scholarly journals Analytic results for oscillatory systems with extremal dynamic properties

2014 ◽  
Vol 24 (4) ◽  
pp. 771-784
Author(s):  
Henryk Górecki ◽  
Mieczysław Zaczyk
Keyword(s):  
System Design ◽  
Characteristic Equation ◽  
Extreme Values ◽  
Dynamic Properties ◽  
Extreme Value ◽  
Important Criterion ◽  
Oscillatory Systems ◽  
Analytical Formulae

Abstract The maximal value of the error is the most important criterion in system design. It is also the most difficult one. For that reason there exist many other criteria. The extreme value of the error represents the attainable accuracy which can be obtained and the corresponding extreme time gives information about how fast the transients are. The extreme values of the error and the corresponding time are treated here as functions of the roots of the characteristic equation. The proposed analytical formulae allow designing systems with prescribed dynamic properties.

scholarly journals Design of the oscillatory systems with the extremal dynamic properties

2014 ◽  
Vol 62 (2) ◽  
pp. 241-253 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Practical Problem ◽  
Dynamic Properties ◽  
Oscillatory Systems ◽  
Complex Roots ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value

Abstract In this article the problem of determination of such coefficients a1, a2, ..., an and eigenvalues s1, s2, ..., sn of the characteristic equation which yield required extremal values of the solution x(t) at the extremal value τ of time is solved. The extremal values of x(τ ) and τ are treated as functions of the roots s1, s2, ..., sn. The analytical formulae enable us to design the systems with prescribed dynamic properties. For solution of the problem the properties of symmetrical equations are used. The method is illustrated by an example of the equation of 4-th degree. The regions of the different kinds of the roots: real, with one pair of complex and two pairs of complex roots are illustrated. A practical problem is shown.

Design of systems with extremal dynamic properties

2013 ◽  
Vol 61 (3) ◽  
pp. 563-567 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Chemical Industry ◽  
Dynamic Properties ◽  
Dynamic Error ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value ◽  
Zeros And Poles

Abstract In this article the problem of determination of coefficients a1, a2, . . . , an of the characteristic equation which yield required extremal values of the solution x(t) at extremal values τ of time is solved. The extremal values of x(t) and τ are treated as functions of the roots s1, s2, . . . , sn. The analytical formulae enable to design the systems with prescribed dynamic properties. The zeros and poles can be located using the known method. The extremal dynamic error x(t) is the most important property of the behaviour of the system. This extremal value of the dynamic error has fundamental role in the chemical industry where for example overrising temperature or pressure can lead to an explosion. A second very important property is the extremal time τ connected with the extremal value of the error. This property is essential in the electroenergetic system, which can be destroyed by the overvoltages waves.

scholarly journals Nonparametric Estimation of Multivariate Extreme Value Copulas with Known and Unknown Marginal Distributions

2021 ◽  
Vol 2068 (1) ◽  
pp. 012003
Author(s):  
Ayari Samia ◽  
Mohamed Boutahar
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Extreme Values ◽  
Empirical Distribution ◽  
Extreme Value ◽  
Dependence Function ◽  
Marginal Distributions ◽  
Nonparametric Estimators ◽  
Monte Carlo Experiments ◽  
Multivariate Extreme Values

Abstract The purpose of this paper is estimating the dependence function of multivariate extreme values copulas. Different nonparametric estimators are developed in the literature assuming that marginal distributions are known. However, this assumption is unrealistic in practice. To overcome the drawbacks of these estimators, we substituted the extreme value marginal distribution by the empirical distribution function. Monte Carlo experiments are carried out to compare the performance of the Pickands, Deheuvels, Hall-Tajvidi, Zhang and Gudendorf-Segers estimators. Empirical results showed that the empirical distribution function improved the estimators’ performance for different sample sizes.

Consideration of Epistemic Uncertainty in Extreme Wave Height Estimation

2014 ◽  
Author(s):  
Ryota Wada ◽  
Takuji Waseda
Keyword(s):  
Gulf Of Mexico ◽  
Observational Data ◽  
Wave Height ◽  
Extreme Values ◽  
Epistemic Uncertainty ◽  
Extreme Value Distribution ◽  
Predictive Distribution ◽  
Extreme Value ◽  
Posterior Predictive Distribution ◽  
Value Estimation

Extreme value estimation of significant wave height is essential for designing robust and economically efficient ocean structures. But in most cases, the duration of observational wave data is not efficient to make a precise estimation of the extreme value for the desired period. When we focus on hurricane dominated oceans, the situation gets worse. The uncertainty of the extreme value estimation is the main topic of this paper. We use Likelihood-Weighted Method (LWM), a method that can quantify the uncertainty of extreme value estimation in terms of aleatory and epistemic uncertainty. We considered the extreme values of hurricane-dominated regions such as Japan and Gulf of Mexico. Though observational data is available for more than 30 years in Gulf of Mexico, the epistemic uncertainty for 100-year return period value is notably large. Extreme value estimation from 10-year duration of observational data, which is a typical case in Japan, gave a Coefficient of Variance of 43%. This may have impact on the design rules of ocean structures. Also, the consideration of epistemic uncertainty gives rational explanation for the past extreme events, which were considered as abnormal. Expected Extreme Value distribution (EEV), which is the posterior predictive distribution, defined better extreme values considering the epistemic uncertainty.

