scholarly journals On the Accuracy of Error Propagation Calculations by Analytic Formulas Obtained for the Inverse Transformation

2019 ◽  
Vol 64 (3) ◽  
pp. 217
Author(s):  
V. I. Romanenko ◽  
N. V. Kornilovska
Keyword(s):  
Error Propagation ◽  
Random Variable ◽  
Mean Value ◽  
Inverse Transformation ◽  
First Order ◽  
Order Of Magnitude ◽  
The Mean ◽  
Normally Distributed

The accuracy of error propagation calculations is estimated for the transformation x → y = f(x) of the normally distributed random variable x. The estimation is based on the formulas for the error propagation obtained for the inverse transformation y → x of the normally distributed random variable y. In the general case, the calculation accuracy for the mean value and the variance of the random variable y is shown to be of the first order of magnitude in the variance of the random variable x.

Unified uncertainty analysis by the mean value first order saddlepoint approximation

2012 ◽  
Vol 46 (6) ◽  
pp. 803-812 ◽  
Author(s):  
Ning-Cong Xiao ◽  
Hong-Zhong Huang ◽  
Zhonglai Wang ◽  
Yu Liu ◽  
Xiao-Ling Zhang
Keyword(s):  
Uncertainty Analysis ◽  
Saddlepoint Approximation ◽  
Mean Value ◽  
First Order ◽  
The Mean

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Design Variable ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
The Mean ◽  
Sensitivity Method

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

Liquid helium films. I. The thickness of the film

1950 ◽  
Vol 203 (1072) ◽  
pp. 119-132 ◽  
Keyword(s):  
Liquid Helium ◽  
Dynamic Method ◽  
Mean Value ◽  
Helium Films ◽  
Helium Ii ◽  
Experimental Factor ◽  
Order Of Magnitude ◽  
The Mean

The thickness ( d ) of the helium II film and its variation with height ( H ) and temperature were measured by a dynamic method involving the oscillations of a meniscus in a capillary. The variation with height could be represented only approximately by the equation d = k/H n , as the effective value of n was greater at smaller values of H . The mean value of n over a range of heights from 0·5 to 5 cm. was 0·14, which is appreciably smaller than the values predicted by the theories so far advanced to explain the formation of the film. The order of magnitude of k was 2 x 10 -6 cm., but it varied slightly with the nature of the surface or some other experimental factor.

scholarly journals A New Conformable Fractional Derivative and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ahmed Kajouni ◽  
Ahmed Chafiki ◽  
Khalid Hilal ◽  
Mohamed Oukessou
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Conformable Fractional Derivative ◽  
First Order ◽  
Rolle’S Theorem ◽  
Power Rule ◽  
The Mean ◽  
Definition Of

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Uniqueness Theorem ◽  
Differential Operators ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Initial Value ◽  
First Order ◽  
Fractional Differential ◽  
The Mean ◽  
Fractional Differential Operators ◽  
First Order Differential Equations

AbstractWe generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

On Two First Order Reliability Methods for Computing the Non-Probabilistic Reliability Index

2014 ◽  
Vol 551 ◽  
pp. 648-652
Author(s):  
Xin Zhou Qiao
Keyword(s):  
Reliability Index ◽  
Limit State ◽  
Mean Value ◽  
Point Method ◽  
State Function ◽  
Design Point ◽  
First Order ◽  
Reliability Methods ◽  
The Mean ◽  
Mean Value Method

The two first order reliability methods (FORM) for computing the non-probabilistic reliability index, namely the mean-value method and the design-point method, are investigated. A performance comparison is presented between these two methods. The results show that: (1) the value of the reliability index of the mean-value method depends on the specific form of the limit state function, whereas the value of the reliability index of the design-point one does not;(2) the design-point method should be preferentially used in structural reliability assessment. The conclusions are verified by a numerical example.

