Probability Distribution and Spectral Analysis of Nonstationary Random Processes

2009 ◽  
pp. 19-24 ◽  
Author(s):  
P. F. Ribeiro ◽  
C. A. Duque
Keyword(s):  
Spectral Analysis ◽  
Probability Distribution ◽  
Random Processes

Extreme Values of Waves and Ship Responses Subject to the Markov Chain Condition

1979 ◽  
Vol 23 (03) ◽  
pp. 188-197
Author(s):  
Michel K. Ochi
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Probability Distribution ◽  
Random Process ◽  
Probability Distribution Function ◽  
Extreme Values ◽  
Random Processes ◽  
Chain Condition ◽  
Extreme Value ◽  
Spectral Bandwidth

This paper discusses the effect of statistical dependence of the maxima (peak values) of a stationary random process on the magnitude of the extreme values. A theoretical analysis of the extreme values of a stationary normal random process is made, assuming the maxima are subject to the Markov chain condition. For this, the probability distribution function of maxima as well as the joint probability distribution function of two successive maxima of a normal process having an arbitrary spectral bandwidth are applied to Epstein's theorem for evaluating the extreme values in a given sample under the Markov chain condition. A numerical evaluation of the extreme values is then carried out for a total of 14 random processes, including nine ocean wave records, with various spectral bandwidth parameters ranging from 0.11 to 0.78. From the results of the computations, it is concluded that the Markov concept is applicable to the maxima of random processes whose spectral bandwidth parameter, ɛ, is less than 0.5, and that the extreme values with and without the Markov concept are constant irrespective of the e-value, and the former is approximately 10 percent greater than the latter. It is also found that the sample size for which the extreme value reaches a certain level with the Markov concept is much less than that without the Markov concept. For example, the extreme value will reach a level of 4.0 (nondimensional value) in 1100 observations of the maxima with the Markov concept, while the extreme value will reach the same level in 3200 observations of the maxima without the Markov concept.

A Fast Calculation Procedure for Fatigue Reliability Estimates of Offshore Structures

2003 ◽  
Author(s):  
H. Karadeniz
Keyword(s):  
Spectral Analysis ◽  
Probability Distribution ◽  
Offshore Structures ◽  
Inertia Force ◽  
Stress Process ◽  
Fatigue Reliability ◽  
Uncertainty Modelling ◽  
Reliability Estimates ◽  
Reliability Method ◽  
Stress Spectrum

This paper presents formulations and procedure of a fast and efficient computation of fatigue reliability estimates of offshore structures, which eliminates repetitive execution of spectral analysis procedure so that it is performed only once for all reliability iterations. This is archived by a suitable uncertainty modelling and spectral formulation of the stress process. For this purpose, a new uncertainty variable is defined to represent all uncertainties in the stress spectrum, except those in the damping and inertia force coefficients, thicknesses of marine growths and structural members, which are represented by their own uncertainty variables. Apart from uncertainties in the stress spectrum, a detailed modelling of the fatigue-related uncertainties is presented. Uncertainties in SCF, damage model (S-N line), analytical modelling of the probability distribution of non-narrow banded stress process, long-term probability distribution of sea states and in the reference damage at which failure occurs, are all considered in the group of fatigue-related uncertainties. Formulation of the stress spectrum and stress spectral moments is presented explicitly in the idealized uncertainty space. Then, the failure function of the reliability analysis is expressed in terms of uncertainty variables as being independent of the spectral analysis. The advanced FORM reliability method is used to calculate the reliability index and to identify important uncertainty origins. The procedure presented in the paper is demonstrated by an example jacket type structure and the results are compared with previously calculated results using more sophisticated uncertainty modelling of the stress spectrum.

