A general frame for uncertainty propagation under multimodally distributed random variables

2020 ◽  
Vol 367 ◽  
pp. 113109 ◽  
Author(s):  
Xianghua Meng ◽  
Jie Liu ◽  
Lixiong Cao ◽  
Zhongbo Yu ◽  
Dongmin Yang
Keyword(s):  
Uncertainty Propagation ◽  
Random Variables ◽  
General Frame

An entropy-based global sensitivity analysis for the structures with both fuzzy variables and random variables

2012 ◽  
Vol 227 (2) ◽  
pp. 195-212 ◽  
Author(s):  
Tang Zhangchun ◽  
Lu Zhenzhou ◽  
Pan Wang ◽  
Zhang Feng
Keyword(s):  
Failure Probability ◽  
Uncertainty Propagation ◽  
Random Variables ◽  
Uncertain Variable ◽  
Importance Measure ◽  
Similar Measure ◽  
Practical Applications ◽  
Fuzzy Variables ◽  
Global Sensitivity ◽  
Uncertain Variables

Based on the entropy of the uncertain variable, a novel importance measure is proposed to identify the effect of the uncertain variables on the system, which is subjected to the combination of random variables and fuzzy variables. For the system with the mixture of random variables and fuzzy variables, the membership function of the failure probability can be obtained by the uncertainty propagation theory first. And then the effect of each input variable on the output response of the system can be evaluated by measuring the shift between entropies of two membership functions of the failure probability, obtained before and after the uncertainty elimination of the input variable. The intersecting effect of the multiple input variables can be calculated by the similar measure. The mathematical properties of the proposed global sensitivity indicators are investigated and proved in detail. A simple example is first employed to demonstrate the procedure of solving the proposed global sensitivity indicators and then the influential variables of four practical applications are identified by the proposed global sensitivity indicators.

Safety Envelope for Load Tolerance and its Application to Fatigue Reliability Design

2005 ◽  
Vol 128 (4) ◽  
pp. 919-927 ◽  
Author(s):  
Haoyu Wang ◽  
Nam H. Kim ◽  
Yoon-Jun Kim
Keyword(s):  
Load Capacity ◽  
Uncertainty Propagation ◽  
Random Variables ◽  
Function Evaluation ◽  
Fatigue Reliability ◽  
Reliability Design ◽  
Structural Part ◽  
Average Value ◽  
Current Design ◽  
Load Tolerance

In this paper, a safety envelope concept for load tolerance is introduced. This shows the capacity of the current design as a future reference for design upgrade, maintenance, and control. The safety envelope is applied to estimate the load tolerance of a structural part with respect to the fatigue reliability. First, the dynamic load history is decomposed into the average value and amplitude, which are modeled as random variables. Second, through fatigue analysis and uncertainty propagation, the reliability is calculated. Last, based on the implicit function evaluation for the reliability, the boundary of the safety envelope is calculated numerically. The effect of different distribution types of random variables is then investigated to identify the conservative envelope. In order to improve the efficiency of searching the boundary, probabilistic sensitivity information is utilized. When the relationship between the safety of the system and the load tolerance is linear or mildly nonlinear, the linear estimation of the safety envelope turns out to be accurate and efficient. During the application of the algorithm, a stochastic response surface of logarithmic fatigue life with respect to the load capacity coefficient is constructed, and the Monte Carlo simulation is utilized to calculate the reliability and its sensitivities.

A Comparative Study of Uncertainty Propagation Methods for Black-Box Type Functions

2007 ◽  
Author(s):  
Sang Hoon Lee ◽  
Wei Chen
Keyword(s):  
Comparative Study ◽  
Uncertainty Propagation ◽  
Computational Cost ◽  
Random Variables ◽  
Quadrature Rule ◽  
Tail Probability ◽  
Black Box ◽  
Test Problems ◽  
Significant Interaction Effect ◽  
Propagation Methods

It is an important step in deign under uncertainty to select an appropriate uncertainty propagation (UP) method considering the characteristics of the engineering systems at hand, the required level of UP associated with the probabilistic design scenario, and the required accuracy and efficiency levels. Many uncertainty propagation methods have been developed in various fields, however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several UP methods, including a few recent methods that have received growing attention, is performed. The full factorial numerical integration (FFNI), the univariate dimension reduction method (UDR), and the polynomial chaos expansion (PCE) are implemented and applied to several test problems with different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function (PDF) construction. It is found that the FFNI with the moment matching quadrature rule shows good accuracy but the computational cost becomes prohibitive as the number of input random variables increases. The accuracy and efficiency of the UDR method for moment estimations appear to be superior when there is no significant interaction effect in the performance function. Both FFNI and UDR are very robust against the non-normality of input variables. The PCE is implemented in combination with FFNI for coefficients estimation. The PCE method is shown to be a useful approach when a complete PDF description is desired. Inverse Rosenblatt transformation is used to treat non-normal inputs of PCE, however, it is shown that the transformation may result in the degradation of accuracy of PCE. It is also shown that in black-box type of system the performance and convergence of PCE highly depend on the method adopted to estimate its coefficients.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

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