On estimating the mean energy of sea waves from the highest waves in a record

1958 ◽  
Vol 247 (1248) ◽  
pp. 22-48 ◽  
Keyword(s):  
Probability Distribution ◽  
Standard Error ◽  
Random Variable ◽  
Ocean Waves ◽  
Mean Value ◽  
Sea Waves ◽  
Mutual Correlation ◽  
Narrow Spectrum ◽  
Asymptotic Formulae ◽  
The Mean

An analysis is made of the probability distribution of the largest values attained by a stationary random variable f ( t ) over a period of time containing several oscillations. Exact computations are made and asymptotic formulae are derived for the expectation and standard error of the first, second and third greatest maxima in terms of √ m 0 , the r. m. s. deviation of f ( t ) about its mean value, on the assumption that successive waves are uncorrelated; an analysis is also made of the corrections necessary to allow for mutual correlation when f ( t ) has a narrow spectrum. The results are applied to measurements from a 24 h record of ocean waves containing some 10000 oscillations.

scholarly journals Polynomial Representations of High-Dimensional Observations of Random Processes

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 123
Author(s):  
Pavel Loskot
Keyword(s):  
Random Variables ◽  
Mean Value ◽  
Weighted Sum ◽  
Computationally Efficient ◽  
First Order ◽  
Statistical Measures ◽  
Polynomial Representations ◽  
Univariate Polynomial ◽  
The Mean ◽  
Numerical Complexity

The paper investigates the problem of performing a correlation analysis when the number of observations is large. In such a case, it is often necessary to combine random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is obtained by approximating the Taylor expansion of a general multivariate scalar symmetric function by a univariate polynomial in the variable given as a simple sum of the original random variables. The mean value of the polynomial is then a weighted sum of statistical central sum-moments with the weights being application dependent. Computing the sum-moments is computationally efficient and amenable to mathematical analysis, provided that the distribution of the sum of random variables can be obtained. Among several auxiliary results also obtained, the first order sum-moments corresponding to sample means are used to reduce the numerical complexity of linear regression by partitioning the data into disjoint subsets. Illustrative examples provided assume the first and the second order Markov processes.

scholarly journals Research on Transmission Capacity Characteristics of the Cluster Flight Spacecraft Network

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Li Su ◽  
Jian Wu ◽  
Jinrong Mo
Keyword(s):  
Outage Probability ◽  
Random Variable ◽  
Mean Value ◽  
Transmission Capacity ◽  
Mean Value Theorem ◽  
Amplify And Forward ◽  
Decode And Forward ◽  
Different Types ◽  
The Mean ◽  
Capacity Performance

In order to analyze the transmission capacity performance of the cluster flight spacecraft network, there are two different types of outage performance theory which are derived in this paper. First of all, by applying the mean value theorem of integrals, the expression of the outage probability of decode-and-forward relaying is derived. Subsequently, according to the Macdonald random variable form, the expression of the outage probability of amplify-and-forward is derived. By simulating the transmission capacity of decode-and-forward, the transmission capacity characteristics of a single hop and dual hops are analyzed. The simulation results showed that transmission capacity performance changes with the change of the time slot in the orbital hyperperiod, and the transmission capacity of a dual-hop relay has better performance than a single-hop transmission in the cluster flight spacecraft network.

On Monte Carlo Estimates in Network Reliability

1994 ◽  
Vol 8 (2) ◽  
pp. 245-264 ◽  
Author(s):  
M. Lomonosov
Keyword(s):  
Monte Carlo ◽  
Markov Process ◽  
Critical Point ◽  
Natural Condition ◽  
Network Reliability ◽  
Random Variable ◽  
Mean Value ◽  
Reliability Measures ◽  
The Mean ◽  
Edge Failure

The paper considers representations of network reliability measures as the mean value of a random variable defined on the trajectories of a certain Markov process and investigates utility of such formulae for Monte Carlo (MC) estimating. Such an MC estimator is called (ε,δ)-polynomial if its relative error is less than ε with probability >1 – δ, for any sample size equal to or greater than a polynomial of ε-1, δ-1, and the size of the network. One of the main results: The suggested MC estimator for the disconnectedness probability of a multiterminal network is (ε,δ)-polynomial, under a certain natural condition on the edge failure probabilities. The method applies also to estimating the percolation critical point and certain equilibrium characteristics of renewal networks.

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