scholarly journals RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

2021 ◽  
pp. 97-103
Author(s):  
Olga Prishchenko ◽  
Nadezhda Cheremskaya
Keyword(s):  
Random Process ◽  
Adjoint Operator ◽  
Random Processes ◽  
Real Spectrum ◽  
Nonstationary Random Process ◽  
Finite Segment ◽  
The Real ◽  
Random Functions ◽  
Internal States ◽  
Infinite Multiplicity

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0,  a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator  promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Information Theoretic Equivalent Bandwidths of Random Processes and Their Applications

2007 ◽  
Vol 46 (02) ◽  
pp. 110-116 ◽  
Author(s):  
S. Kikkawa ◽  
H. Yoshida
Keyword(s):  
Random Process ◽  
Random Processes ◽  
Systolic Murmur ◽  
Nonstationary Random Process ◽  
Frequency Distributions ◽  
Copula Theory ◽  
Information Theoretic ◽  
Time Frequency ◽  
New Class ◽  
Stationary Random Processes

Summary Objectives : Since most of the biomedical signals, such as electroencephalogram (EEG), electromyogram (EMG) and phonocardiogram (PCG), are nonstationary random processes, the time-frequency analysis has recently been extensively applied to those signals in order to achieve precise characterization and classification. In this paper, we have first defined a new class of information theoretic equivalent bandwidths (EBWs) of stationary random processes, then instantaneous EBWs (IEBWs) using nonnegative time-frequenc distributions have been defined in order to track the change of the EBW of a nonstationary random process. Methods : The new class of EBWs which includes spectral flatness measure (SFM) for stationary random processes is defined by using generalized Burg entropy. Generalized Burg entropy is derived from the relation between Rényi entropy and Rényi information divergence of order α. In order to track the change of EBWs of a nonstationary random process, the IEBWs are defined on the nonnegative time-frequency distributions, which are constructed by the Copula theory. Results : We evaluate the IEBWs for a first order stationary auto-regressive (AR) process and three types of time-varying AR processes. The results show that the IEBWs proposed here properly represent a signal bandwidth. In practical application to PCGs, the proposed method was successful in extracting the information that the bandwidth of the innocent systolic murmur was much smaller than that of the abnormal systolic murmur. Conclusions : We have defined new information theoretic EBWs and have proposed a novel method to track the change of the IEBWs. Some computer simulation showed effectiveness of the methods. Applying the IEBWs to PCGs, we could extract some features of a systolic murmur.

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.

On the two sided barrier problem

1965 ◽  
Vol 2 (01) ◽  
pp. 79-87
Author(s):  
Masanobu Shinozuka
Keyword(s):  
Random Process ◽  
Ground Motion ◽  
Lower Bounds ◽  
Present Method ◽  
Structural Response ◽  
Random Processes ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Long Run ◽  
Barrier Problem

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ 1, λ 2) for 0 ≦ t ≦ T, where λ 1 and λ 2 are positive constants. The random process needs to be neither stationary, Gaussian nor purely random (white noise). In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake. Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

scholarly journals Real Tropicalization and Analytification of Semialgebraic Sets

2020 ◽  
Author(s):  
Philipp Jell ◽  
Claus Scheiderer ◽  
Josephine Yu
Keyword(s):  
Inverse Limit ◽  
Point Of View ◽  
General Point ◽  
Real Spectrum ◽  
Closed Field ◽  
Topological Properties ◽  
Absolute Value ◽  
The Real ◽  
Semialgebraic Set ◽  
Refined Version

Abstract Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subseteq K^n$ we define a space $S_r^{{\operatorname{an}}}$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $X$ is an algebraic variety, we show that $X_r^{{\operatorname{an}}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

On random motions with velocities alternating at Erlang-distributed random times

2001 ◽  
Vol 33 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Stochastic Process ◽  
Random Process ◽  
Bessel Function ◽  
Real Line ◽  
Renewal Process ◽  
The Real ◽  
Alternating Renewal Process ◽  
Transition Densities ◽  
Random Motions ◽  
Random Times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

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