Random Processes and Fields

1996 ◽  
Author(s):  
Ilya Polyak
Keyword(s):  
Spectral Analysis ◽  
Correlation Analysis ◽  
Random Function ◽  
Numerical Algorithms ◽  
Random Processes ◽  
Multidimensional Case ◽  
Stationary Processes ◽  
Spectral Window ◽  
Estimation Procedures ◽  
Analogous Reasoning

In this chapter, the nonparametric methods of estimating the spectra and correlation functions of stationary processes and homogeneous fields are considered. It is assumed that the principal concepts and definitions of the corresponding theory are known (see Anderson, 1971; Box and Jenkins, 1976; Jenkins and Watts, 1968; Kendall and Stuart, 1967; Loeve, 1960; Parzen, 1966; Yaglom, 1986); therefore, only questions connected with the construction of numerical algorithms are studied. The basic results ranged from univariate process to multidimensional field are presented in Tables 3.1 and 3.2. These formulas make it possible to compare and trace the formal character of developing estimation procedures when the dimensionality is increasing. The schemes in these tables, as well as the formulas in the previous chapters, can be used for software development without any rearrangement. In part, this approach presents the application of the methods of Chapters 1 and 2 in evaluating random function characteristics. Of course, the final identification of the algorithm parameters (for example, the spectral window widths) can be made only through trial and error and by taking into account the character of the problem under study, that is, the physical properties of the processes and fields observed. The last section of this chapter presents results of the application of these methods to the analysis of some climatological fields. Here the basic results of the univariate spectral analysis are briefly discussed in order to develop algorithms for a multidimensional case by analogous reasoning. The complete description of the estimation procedures of the spectral and correlation analysis for univariate stationary process can be found, for example, in Jenkins and Watts, 1968.

APPLICATION OF KOTELNIKOV’S THEOREM TO ANALYSIS OF RANDOM PROCESSES

2019 ◽  
pp. 28-34
Author(s):  
O. V. Goriunov ◽  
S. V. Slovtsov
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Random Process ◽  
Spectral Characteristics ◽  
Random Processes ◽  
Statistical Characteristics ◽  
Signal Spectrum ◽  
The Fourier Transform ◽  
The Stability ◽  
Correlation And Spectral Analysis

Analysis of many dynamic tasks arising in engineering applications is associated with the construction of spectral characteristics. However, the application of spectral analysis to random oscillations, which in most cases describe real processes (technical, technological, etc.), has a number of features and limitations associated, in particular, with the anconvergence of the Fourier transform. The substantiated metrological evaluation of the spectra associated with the reliability of the applied results is complicated by the absence of a rigorous mathematical model of a random process. The above remarks were solved on the basis of application of Kotelnikov's theorem at decomposition of a random process on known eigenfunctions. The obtained decomposition allowed us to obtain a number of results in the field of correlation and spectral analysis of random processes: the stability of the ACF and the relationship with the statistical characteristics of the implementation is proved, the orthogonal decomposition of the random process in the form of a continuous function is presented, which allows us to consider the evaluation and analyze the characteristics of the realizations without the use of a fast Fourier transform; the natural relationship between ACF and spectral density for a time-limited signal is shown, and the symmetric form of recording the signal spectrum is justified.

A mathematical model of an oil and gas field development process

2010 ◽  
Vol 21 (3) ◽  
pp. 205-227 ◽  
Author(s):  
C. ATKINSON ◽  
R. ISANGULOV
Keyword(s):  
Mathematical Model ◽  
Probability Distribution ◽  
Development Process ◽  
Oil And Gas ◽  
Renewal Theory ◽  
Random Processes ◽  
Gas Field ◽  
Field Development ◽  
Oil And Gas Field ◽  
Complete Probability

A mathematical model of the development of an oil and gas field is presented. The field development process is treated as sequential in nature. Completion of a well and its production are considered to be random processes. The model uses results from renewal theory where the completion of a well and failure to produce economical amount of oil or gas are analogous to the failure of a component. In principle, the theory described can give the complete probability distribution associated with a field development. Explicit expressions are given for the expected value and variance of the number of completed wells.

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