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

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 .

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.

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.

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.

A Mean Value Reliability Method for Bimodal Distributions

2017 ◽  
Author(s):  
Zhangli Hu ◽  
Xiaoping Du
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Random Variable ◽  
Mean Value ◽  
Traffic Load ◽  
First Order ◽  
Reliability Method ◽  
Basic Random Variable ◽  
Two Peaks

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. For example, the random load of a bridge may have two peaks, with a distribution consisting of a weighted sum of two normal distributions, suggested by traffic load data. When binomial variables are involved, traditional reliability methods, such as the First Order Second Moment (FOSM) method and the First Order Reliability Method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation for bimodal variables and then employs a mean value reliability method to accurately predict the reliability. A limit-state function is at first approximated with the first order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The saddlepoint approximation is then applied to estimate the reliability. Examples show that the new method is more accurate than FOSM and FORM.

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