyclical and stochastic drift components is often attempted by alternating autocorrelation analysis and differencing of the trend. Autocorrelation analysis helps to identify the correct phase of the fitted model while the successive differencing transforms the stochastic drift component into white noise . Stochastic drift can also occur in population genetics where it is known as Genetic drift . A finite population of randomly reproducing organisms would experience changes from generation to generation in the frequencies of the different genotypes. This may lead to the fixation of one of the genotypes, and even the emergence of a speciation new species . In sufficiently small populations, drift can also neutralize the effect of deterministic natural selection on the population. Stochastic ... variable. In this case the stochastic drift can be removed from the data by regressing math y t math ... where math u t math is a zero long run mean stationary random variable here c is a non stochastic ... any stochastic change to the price level permanently affects the expected values of the price level ... analysis Category Stochastic processes Category Economics Category Finance ... more details
about iterative method s the modeling and optimization of decisions under uncertainty stochastic programming Stochastic optimization SO methods are optimization mathematics optimization iterative method method s that generate and use random variable s. For stochastic problems, the random variables appear ... s or random constraints, for example. Stochastic optimization methods also include methods with random iterates. Some stochastic optimization methods use random iterates to solve stochastic problems, combining both meanings of stochastic optimization. ref name spall2003 Cite book author Spall, J. C. title Introduction to Stochastic Search and Optimization year 2003 publisher Wiley url http www.jhuapl.edu ISSO isbn 0471330523 ref Stochastic optimization methods generalize deterministic system mathematics deterministic methods for deterministic problems. Methods for stochastic functions Partly random ... decisions about the next steps. Methods of this class include stochastic approximation SA , by Herbert ..., S. title A Stochastic Approximation Method journal Annals of Mathematical Statistics year 1951 volume 22 pages 400 407 doi 10.1214 aoms 1177729586 issue 3 ref stochastic gradient descent inventor and reference needed finite difference stochastic approximation finite difference SA by Kiefer and Wolfowitz ... J. Wolfowitz title Stochastic Estimation of the Maximum of a Regression Function journal Annals ... ref simultaneous perturbation stochastic approximation simultaneous perturbation SA by Spall 1992 ref name spall1992 cite journal author Spall, J. C. title Multivariate Stochastic Approximation Using ... ref Holger H. Hoos and Thomas St tzle, http www.sls book.net Stochastic Local Search Foundations and Applications ... performance uniformly across many data sets, for many sorts of problems. Stochastic optimization ... Random Search year 1991 publisher Kluwer Academic isbn 0792311221 ref stochastic tunneling ref name wenz1999 cite journal author W. Wenzel coauthors K. Hamacher title Stochastic tunneling approach ... more details
For a matrix whose elements are stochastic, see Random matrix In mathematics , a stochastic matrix also termed probability matrix , transition matrix , substitution matrix , or Markov matrix is a matrix mathematics matrix used to describe the transitions of a Markov chain . It has found use in probability theory , statistics and linear algebra , as well as computer science . There are several different definitions and types of stochastic matrices A right stochastic matrix is a square matrix each of whose rows consists of nonnegative real numbers, with each row summing to 1. A left stochastic matrix ... summing to 1. A doubly stochastic matrix is a square matrix where all entries are nonnegative and all rows and all columns sum to 1. In the same vein, one may define a stochastic vector as a Euclidean ... of a stochastic matrix is a probability vector , which are sometimes called stochastic vectors ... and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices this article follows that convention. Definition and properties A stochastic matrix describes ... math , the stochastic matrix P is given by using math P i,j math as the math i th math row and math ... i math to some state must be 1, this matrix is a right stochastic matrix, so that math sum j P i ... . math The Perron Frobenius theorem ensures that every stochastic matrix has such a vector, and that the largest ... state . Intuitively, a stochastic matrix represents a Markov chain with no sink states, this implies that the application of the stochastic matrix to a probability distribution would redistribute ... ate the mouse and the game ended F. We use a stochastic matrix to represent the transition probabilities ... be ignored. Let math boldsymbol tau 0,1,0,0 math and remove state five to make a sub stochastic matrix ... 1 ,. math See also Muirhead s inequality Perron Frobenius theorem Doubly stochastic matrix Discrete .... Introduction to Matrix Analytic Methods in Stochastic Modeling , 1st edition. Chapter 2 PH ... more details
