On the degree of second-order non-circularity of complex random variables

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

Download Full-text

Characterization of a class of second-order density functions

Journal of Applied Probability ◽

10.2307/3212304 ◽

1967 ◽

Vol 4 (1) ◽

pp. 123-129 ◽

Cited By ~ 1

Author(s):

C. B. Mehr

Keyword(s):

Exponential Distribution ◽

Gamma Distribution ◽

Gaussian Distribution ◽

Random Variables ◽

Second Order ◽

Independent Random Variables ◽

Density Functions ◽

Linear Filtering ◽

Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

Download Full-text

A note on second-order stochastic dominance for linear combinations of dependent Bernoulli random variables

Communication in Statistics- Theory and Methods ◽

10.1080/03610926.2021.1971248 ◽

2021 ◽

pp. 1-7

Author(s):

Martín Egozcue ◽

Luis Fuentes García

Keyword(s):

Stochastic Dominance ◽

Random Variables ◽

Second Order ◽

Linear Combinations ◽

Bernoulli Random Variables ◽

Order Stochastic Dominance ◽

Second Order Stochastic Dominance

Download Full-text

Resonance Reliability Analysis of Aeroengine Compressor Rotor Blade

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.472.79 ◽

2014 ◽

Vol 472 ◽

pp. 79-84

Author(s):

Hai Feng Gao ◽

Guang Chen Bai

Keyword(s):

Vibration Frequency ◽

Rotor Blade ◽

Random Variables ◽

Second Order ◽

Rotor Blades ◽

Rotor Speed ◽

Blade Vibration ◽

Compressor Rotor ◽

Set Up ◽

The Impact

To describe the frequency distribution of the rotor blades and improve the optimization, resonance reliability of the rotor blades was analyzed in this paper. Considering the variety of rand-om variables, we jointly used finite element method and response surface method. The Campbell diagram was set up to describe blade resonance by analyzing the compressor rotor blade vibration characteristics. For the second-order vibration failure of the rotor blade, we considered the impact of random variables with the rotor blade material, the blade dimension and the rotor speed. The pro-bability distribution and allowable reliability of the second-order vibration frequency was calculated, and the sensitivity of the random variables influencing vibration frequency was completed. The res-ults show that the resonance reliability with the confidence level 0.95 of the rotor blade are = 0.99753 with the excited order =4 and =0.99767 with the excited order =5,and basically ag-ree with the design requirements when the rotor speed =9916.2, and the factors mainly affe-cting the distribution of the second-order vibration frequency of the blades include elastic modulus, density and the rotor speed, with the sensitivity probabilities 35.09%,34.56% and 24.15% respecti-vely.

Download Full-text

General Robust Design by Semi-Second-Order Taylor Expansion

Volume 4: Design and Manufacturing ◽

10.1115/imece2008-66686 ◽

2008 ◽

Cited By ~ 1

Author(s):

Xiaoping Du

Keyword(s):

Design Optimization ◽

Robust Design ◽

Taylor Expansion ◽

Random Variables ◽

Second Order ◽

Robust Design Optimization ◽

Integration Strategy ◽

Interval Variables ◽

Standard Deviations ◽

Robustness Assessment

The purpose of robust design optimization is to minimize variations in design performances and therefore to make the design insensitive to uncertainties. Current robust design methods fall into two types — probabilistic robust design and worst-case (interval) robust design. The former method is used when random variables are involved. In this method, robustness is measure by standard deviations of design performances. The later method is used when uncertainties are represented by intervals. The widths of design performances are then used to measure robustness. In many engineering application, both random variables and interval variables may exist simultaneously. In this paper, a general approach to robust design optimization is proposed. The generality comes from the ability to handle both random and interval variables. To alleviate the computational burden, we employ a previously developed general robustness assessment method — semi-second-order Taylor expansion method, to evaluate the maximum and minimum standard deviations of a design performance. An efficient integration strategy of the general robustness assessment and optimization is proposed. With the integration strategy, the number of function calls can be reduced while good accuracy is maintained. A robust shaft design problem is given for demonstration.

Download Full-text

An introduction to second‐order random variables in human health risk assessments

Human and Ecological Risk Assessment An International Journal ◽

10.1080/10807039609383655 ◽

1996 ◽

Vol 2 (4) ◽

pp. 892-919 ◽

Cited By ~ 41

Author(s):

David E. Burmaster ◽

Andrew M. Wilson

Keyword(s):

Health Risk ◽

Human Health ◽

Human Health Risk ◽

Random Variables ◽

Second Order ◽

Risk Assessments ◽

Health Risk Assessments

Download Full-text

IMPRECISE SECOND-ORDER MODEL FOR A SYSTEM OF INDEPENDENT RANDOM VARIABLES

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488505003382 ◽

2005 ◽

Vol 13 (02) ◽

pp. 177-193 ◽

Cited By ~ 2

Author(s):

LEV V. UTKIN

Keyword(s):

Random Variables ◽

Random Variable ◽

Second Order ◽

Independent Random Variables ◽

Order Model ◽

First Order ◽

Probability Bounds ◽

Uncertainty Model ◽

Lower And Upper Probabilities ◽

Interval Valued

A new hierarchical uncertainty model for combining different evidence about a system of statistically independent random variable is studied in the paper. It is assumed that the first-order level of the model is represented by sets of lower and upper previsions (expectations) of random variables and the second-order level is represented by sets of lower and upper probabilities which can be viewed as confidence weights for interval-valued expectations of the first-order level. The model is rather general and allows us to compute probability bounds and "average" bounds for previsions of a function of random variables. A numerical example illustrates this model.

Download Full-text

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

Entropy ◽

10.3390/e23101313 ◽

2021 ◽

Vol 23 (10) ◽

pp. 1313

Author(s):

Wei Liu ◽

Yong Zhang

Keyword(s):

Independent Random Variable ◽

Random Variables ◽

Random Variable ◽

Second Order ◽

Law Of Iterated Logarithm ◽

Independent Random Variables ◽

Linear Processes ◽

Iterated Logarithm ◽

The Law

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

Download Full-text