Simulation of Stochastic Processes by Spectral Representation

1991 ◽  
Vol 44 (4) ◽  
pp. 191-204 ◽  
Author(s):  
Masanobu Shinozuka ◽  
George Deodatis
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Autocorrelation Function ◽  
Structural Engineering ◽  
Spectral Representation ◽  
Target Function ◽  
Mean Value ◽  
Random Loading ◽  
Cosine Series ◽  
Spectral Representation Method

The subject of this paper is the simulation of one-dimensional, uni-variate, stationary, Gaussian stochastic processes using the spectral representation method. Following this methodology, sample functions of the stochastic process can be generated with great computational efficiency using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number N of the terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the temporally-averaged mean value and the autocorrelation function are identical with the corresponding targets, when the averaging takes place over the fundamental period of the cosine series. The most important property of the simulated stochastic process is that it is asymptotically Gaussian as N → ∞. Another attractive feature of the method is that the cosine series formula can be numerically computed efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in engineering mechanics and structural engineering. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).

Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation

1996 ◽  
Vol 49 (1) ◽  
pp. 29-53 ◽  
Author(s):  
Masanobu Shinozuka ◽  
George Deodatis
Keyword(s):  
Spectral Density ◽  
Power Spectral Density ◽  
Autocorrelation Function ◽  
Structural Engineering ◽  
Spectral Representation ◽  
Target Function ◽  
Stochastic Field ◽  
Cosine Series ◽  
Stochastic Fields ◽  
Power Spectral

The subject of this paper is the simulation of multi-dimensional, homogeneous, Gaussian stochastic fields using the spectral representation method. Following this methodology, sample functions of the stochastic field can be generated using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic field when the number of terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the spatially-averaged mean value, autocorrelation function and power spectral density function are identical with the corresponding targets, when the averaging takes place over the multi-dimensional domain associated with the fundamental period of the cosine series. Another property of the simulated stochastic field is that it is asymptotically Gaussian as the number of terms in the cosine series approaches infinity. The most important feature of the method is that the cosine series formula can be numerically computed very efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in structural engineering, engineering mechanics and physics. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).

scholarly journals Evolution and mass extinctions as lognormal stochastic processes

2014 ◽  
Vol 13 (4) ◽  
pp. 290-309 ◽  
Author(s):  
Claudio Maccone
Keyword(s):  
Mathematical Model ◽  
Stochastic Process ◽  
Molecular Evolution ◽  
Stochastic Processes ◽  
Neutral Theory ◽  
Mean Value ◽  
Darwinian Evolution ◽  
List Type ◽  
Mass Extinctions ◽  
Human History

AbstractIn a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely calledEvo-SETI. The relevant mathematical tools are:(1)Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black–Sholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time.(2)The probability distributions known asb-lognormals, i.e. lognormals starting at a certain positive instantb>0 rather than at the origin. Theseb-lognormals were then forced by us to have their peak value located on the exponential mean-value curve of the GBM (Peak-Locus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists callCladistics.(3)The (Shannon)entropyof suchb-lognormals is then seen to represent the ‘degree of progress’ reached by each living organism or by each big set of living organisms, like historic human civilizations. Having understood this fact, human history may then be cast into the language ofb-lognormals that are more and more organized in time (i.e. having smaller and smaller entropy, or smaller and smaller ‘chaos’), and have their peaks on the increasing GBM exponential. This exponential is thus the ‘trend of progress’ in human history.(4)All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics).(5)But the most striking new result is that the well-known ‘Molecular Clock of Evolution’, namely the ‘constant rate of Evolution at the molecular level’ as shown by Kimura's Neutral Theory of Molecular Evolution,identifieswith growth rate of the entropy of our Evo-SETI model, because they both grewlinearlyin time since the origin of life.(6)Furthermore, we apply our Evo-SETI model to lognormal stochastic processesother than GBMs.For instance, we provide two models for the mass extinctions that occurred in the past: (a) one based on GBMs and (b) the other based on aparabolicmean value capable of covering both the extinction and the subsequent recovery of life forms.(7)Finally, we show that the Markov & Korotayev (2007, 2008) model for Darwinian Evolution identifies with an Evo-SETI model for which the mean value of the underlying lognormal stochastic process is acubicfunction of the time.In conclusion: we have provided a new mathematical model capable of embracing molecular evolution, SETI and entropy into a simple set of statistical equations based uponb-lognormals and lognormal stochastic processes with arbitrary mean, of which the GBMs are theparticular case of exponential growth.