No footnotes date November 2010 In probability theory , a stochastic process IPAc en pron s t k ... equation , in a stochastic or random process there is some indeterminacy even if the initial condition ... may evolve. In the simple case of discrete time discrete time , a stochastic process amounts ... chain . Another basic type of a stochastic process is a random field , whose domain is a region of space ... values. One approach to stochastic processes treats them as function mathematics function s of one ... type. Type refers to the codomain of the function. Although the random values of a stochastic process ... modeled as stochastic time series include stock market and exchange rate fluctuations, signals ... math and a measurable space math S, mathcal S math , an S valued stochastic process is a collection of S valued random variable s on math Omega math , indexed by a totally ordered set T time . That is, a stochastic ... dimensional distributions Let X be an S valued stochastic process. For every finite subset math ... of finite dimensional distributions can be used to define a stochastic process see Kolmogorov ... ref Karlin, Samuel & Taylor, Howard M. 1998 . An Introduction to Stochastic Modeling , Academic ... derived from the full blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a Wiener process in which case the marginals are all gaussian distributions of the exponential class but not in general for all stochastic processes. When ... of a stochastic process with a given family of finite dimensional probability distribution ... news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly ... many values of the function. One solution to this problem is to require that the stochastic process ... F t subseteq mathcal F math . A stochastic process X on the same time set T is said to be adapted ... Press, 2010. ref The natural filtration Given a stochastic process math X X t t in T math , the natural ... more details
Other uses Dominance disambiguation Dominance Stochastic dominance ref Hadar and Russell, Rules for Ordering ... is a form of stochastic ordering . The term is used in decision theory and decision analysis to refer ... aversion is a factor only in second order stochastic dominance. Stochastic dominance does not give a order .... A related concept not included under stochastic dominance is deterministic dominance , which ... outcome of gamble B. Statewise dominance The simplest case of stochastic dominance is statewise dominance ... dominant gamble. First order stochastic dominance Statewise dominance is a special case of the canonical first order stochastic dominance , defined as follows gamble A has first order stochastic dominance ... toss outcome by value won, but gamble C has first order stochastic dominance over B without statewise ... to stochastic dominance simply by comparing the means of their probability distributions. Every ... will prefer gamble A over gamble B if A first order stochastically dominates B. First order stochastic ..., pushing some of the probability mass to the left. Second order stochastic dominance The other commonly used type of stochastic dominance is second order stochastic dominance . Roughly speaking, for two gambles A and B, gamble A has second order stochastic dominance over gamble B if the former is more ... for all nondecreasing and concave utility functions math U math . Second order stochastic dominance ... to the fixed number 0 , then B is a mean preserving spread of A. Second order stochastic dominance ... other portfolio. See modern portfolio theory and marginal conditional stochastic dominance . Sufficient conditions for second order stochastic dominance First order stochastic dominance is a sufficient condition. Necessary conditions for second order stochastic dominance math E A x geq E B x math ... order stochastic dominance Let math F A math and math F B math be the cumulative distribution functions ... a positive third derivative throughout . Sufficient condition for third order stochastic dominance ... more details
In game theory , a stochastic game , introduced by Lloyd Shapley in the early 1950s, is a dynamic game with probabilistic transitions played by one or more players. The game is played in a sequence of stages ... payoffs or the limit inferior of the averages of the stage payoffs. Stochastic games generalize both Markov decision process es and repeated game s. Theory The ingredients of a stochastic game are a finite ... to the probability math P cdot mid m t,s t math . A play of the stochastic game, math m 1,s 1, ldots ... lambda m 1 math , of a two person zero sum stochastic game math Gamma n math , respectively math Gamma ... math . The uniform value math v infty math of a two person zero sum stochastic game math Gamma infty ... that every two person zero sum stochastic game with finitely many states and actions has a uniform ..., then a stochastic game with a finite number of stages