Conservative processes with stochastic rates

1975 ◽  
Vol 12 (03) ◽  
pp. 447-456 ◽  
Author(s):  
Roy Saunders
Keyword(s):  
Transition Rate ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Dependent ◽  
Independent Random Variables ◽  
Exchangeable Random Variables

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

Conservative processes with stochastic rates

1975 ◽  
Vol 12 (3) ◽  
pp. 447-456 ◽  
Author(s):  
Roy Saunders
Keyword(s):  
Transition Rate ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Dependent ◽  
Independent Random Variables ◽  
Exchangeable Random Variables

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

On joint exchangeability and conservative processes with stochastic rates

1976 ◽  
Vol 13 (3) ◽  
pp. 584-590 ◽  
Author(s):  
Roy Saunders
Keyword(s):  
Transition Rate ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Dependent ◽  
Previous Article ◽  
Initial State ◽  
Exchangeable Random Variables ◽  
Time Zero ◽  
Initial States

In a previous article Saunders (1975) investigated the form of transition probabilities for a generalization of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The results of that investigation are given in terms of properties of exchangeable random variables and require that the process be in a particular initial state at time zero. This article removes the restriction on the initial state by using some properties of two sequences of jointly exchangeable variables. General results analogous to those obtained previously are shown to hold for general initial states.

On joint exchangeability and conservative processes with stochastic rates

1976 ◽  
Vol 13 (03) ◽  
pp. 584-590
Author(s):  
Roy Saunders
Keyword(s):  
Transition Rate ◽  
Transition Probabilities ◽  
Random Variables ◽  
Time Dependent ◽  
Previous Article ◽  
Initial State ◽  
Exchangeable Random Variables ◽  
Time Zero ◽  
Initial States

In a previous article Saunders (1975) investigated the form of transition probabilities for a generalization of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The results of that investigation are given in terms of properties of exchangeable random variables and require that the process be in a particular initial state at time zero. This article removes the restriction on the initial state by using some properties of two sequences of jointly exchangeable variables. General results analogous to those obtained previously are shown to hold for general initial states.

An LSTM-Based Ensemble Learning Approach for Time-Dependent Reliability Analysis

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Mingyang Li ◽  
Zequn Wang
Keyword(s):  
Reliability Analysis ◽  
Ensemble Learning ◽  
Short Term Memory ◽  
Limit State ◽  
Random Variables ◽  
Time Dependent ◽  
Independent Random Variables ◽  
State Function ◽  
Learning Approach ◽  
Lstm Network

Abstract This paper presents a long short-term memory (LSTM)-based ensemble learning approach for time-dependent reliability analysis. An LSTM network is first adopted to learn system dynamics for a specific setting with a fixed realization of time-independent random variables and stochastic processes. By randomly sampling the time-independent random variables, multiple LSTM networks can be trained and leveraged with the Gaussian process (GP) regression to construct a global surrogate model for the time-dependent limit state function. In detail, a set of augmented data is first generated by the LSTM networks and then utilized for GP modeling to estimate system responses under time-dependent uncertainties. With the GP models, the time-dependent system reliability can be approximated directly by sampling-based methods such as the Monte Carlo simulation (MCS). Three case studies are introduced to demonstrate the efficiency and accuracy of the proposed approach.

scholarly journals Dynamical low rank approximation of time dependent random PDEs

2018 ◽  
Author(s):  
Eva Vidlickova ◽  
Fabio O. de Nobile ◽  
Eleonora Musharbash
Keyword(s):  
Numerical Test ◽  
Random Variables ◽  
Time Dependent ◽  
Low Rank ◽  
Independent Random Variables ◽  
Random Parameters ◽  
Low Rank Approximation ◽  
Linearly Independent ◽  
Geometric Point ◽  
Rank Approximation

Partial differential equations with random coefficients and input data arise in many real world applications. What they often have in common is that the data describing the PDE model are subject to uncertainties. The numerical approximation of statistics of this random solution poses several challenges, in particular when the number of random parameters is large and/or the parameter-to-solution map is complex. Therefore, effective surrogate or reduced models are of great need. We consider a class of time dependent PDEs with random parameters and search for an approximate solution in a separable form, i.e. at each time instant expressed as a linear combination of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation) in the spirit of a truncated Karhunen-Loeve expansion. Since the optimal deterministic and stochastic modes can significantly change over time, static versions, such as proper orthogonal decomposition or polynomial chaos expansion, may lose their effectiveness. Instead, here we consider a dynamical approach in which those modes are computed on-the-fly as solutions of suitable auxiliary evolution equations. From a geometric point of view, this approach corresponds to constraining the original dynamics to the manifold of fixed rank functions. The original equations are projected onto the tangent space of this manifold along the approximate trajectory. In this poster we recall the construction of the DLR method and give some implementation details. The spatial discretization is carried out by the finite element method and the discretization of the random variables relies on an adaptive choice of sparse grid. We will present some numerical test cases including the heat equation with a random diffusion coefficient and initial condition.

scholarly journals Markov Processes Involving q-Stirling Numbers

1997 ◽  
Vol 6 (2) ◽  
pp. 165-178 ◽  
Author(s):  
D. CRIPPA ◽  
K. SIMON ◽  
P. TRUNZ
Keyword(s):  
Markov Process ◽  
Normal Distribution ◽  
Markov Processes ◽  
Random Graphs ◽  
Transition Probabilities ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Stirling Numbers ◽  
Time Dependent

In this paper we consider the Markov process defined byP1,1=1, Pn,[lscr ]=(1−λn,[lscr ]) ·Pn−1,[lscr ] +λn,[lscr ]−1 ·Pn−1,[lscr ]−1for transition probabilities λn,[lscr ]=q[lscr ] and λn,[lscr ]=qn−1. We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q-Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

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