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Modeling with Stochastic Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
A Gaussian Process is a stochastic process such that the joint distribution of every finite subset of random variables is a multivariate normal distribution. Thus, a random process {X(t),t∈T} is said to be a Gaussian (Normal) process if, for all t1,t2,...,tn∈T, the random variables X(t1), X(t2),…,X(tn) are jointly normal.
The Modeling Aspect of Simulation
Published in William Delaney, Erminia Vaccari, Dynamic Models and Discrete Event Simulation, 2020
William Delaney, Erminia Vaccari
A fundamental consideration regards the type of mathematical variable that should be used to represent each of the input, output, and state variables as time functions. If all such variables can be represented as deterministic time functions, the model will be a deterministic one; if some variable must be represented as a stochastic process (its value at a generic time being a random variable), the model will be stochastic. Such decisions will typically be made quite readily on the basis of the conceptual model, the relevant question being whether the value of a variable is affected significantly (relative to the accuracy requirements of the model) by uncontrollable random influences. For example, a realistic representation of the broken machine arrivals to the repair shop will require modeling this input as a stochastic process.
Regularization and Kernel Methods
Published in Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman, Data Science and Machine Learning, 2019
Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman
Another application of the kernel machinery is to Gaussian process regression. A Gaussian process (GP) on a space 𝒳 is a stochastic process {Zx, x ∈ 𝒳} where, for any choice of indices x1, …, xn, the vector [Zx1,…Zxn]⊤ has a multivariate Gaussian distribution. As such, the distribution of a GP is completely specified by its mean and covariance functions μ : 𝒳 → ℝ and κ : 𝒳 × 𝒳 → ℝ, respectively. The covariance function is a finite positive semidefinite function, and hence, in view of Theorem 6.2, can be viewed as a reproducing kernel on 𝒳. Gaussian process
Sensitivity analysis of risk assessment for continuous Markov process Monte Carlo method using correlated sampling method
Published in Journal of Nuclear Science and Technology, 2023
Yuki Morishita, Akio Yamamoto, Tomohiro Endo
The CMMC method couples the Markov processes and the Monte Carlo method. The Markov process is a stochastic process assuming that the future state of a system depends only on its current state, not on its past states. In other words, parameters at the next time step are calculated only from the parameters of the current time step. Continuous Markov process means that the Markov process is treated continuously in time. In the Monte Carlo method, random numbers are used to solve mathematical equations that model phenomena, and the solution is calculated by statistical processing of multiple results with different random seeds. In summary, the transition probabilities are determined by the current state information, and the future state transitions are determined by random numbers.
Valuing the unknown: could the real options have redeemed the ailing Western Australian junior iron ore operations in 2013–2016 iron price crash
Published in International Journal of Mining, Reclamation and Environment, 2019
Ajak Duany Ajak, Eric Lilford, Erkan Topal
A process is an event that evolves over time with the intention of achieving a goal. Normally, the time period for a process is from 0 to T. During this time, events may be happening at various points along the path that may have an effect on the eventual value of the process. A stochastic process is therefore formally defined as a process that can be described by the change of some random variable over time, which may be either discrete or continuous. A stochastic process with the expression {Wt: 0 ≤ t ≤ ∞} is a standard Brownian motion if W0 = 0. It has a continuous sample path and independent and normally distributed increments. When an independent increment has a distribution Wt − Ws~ N(0,t − s), it is then referred to as a Wiener process which is also normally distributed, with an expected value of µ and sample variance of σ. In a situation where two or more independent variables need to be correlated, a modified Wiener process applies, known as Geometric Brownian Motion (GBM), which is represented as:
An introductory survey of probability density function control
Published in Systems Science & Control Engineering, 2019
Mifeng Ren, Qichun Zhang, Jianhua Zhang
Since the random noises widely exist in industrial processes, relevant research has been performed to investigate modelling, control and application of stochastic processes. To simplify the system model, we can assume that all the system variables are Gaussian noises. Based on this assumption, many theoretical results and applications have been presented, for example, self-turning control, minimum variance control, linear quadratic Gaussian control and Markov jumping parameter system stochastic control have been developed by Astrom (1971), Goodwin and Sin (2014), Sain and Liberty (1971), Sworder (1969). The design objectives of all the mentioned methods only focus on the minimizing mean and variance of the system variables. As far as the linear stochastic system with Gaussian random variables is concerned, the shape of the system variable probability density function (PDF) can be fully determined by its mean and variance.