always has a Nash equilibrium . The same is true ... has shown that all two person stochastic games with finite state and action spaces have Epsilon equilibrium ... open question. Applications Stochastic games have applications in economics, evolutionary biology and computer networks. ref http www net.cs.umass.edu sadoc mdp main.pdf Constrained Stochastic Games ... of Markov Decision Process es and two person stochastic games. They coin the term Competitive MDPs to encompass both one and two player stochastic games. Notes reflist Further reading cite journal first A. last Condon authorlink Anne Condon title The complexity of stochastic games journal ... A. last2 Neyman title Stochastic Games journal International Journal of Game Theory volume 10 issue ... first2 S. last2 Sorin title Stochastic Games and Applications location Dordrecht publisher Kluwer Academic Press year 2003 isbn 1402014929 cite journal first L. S. last Shapley title Stochastic games ... Vieille chapter Stochastic games Recent results title Handbook of Game Theory pages 1833 1850 location ... main results, no proofs Game theory DEFAULTSORT Stochastic Game Category Game theory ru ... more details
In probability theory and statistics , a stochastic order quantifies the concept of one random variable being bigger than another. These are usually partial order s, so that one random variable math A math may be neither stochastically greater than, less than nor equal to another random variable math B math . Many different orders exist, which have different applications. Usual stochastic order A real random variable math A math is less than a random variable math B math in the usual stochastic order if math Pr A x le Pr B x text for all x in infty, infty , math where math Pr cdot math denotes the probability of an event. This is sometimes denoted math A preceq B math or math A le st B math . If additionally math Pr A x Pr B x math for some math x math , then math A math is stochastically strictly ... in distribution. Stochastic dominance Stochastic dominance ref http www.mcgill.ca files economics stochasticdominance.pdf ref is a stochastic ordering used in decision theory . Several orders of stochastic dominance are defined. Zeroth order stochastic dominance consists of simple inequality math A preceq 0 B math if math A le B math for all state of nature states of nature . First order stochastic dominance is equivalent to the usual stochastic order above. Higher order stochastic dominance is defined in terms of integrals of the distribution function . Lower order stochastic dominance implies higher order stochastic dominance. Multivariate stochastic order Empty section date July 2010 Other stochastic orders Hazard rate order The hazard rate of a non negative random variable math X math ... are. This is captured to a limited extent by the variance , but more fully by a range of stochastic .... The converse is not true. See also Stochastic dominance References refimprove date February 2012 M. Shaked and J. G. Shanthikumar, Stochastic Orders and their Applications , Associated Press, 1994. E ... 419, 1955. reflist DEFAULTSORT Stochastic Ordering Category Theory of probability distributions de ... more details
Stochastic simulation algorithms and methods were initially developed to analyse chemical reactions involving large numbers of species with complex reaction kinetics ref cite journal last Bradley first Jeremy authorlink Jeremy Bradley coauthors Stephen Gilmore year 2005 title Stochastic simulation methods applied to a secure electronic voting model journal Electronic Notes in Theoretical Computer Science ref . The first algorithm, the Gillespie algorithm was proposed by Dan Gillespie in 1977. It is an exact procedure for numerically simulating the time evolution of a well stirred chemically reacting system. The algorithm is a Monte Carlo method Monte Carlo type method. Discrete, exact variants 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 ... stochastic oscillations in gene regulation journal PNAS volume 102 issue 41 pages 14593 8 year 2005 ... stochastic simulation of coupled chemical reactions with delays journal J. Chem. Phys. volume 126 ... place New York isbn 978 0 521 88068 8 chapter Section 17.7. Stochastic Simulation of Chemical ... exact stochastic simulation algorithms for chemical reaction networks journal J. Chem. Phys. volume ... stochastic simulation algorithm for chemical reaction networks journal J. Chem. Phys. volume ... author R. Ramaswamy, I. F. Sbalzarini, title A partial propensity formulation of the stochastic simulation ... Cain Stochastic simulation of chemical kinetics. Direct, next reaction, tau leaping, hybrid .... http stompy.sourceforge.net StochPy Stochastic modelling in Python Category Stochastic processes ... more details