Analysis and Probabilistic Modeling of the Stationary Ice Loads Stochastic Process With Lognormal Distribution

2014 ◽  
Author(s):  
Petr Zvyagin ◽  
Kirill Sazonov
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Normal Distribution ◽  
Autocorrelation Function ◽  
Strain Gauge ◽  
Lognormal Distribution ◽  
Probabilistic Modeling ◽  
Stationary Stochastic Process ◽  
Strain Gauge Transducer ◽  
Ice Loads

Until recent times researchers who investigated ice loads stochastic processes usually stated the fact of normal distribution for them. In the paper the model of a stationary stochastic process with a lognormal distribution for ice loads is offered. This model relates to the strain gauge transducer ice loads measurements as well as to some examples considered in different papers that were published earlier. For this model dependencies of the autocorrelation function were found that allows to simulate the ice loads process relatively easily. The procedure of such a simulation is described in details and the example of the analysis and simulation ice loads measurements is provided.

scholarly journals Über Linear Passive Transformationen Stochastischer Prozesse I

1968 ◽  
Vol 23 (10) ◽  
pp. 1430-1438 ◽  
Author(s):  
J. Keller
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Mean Value ◽  
Output Process ◽  
Passive Systems ◽  
Physical Systems ◽  
General Properties ◽  
Input And Output ◽  
The Mean

The theory of linear passive systems, developed by KÖNIG and MEIXNER, is extended to the case where the input is not a well determined function of time but rather a stochastic process. In this case the answer of the system generally will also be a stochastic process. The input and output processes are connected by a linear passive transformation (LPT). Some examples are given of physical systems which may be described by LPT of stochastic processes. General properties of the mean value and the dispersion of the output process are derived.

Simulation of Stochastic Processes and Fields for Monte Carlo Simulation Applications: Some Recent Developments

1995 ◽  
Author(s):  
George Deodatis ◽  
Radu Popescu ◽  
Jean H. Prevost
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Stochastic Processes ◽  
Spectral Representation ◽  
Distribution Functions ◽  
Spectral Density Matrix ◽  
Spectral Representation Method ◽  
Cross Spectral Density ◽  
Representation Method ◽  
Non Gaussian

Abstract Two of the latest developments concerning the spectral representation method (used to simulate stochastic processes and fields) are presented in this paper. The first one introduces an extension of the spectral representation method to simulate non-stationary stochastic vector processes with evolutionary power. The proposed simulation formula is simple and straightforward and generates sample functions of the vector process according to a prescribed non-stationary cross-spectral density matrix. The second development introduces another extension of the spectral representation method to simulate multi-dimensional, multi-variate, non-Gaussian stochastic fields. In this case, sample functions are generated according to a prescribed cross-spectral density matrix and prescribed (non-Gaussian) probability distribution functions. Numerical examples are provided for both developments.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

An efficient Cholesky decomposition and applications for the simulation of large-scale random wind velocity fields

2018 ◽  
Vol 22 (6) ◽  
pp. 1255-1265 ◽  
Author(s):  
Yongle Li ◽  
Chuanjin Yu ◽  
Xingyu Chen ◽  
Xinyu Xu ◽  
Koffi Togbenou ◽  
...  
Keyword(s):  
Wind Velocity ◽  
Large Scale ◽  
Spectral Representation ◽  
Cholesky Decomposition ◽  
Velocity Fields ◽  
Data Driven ◽  
Long Span Bridges ◽  
Long Span ◽  
Spectral Representation Method ◽  
Representation Method

A growing number of long-span bridges are under construction across straits or through valleys, where the wind characteristics are complex and inhomogeneous. The simulation of inhomogeneous random wind velocity fields on such long-span bridges with the spectral representation method will require significant computation resources due to the time-consuming issues associated with the Cholesky decomposition of the power spectrum density matrixes. In order to improve the efficiency of the decomposition, a novel and efficient formulation of the Cholesky decomposition, called “Band-Limited Cholesky decomposition,” is proposed and corresponding simulation schemes are suggested. The key idea is to convert the coherence matrixes into band matrixes whose decomposition requires less computational cost and storage. Subsequently, each decomposed coherence matrix is also a band matrix with high sparsity. As the zero-valued elements have no contribution to the simulation calculation, the proposed method is further expedited by limiting the calculation to the non-zero elements only. The proposed methods are data-driven ones, which can be applicable broadly for simulating many complicated large-scale random wind velocity fields, especially for the inhomogeneous ones. Through the data-driven strategies presented in the study, a numerical example involving inhomogeneous random wind velocity field simulation on a long-span bridge is performed. Compared to the traditional spectral representation method, the simulation results are with high accuracy and the entire simulation procedure is about 2.5 times faster by the proposed method for the simulation of one hundred wind velocity processes.

Sign in / Sign up

Export Citation Format

Share Document