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###### Ancestral reconstruction
Ornstein-Uhlenbeck process: in brief, an Ornstein-Uhlenbeck process is a continuous stochastic process that behaves like a ... Such models assume that the evolution of a trait through time may be modelled as a stochastic process. For discrete-valued ... that infer the stochastic process of evolution as it unfolds along each branch of a tree. Statistical justification. Without a ... the process is frequently taken to be a Brownian motion or an Ornstein-Uhlenbeck process. Using this model as the basis for ...
###### Law (stochasticprocesses)
In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the ... Let X : T × Ω → S be a stochastic process (so the map X t : Ω → S : ω ↦ X ( t , ω ) {\displaystyle X_{t}:\Omega \to S:\omega \ ... The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability ... Indeed, many authors define Brownian motion to be a sample continuous process starting at the origin whose law is Wiener ...
###### Stochasticprocess
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes ... then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the ... to denote the stochastic process. Markov processes are stochastic processes, traditionally in discrete or continuous time, that ... Lévy processes, Gaussian processes, and random fields, renewal processes and branching processes. The study of stochastic ...
###### Filtering problem (stochasticprocesses)
Stochastic Processes and Filtering Theory. New York: Academic Press. ISBN 0-12-381550-9. Øksendal, Bernt K. (2003). Stochastic ... In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems ... Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series Mathematics in Science and Engineering, 1979 ... Filtering (disambiguation) Not to be confused with Filter (signal processing) Kalman filter most famous filtering algorithm in ...
###### Smoothing problem (stochasticprocesses)
Especially non-stochastic and non-Bayesian signal processing, without any hidden variables. 2. Estimation: The smoothing ... Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on ... especially non-stochastic signal processing, often a name of various types of convolution). These names are used in the context ... whereas smoothing is batch processing of the given data. Filtering is the estimation of a (hidden) time-series process based on ...
###### Continuous stochasticprocess
In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a ... this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, ... Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For ... It is implicit here that the index of the stochastic process is a continuous variable. Some authors define a "continuous ( ...
###### List of stochasticprocesses topics
Markov chain Continuous-time Markov process Markov process Semi-Markov process Gauss-Markov processes: processes that are both ... bridge Brownian motion Chinese restaurant process CIR process Continuous stochastic process Cox process Dirichlet processes ... Stochastic control Stochastic differential equation Stochastic process Telegraph process Time series Wald's martingale Wiener ... See also Category:Stochastic processes Basic affine jump diffusion Bernoulli process: discrete-time processes with two possible ...
###### Continuous-time stochasticprocess
... a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index ... Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are ... An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. An example ... A more restricted class of processes are the continuous stochastic processes: here the term often (but not always) implies both ...
###### Doob-Meyer decomposition theorem
Doob decomposition theorem Doob 1953 Meyer 1952 Meyer 1963 Protter 2005 Protter (2005) Doob, J. L. (1953). Stochastic Processes ... Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107-113. ISBN 3-540-00313-4. ... Let Z {\displaystyle Z} be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process A {\ ... The Doob-Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may ...
###### Markov renewal process
Stochastic processes. New York: Wiley & Sons. ISBN 978-0-470-27000-4. Ross, Sheldon M. (1999). Stochastic processes (2nd ed.). ... Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP ( ... In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes ... The entire process is not Markovian, i.e., memoryless, as happens in a continuous time Markov chain/process (CTMC). Instead the ...
###### Markov chain
... and the Poisson process, which are considered the most important and central stochastic processes in the theory of stochastic ... The process described here is an approximation of a Poisson point process - Poisson processes are also Markov processes. ... Two important examples of Markov processes are the Wiener process, also known as the Brownian motion process, ... A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration. In the ...
###### Stein's method
Barbour, A. D. & Brown, T. C. (1992). "Stein's method and point process approximation". Stochastic Processes and their ... such as Gaussian processes by Barbour (1990), the binomial distribution by Ehm (1991), Poisson processes by Barbour and Brown ( ... If A {\displaystyle {\mathcal {A}}} is the generator of a Markov process ( Z t ) t ≥ 0 {\displaystyle (Z_{t})_{t\geq 0}} (see ... ISBN 978-1-43983-574-6. Barbour, A. D. (1988). "Stein's method and Poisson process convergence". Journal of Applied Probability ...
###### Chinese restaurant process
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at ... The Generalized Chinese Restaurant Process is closely related to Pitman-Yor process. These processes have been used in many ... The Chinese restaurant process is closely connected to Dirichlet processes and Pólya's urn scheme, and therefore useful in ... At time n, the n customers have been partitioned among m ≤ n tables (or blocks of the partition). The results of this process ...
###### Regenerative process
A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process ... Else, the process is called a delayed regenerative process. Renewal processes are regenerative processes, with T1 being the ... a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated ... These time point may themselves be determined by the evolution of the process. That is to say, the process {X(t), t ≥ 0} is a ...
###### Uncertainty quantification
Module 1: Gaussian process modeling for the computer model To address the issue from lack of simulation results, the computer ... Uncertainty quantification in stochastic systems using polynomial chaos expansion, International Journal of Applied Mechanics, ... Module 2: Gaussian process modeling for the discrepancy function Similarly with the first module, the discrepancy function is ... For example, the dimensions of a work piece in a process of manufacture may not be exactly as designed and instructed, which ...