Quantum mechanics cTopic Interpretation of quantum mechanics Interpretations The stochastic interpretation is an interpretation of quantum mechanics . The modern application of stochastics to quantum mechanics involves the assumption of spacetime stochasticity , the idea that the small scale structure of spacetime is undergoing both metric and topological fluctuations John Archibald Wheeler s quantum foam , and that the averaged result of these fluctuations recreates a more conventional looking metric at larger scales that can be described using classical physics, along with an element of nonlocality that can be described using quantum mechanics. A stochastic interpretation of quantum mechanics due to persistent vacuum fluctuations is suggested by Roumen Tsekov. The main idea is that vacuum or spacetime fluctuations are the reason for quantum mechanics and not a result of it how it is usually considered. See also Quantum foam Interpretation of quantum mechanics Interpretations of quantum mechanics References cite journal author Edward Nelson title Derivation of the Schr dinger Equation from Newtonian Mechanics journal Physical Review volume 150 page 1079 1085 year 1966 bibcode 1966PhRv..150.1079N doi 10.1103 PhysRev.150.1079 cite book author Khavtain Namsrai title Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics publisher Springer year 1985 isbn 9027720010 cite journal author Roumen Tsekov title Dissipative and Quantum Mechanics journal New Adv. Phys. volume 3 page 35 44 year 2009 Category Interpretations of quantum mechanics quantum stub ... more details
context date June 2011 Stochastic approximation methods are a family of iterative stochastic optimization ... by Herbert Robbins and Sutton Monro, ref name rm A Stochastic Approximation Method, Herbert Robbins ... math x in Theta math . ref name jsacks Asymptotic distribution of Stochastic Approximation, J. Sacks, Annals of Mathematical Statistics 29 , 1958 , pp. 373 409. ref ref name NJLS Robust Stochastic Approximation Approach to Stochastic Programming, A. Nemirovski, A. Juditsky, G. Lan and A. Shapiro .... ref name NJLS ref name jcsbook Introduction to Stochastic Search and Optimization Estimation ..., Polyak and Juditsky, ref name pj Acceleration of Stochastic Approximation by Averaging, B.T. ... 1978 English . ref for the cases of solving the stochastic optimization problem with continuous convex ..., ref name KW Stochastic Estimation of the Maximum of a Regression Function, Jack Kiefer mathematician ..., proposed the use of Simultaneous perturbation stochastic approximation simultaneous perturbations ... of the dimension math d math . ref name Jsp Adaptive Stochastic Approximation by the Simultaneous Perturbation ... can be fairly restrictive and highly unrealistic. Further Developments in Stochastic Approximation ..., possible noise models, and so on. ref name kushneryin Stochastic Approximation Algorithms and Applications ... ed., titled Stochastic Approximation and Recursive Algorithms and Applications , 2003, ISBN 0387008942. ref ref Stochastic Approximation and Recursive Estimation , Mikhail Borisovich Nevel son and Rafail ... C. Johan Masreliez and R. Douglas Martin were the first to apply stochastic approximation to Robust statistics robust estimation . ref R.D. Martin & C.J. Masreliez, Robust estimation via stochastic approximation . IEEE Trans. Inform. Theory, 21 pp.263 271 1975 . ref See also Stochastic gradient descent Stochastic optimization Simultaneous perturbation stochastic approximation References reflist DEFAULTSORT Stochastic Approximation Category Stochastic optimization Category Statistical approximations ... more details
Hatnote See also Volatility finance . Stochastic Volatility finance volatility models are used in the field of mathematical finance to evaluate derivative finance derivative securities , such as option finance options . The name derives from the models treatment of the underlying security s volatility ... process itself, among others. Stochastic volatility models are one approach to resolve a shortcoming ... price is a stochastic process rather than a constant, it becomes possible to model derivatives more ... Wiener process with zero mean and unit rate of variance . The explicit solution of this stochastic differential ... stochastic volatility models such as Black Scholes and Cox Ross Rubinstein . For a stochastic volatility ... between volatility and price, introducing stochastic volatility math dS t mu S t d t sigma S t gamma ... 1 math . Some argue that because the CEV model does not incorporate its own stochastic process for volatility, it is not truly a stochastic volatility model. Instead, they call it a local volatility model. SABR volatility model Main SABR Volatility Model The SABR model Stochastic Alpha, Beta, Rho ... or equity under stochastic volatility math sigma math math dF t sigma t F beta t , dW t, math math ... model for estimating stochastic volatility. It assumes that the randomness of the variance process ..., mean reverting and volatility of variance parameters, are stochastic quantities given by math theta ... developed the first stochastic mean and stochastic volatility model, Chen model . Specifically, the dynamics of the instantaneous interest rate are given by following the stochastic differential ... and ambiguity Black Scholes References http www.wilmott.com detail.cfm?articleID 245 Stochastic Volatility ... Heston original.pdf A closed form solution for options with stochastic volatility , SL Heston, 1993 ... of Stochastic Volatility Models , Kilin, Fiodar 2006 . cite book title Stochastic Mean and Stochastic Volatility A Three Factor Model of the Term Structure of Interest Rates and Its Application ... more details