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.

scholarly journals Implicit max-stable extremal integrals

Extremes ◽  
2020 ◽  
Author(s):  
D. Kremer
Keyword(s):  
Loss Function ◽  
Open Problem ◽  
Stochastic Integral ◽  
Extreme Values ◽  
Stable Distributions ◽  
Extreme Value ◽  
Point Of View ◽  
Extreme Value Distributions ◽  
Application Point ◽  
Deterministic Function

Abstract Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α-Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach (2016) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8, 237–266 (2005)) and, at the same time, reveals striking parallels.

Covariate Extreme Value Analysis Using Wave Spectral Partitioning

2020 ◽  
Vol 37 (5) ◽  
pp. 873-888 ◽  
Author(s):  
Jesús Portilla-Yandún ◽  
Edwin Jácome
Keyword(s):  
Extreme Values ◽  
Wind Waves ◽  
Spectral Energy Distribution ◽  
Wave Spectrum ◽  
Wave Climate ◽  
Extreme Value ◽  
Extreme Value Analysis ◽  
Spectral Energy ◽  
Value Analysis ◽  
Spectral Partitioning

AbstractAn important requirement in extreme value analysis (EVA) is for the working variable to be identically distributed. However, this is typically not the case in wind waves, because energy components with different origins belong to separate data populations, with different statistical properties. Although this information is available in the wave spectrum, the working variable in EVA is typically the total significant wave height Hs, a parameter that does not contain information of the spectral energy distribution, and therefore does not fulfill this requirement. To gain insight in this aspect, we develop here a covariate EVA application based on spectral partitioning. We observe that in general the total Hs is inappropriate for EVA, leading to potential over- or underestimation of the projected extremes. This is illustrated with three representative cases under significantly different wave climate conditions. It is shown that the covariate analysis provides a meaningful understanding of the individual behavior of the wave components, in regard to the consequences for projecting extreme values.

Tree-dependent extreme values: the exponential case

1990 ◽  
Vol 27 (01) ◽  
pp. 124-133 ◽  
Author(s):  
Vijay K. Gupta ◽  
Oscar J. Mesa ◽  
E. Waymire
Keyword(s):  
Branching Process ◽  
Extreme Values ◽  
Rooted Tree ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Extinction Time ◽  
Brownian Excursion ◽  
Weighted Trees

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

scholarly journals EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?

2019 ◽  
Vol 34 (2) ◽  
pp. 200-220
Author(s):  
Jingjing Zou ◽  
Richard A. Davis ◽  
Gennady Samorodnitsky
Keyword(s):  
Order Statistics ◽  
Extreme Values ◽  
Sample Path ◽  
Estimation Procedure ◽  
Extreme Value ◽  
Hill Estimator ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Heavy Tailed ◽  
The Hill

AbstractIn this paper, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for these data exhibited a smooth and increasing “sample path” as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill's estimator. We illustrate how this approach works in both simulations and real data examples.

scholarly journals Statistics of the largest geomagnetic storms per solar cycle (1844-1993)

1997 ◽  
Vol 15 (6) ◽  
pp. 719-728 ◽  
Author(s):  
D. M. Willis ◽  
P. R. Stevens ◽  
S. R. Crothers
Keyword(s):  
Statistical Analysis ◽  
Solar Cycle ◽  
Geomagnetic Storm ◽  
Extreme Values ◽  
Geomagnetic Storms ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Relative Dispersion ◽  
Solar Cycles ◽  
Aa Index

Abstract. A previous application of extreme-value statistics to the first, second and third largest geomagnetic storms per solar cycle for nine solar cycles is extended to fourteen solar cycles (1844–1993). The intensity of a geomagnetic storm is measured by the magnitude of the daily aa index, rather than the half-daily aa index used previously. Values of the conventional aa index (1868–1993), supplemented by the Helsinki Ak index (1844–1880), provide an almost continuous, and largely homogeneous, daily measure of geomagnetic activity over an interval of 150 years. As in the earlier investigation, analytic expressions giving the probabilities of the three greatest storms (extreme values) per solar cycle, as continuous functions of storm magnitude (aa), are obtained by least-squares fitting of the observations to the appropriate theoretical extreme-value probability functions. These expressions are used to obtain the statistical characteristics of the extreme values; namely, the mode, median, mean, standard deviation and relative dispersion. Since the Ak index may not provide an entirely homogeneous extension of the aa index, the statistical analysis is performed separately for twelve solar cycles (1868–1993), as well as nine solar cycles (1868–1967). The results are utilized to determine the expected ranges of the extreme values as a function of the number of solar cycles. For fourteen solar cycles, the expected ranges of the daily aa index for the first, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude, contrary to the situation for the half-daily aa index over nine solar cycles. The observed range of the first extreme daily aa index for fourteen solar cycles is 159–352 nT and for twelve solar cycles is 215–352 nT. In a group of 100 solar cycles the expected ranges are expanded to 137–539 and 177–511 nT, which represent increases of 108% and 144% in the respective ranges. Thus there is at least a 99% probability that the daily aa index will satisfy the condition aa < 550 for the largest geomagnetic storm in the next 100 solar cycles. The statistical analysis is used to infer that remarkable conjugate auroral observations on the night of 16 September 1770, which were recorded during the first voyage of Captain Cook to Australia, occurred during an intense geomagnetic storm.

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