scholarly journals First passage probability assessment of stationary non-Gaussian process using the third-order polynomial transformation

2018 ◽  
Vol 22 (1) ◽  
pp. 187-201 ◽  
Author(s):  
Yan-Gang Zhao ◽  
Long-Wen Zhang ◽  
Zhao-Hui Lu ◽  
Jun He
Keyword(s):  
Gaussian Process ◽  
Probability Assessment ◽  
Third Order ◽  
First Passage ◽  
The Third ◽  
Polynomial Transformation ◽  
Order Polynomial ◽  
First Passage Probability ◽  
Non Gaussian ◽  
First Four Moments

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosis > 3) and “hardening” responses (with narrower tails; for example, kurtosis < 3). It is found that the proposed method has better accuracy for estimating the first passage probabilities than the existing methods, which provides an efficient and rational tool for the first passage probability assessment of stationary non-Gaussian process.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (01) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x 0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ &lt; τ &lt; ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

The influence of the regression model and final speed criteria on the reliability of lactate threshold determined by the Dmax method in endurance-trained runners

2016 ◽  
Vol 41 (10) ◽  
pp. 1039-1044 ◽  
Author(s):  
Júlio César Camargo Alves ◽  
Cecília Segabinazi Peserico ◽  
Geraldo Angelo Nogueira ◽  
Fabiana Andrade Machado
Keyword(s):  
Regression Model ◽  
Regression Models ◽  
Lactate Threshold ◽  
Polynomial Regression ◽  
Third Order ◽  
The Third ◽  
Constant Regression ◽  
Order Polynomial ◽  
Final Speed ◽  
Polynomial Regression Model

Few studies verified the reliability of the lactate threshold determined by Dmax method (LTDmax) in runners and it remains unclear the effect of the regression model and the final speed on the reliability of LTDmax. This study aimed to examine the test–retest reliability of the speed at LTDmax in runners, considering the effects of the regression models (exponential-plus-constant vs third-order polynomial) and final speed criteria (complete vs proportional). Seventeen male, recreational runners performed 2 identical incremental exercise tests, with increments of 1 km·h–1 each for 3 min on treadmill to determine peak treadmill speed (Vpeak) and lactate threshold. Earlobe capillary blood samples were collected during rest between the stages. The Vpeak was defined as the speed of the last complete stage (complete final speed criterion) and as the speed of the last complete stage added to the fraction of the incomplete stage (proportional final speed criterion). Lactate threshold was determined from exponential-plus-constant and from third-order polynomial regression models with both complete and proportional final speed criteria and from fixed blood lactate level of 3.5 mmol·L−1 (LT3.5mM). The LTDmax obtained from the exponential-plus-constant regression model presented higher reliability (coefficient of variation (CV) ≤ 3.7%) than the LTDmax calculated from the third-order polynomial regression model (CV ≤ 5.8%) and LT3.5mM (CV = 5.4%). The proportional final speed criterion is more appropriate when using the exponential-plus-constant regression model, but less appropriate when using the third-order polynomial regression model. In conclusion, exponential-plus-constant using the proportional final speed criterion is preferred over LT3.5mM and over third-order polynomial regression model to determine a reliable LTDmax.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (1) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

scholarly journals More on the Non-Gaussianity of Perturbations in a Nonminimal Inflationary Model

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
R. Shojaee ◽  
K. Nozari ◽  
F. Darabi
Keyword(s):  
Scalar Field ◽  
Observational Data ◽  
Coupling Parameter ◽  
Extended Model ◽  
Inflationary Model ◽  
Cosmological Perturbations ◽  
Third Order ◽  
The Third ◽  
Non Gaussian

We study nonlinear cosmological perturbations and their possible non-Gaussian character in an extended nonminimal inflation where gravity is coupled nonminimally to both the scalar field and its derivatives. By expansion of the action up to the third order, we focus on the nonlinearity and non-Gaussianity of perturbations in comparison with recent observational data. By adopting an inflation potential of the form V(ϕ)=1/nλϕn, we show that, for n=4, for instance, this extended model is consistent with observation if 0.013<λ<0.095 in appropriate units. By restricting the equilateral amplitude of non-Gaussianity to the observationally viable values, the coupling parameter λ is constrained to the values λ<0.1.

Some aspects of nonminimal inflation driven by a superpotential

2016 ◽  
Vol 26 (07) ◽  
pp. 1750058
Author(s):  
Kourosh Nozari ◽  
Narges Rashidi
Keyword(s):  
Spectral Index ◽  
Second Order ◽  
Numerical Results ◽  
Nonminimal Coupling ◽  
Third Order ◽  
The Third ◽  
Scalar Spectral Index ◽  
Non Gaussian

We study a nonminimal inflation which is driven by a superpotential. By adopting the Arnowitt–Deser–Misner formalism, we explore the primordial perturbations and its non-Gaussianity in this framework. By expanding the action up to the second-order in perturbations, we seek the scalar spectral index, its running and the tensor-to-scalar ratio. In this regard, we find the ranges of the nonminimal coupling and superpotential parameters which lead to the observationally viable perturbations parameters. The non-Gaussian feature in both the equilateral and orthogonal configurations in this setup, is studied by exploring the third-order action. We show that in some ranges of the nonminimal and superpotential parameters, the model predicts large non-Gaussianity. By comparing the numerical results with Planck2015 data, we test the viability of the model and find some constraints on the model’s parameters space.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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