Stochastic tunneling STUN is an approach to global optimization based on the Monte Carlo method Sampling signal processing sampling of the function to be minimized. Idea image stun.jpg thumb 400px Schematic one dimensional test function black and STUN effective potential red & blue , where the minimum indicated by the arrows is the best minimum found so far. All Potential well well s that lie above the best minimum found are suppressed. If the dynamical process can escape the well around the current minimum estimate it will not be trapped by other local minima that are higher. Wells with deeper minima are enhanced. The dynamical process is accelerated by that. Monte Carlo method based optimization techniques sample the objective function by randomly hopping from the current solution vector to another with a difference in the function value of math Delta E math . The acceptance probability of such a trial jump is in most cases chosen to be math min left 1 exp left beta cdot Delta E right right math Nicholas Metropolis Metropolis criterion with an appropriate parameter math beta math . The general idea of STUN is to circumvent the slow dynamics of ill shaped energy functions that one encounters for example in spin glass es by tunneling through such barriers. This goal is achieved by Monte Carlo sampling of a transformed function that lacks this slow dynamics. In the standard form the transformation reads math f STUN 1 exp left gamma cdot left f x f o right right math where math f o math is the lowest function value found so far. This transformation preserves the Locus mathematics ... author K. Hamacher title Adaptation in Stochastic Tunneling Global Optimization of Complex Potential ... i2006 10058 0 Cite journal author K. Hamacher and W. Wenzel title The Scaling Behaviour of Stochastic ... title A Stochastic tunneling approach for global minimization journal Phys. Rev. Lett. volume 82 issue .....21.1087M Category Stochastic optimization de Stochastisches Tunneln ... more details
A stochastic grammar statistical grammar is a grammar framework with a probabilistic notion of grammaticality Stochastic context free grammar Statistical parsing Data oriented parsing Hidden Markov model Estimation theory Statistical natural language processing uses stochastic , probabilistic and statistical methods, especially to resolve difficulties that arise because longer sentences are highly ambiguous when processed with realistic grammars, yielding thousands or millions of possible analyses. Methods for disambiguation often involve the use of corpus linguistics corpora and Markov model s. A probabilistic model consists of a non probabilistic model plus some numerical quantities it is not true that probabilistic models are inherently simpler or less structural than non probabilistic models. ref John Goldsmith. 2002. Probabilistic Models of Grammar Phonology as Information Minimization. Phonological Studies 5 21&ndash 46. ref The technology for statistical NLP comes mainly from machine learning and data mining , both of which are fields of artificial intelligence that involve learning from data. See also Colorless green ideas sleep furiously Computational linguistics Refimprove date March 2011 More footnotes date March 2011 References references Further reading Christopher D. Manning, Hinrich Sch tze Foundations of Statistical Natural Language Processing , MIT Press 1999 , ISBN 978 0262133609. Stefan Wermter, Ellen Riloff, Gabriele Scheler eds. Connectionist, Statistical and Symbolic Approaches to Learning for Natural Language Processing , Springer 1996 , ISBN 978 3540609254. Category Grammar frameworks Category Statistical natural language processing Category Probabilistic models ling stub nl Stochastische grammatica ... more details
In estimation theory in statistics , stochastic equicontinuity is a property of estimator s or of estimation procedures that is useful in dealing with their Asymptotic theory statistics asymptotic behviour as the amount of data increases. It is a version of equicontinuity used in the context of functions of random variables that is, random function s. The property relates to the rate of convergence of random variables convergence of sequences of random variables and requires that this rate is essentially the same within a region of the parameter space being considered. For instance, stochastic equicontinuity, along with other conditions, can be used to show uniform weak convergence, which can be used to prove the convergence of random variables convergence of extremum estimator s. ref Newey, Whitney K. 1991 Uniform Convergence in Probability and Stochastic Equicontinuity , Econometrica , 59 4 , 1161 1167 jstor 2938179 ref Definition Let math H n theta n geq 1 math be a family of random functions defined from math Theta rightarrow reals math , where math Theta math is any normed metric space. Here math H n theta math might represent a sequence of estimators applied to datasets of size n , given that the data arises from a population for which the parameter indexing the statistical model for the data is &theta . The randomness of the functions arises from the data generating process under which a set of observed data is considered to be a realisation of a probabilistic or statistical model. However, in math H n theta math , &theta relates to the model currently being postulated or fitted rather than to an underlying model which is supposed to represent the mechanism generating the data. Then math H n math is stochastically equicontinuous if, for every math epsilon 0 math , there is a math delta 0 math such that math lim n rightarrow infty Pr left sup theta in Theta sup theta in B theta, delta H n theta H n theta epsilon right delta . math Here B &theta , &delta represents ... more details
refimprove date March 2011 Stochastic screening or FM screening is a halftone process based on Pseudorandomness pseudo random distribution of halftone dots, using frequency modulation FM to change the density of dots according to the gray level desired. Traditional amplitude modulation halftone screening is based on a geometric and fixed spacing of dots, which vary in size depending on the tone color represented for example, from 10 to 200 micrometre s . The stochastic screening or FM screening instead uses a fixed size of dots for example, about 25 micrometres and a distribution density that varies depending on the color s tone. The technique of stochastic screening, which has existed since the seventies, Citation needed date March 2011 has had a revival in recent times thanks to increased use of Computer to plate computer to plate CTP techniques. In previous techniques, computer to film , during the exposure there could be a drastic variation in the quality of the plate. It was a very delicate and difficult procedure that was not much used. Today, with CTP during the creation of the plate you just need to check a few parameters on the density and tonal correction curve. When you make a plate with stochastic screening you must use a tone correction curve, this curve allows one to align the tone reproduction of an FM screen to that of an industry standard. Given the same final presswork tone value, an FM screen utilizes more halftone dots than an AM XM screen. The result is that more light is filtered by the ink and less light simply reflects off the surface of the substrate. The result is that FM screens exhibit a greater color gamut than conventional AM XM halftone screen frequencies. The creation of a plate with stochastic screening is done the same way as is done with an AM XM screen. A tone reproduction compensation curve is typically applied to align the stochastic screening to conventional AM FM tone reproductions targets e.g. ISO 12647 2 . Advantages The screening ... more details
Refimprove date December 2009 Stochastic cooling is a form of particle beam cooling . It is used in some particle accelerator s and storage ring s to control the Beam emittance emittance of the particle beam s in the machine. This process uses the Signal electrical engineering electrical signals that the individual charged particle s generate in a feedback loop to reduce the tendency of individual particles to move away from the other particles in the beam. It is accurate to think of this as thermodynamic cooling, or the reduction of entropy , in much the same way that a refrigerator or an air conditioner cools its contents. The technique was invented and applied at the Intersecting Storage Rings , ref name overview Citation arxiv physics 0308044 title Stochastic Cooling Overview author John Marriner arxiv physics.acc ph 0308044 doi 10.1016 j.nima.2004.06.025 date 2003 08 11 journal Nuclear Instruments and Methods A volume 532 issue 1 2 pages 11 18 bibcode 2004NIMPA.532...11M ref and later the Super Proton Synchrotron , at CERN in Geneva, Switzerland by Simon van der Meer , ref http www.nytimes.com 2011 03 12 science 12vandermeer.html? r 1&scp 1&sq Simon van der Meer&st nyt Simon van ... National Accelerator Laboratory continues to use stochastic cooling in its antiproton source. The accumulated ... Detector at Fermilab CDF and the D0 experiment . Stochastic cooling in the Tevatron at Fermilab ... RHIC . Technical details This section needs to be edited for clarity by a stochastic cooling expert. Stochastic cooling uses the electrical signals produced by individual particles in a group of particles ... on the depth of the cooling that is required. Stochastic cooling is used to reduce the transverse ... spread of each bunch is not affected by this damping. The key to stochastic cooling is to address individual ... and gets smaller. The word stochastic in the title stems from the fact that usually only some of the particles ... reflist Accelerator stub DEFAULTSORT Stochastic Cooling Category Accelerator physics de Stochastische ... more details
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