An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes

In recent years, methods were proposed so as to efficiently perform time-variant reliability analysis. However, importance sampling (IS) for time-variant reliability analysis is barely studied in the literature. In this paper, an IS framework is proposed. A multi-dimensional integral is first derived to define the time-variant cumulative probability of failure, which has the similar expression to the classical definition of time-invariant failure probability. An IS framework is then developed according to the fact that time-invariant random variables are commonly involved in time-variant reliability analysis. The basic idea of the proposed framework is to simultaneously apply time-invariant IS and crude Monte Carlo simulation on time-invariant random variables and stochastic processes, respectively. Thus, the probability of acquiring failure trajectories of time-variant performance function is increased. Two auxiliary probability density functions are proposed to implement the IS framework. However, auxiliary PDFs available for the framework are not limited to the proposed two. Three examples are studied in order to validate the effectiveness of the proposed IS framework.

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Time-Dependent Reliability Analysis by a Sampling Approach to Extreme Values of Stochastic Processes

Volume 3: 38th Design Automation Conference, Parts A and B ◽

10.1115/detc2012-70132 ◽

2012 ◽

Cited By ~ 3

Author(s):

Zhen Hu ◽

Xiaoping Du

Keyword(s):

Stochastic Processes ◽

Reliability Analysis ◽

Extreme Values ◽

Limit State ◽

Time Dependent ◽

State Function ◽

Limit State Function ◽

Reliability Method ◽

Time Invariant ◽

State Functions

Maintaining high accuracy and efficiency is a challenging issue in time-dependent reliability analysis. In this work, an accurate and efficient method is proposed for limit-state functions with the following features: The limit-state function is implicit with respect to time, and its input contains stochastic processes; the stochastic processes include only general strength and stress variables, or the limit-state function is monotonic to these stochastic processes. The new method employs random sampling approaches to estimate the distributions of the extreme values of the stochastic processes. The extreme values are then used to replace the corresponding stochastic processes, and consequently the time-dependent reliability analysis is converted into its time-invariant counterpart. The commonly used time-invariant reliability method, the First Order Reliability Method, is then applied for the time-variant reliability analysis. The results show that the proposed method significantly improves the accuracy and efficiency of time-dependent reliability analysis.

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Reply by the author to the Discussion by J. M. Mendel

Geophysics ◽

10.1190/1.1486999 ◽

1993 ◽

Vol 58 (3) ◽

pp. 443-445

Author(s):

Anton Ziolkowski

Keyword(s):

Stochastic Processes ◽

Seismic Data ◽

Random Variables ◽

The Earth ◽

Definition Of

I thank Professor Mendel for his comments on my paper. I first examine his model of the seismogram and show that losses in the earth cannot be convolutional. I then examine the structure of his “reflectivity sequence” and show that this cannot be random. I suggest that the theory of probability, random variables and stochastic processes is not applicable to our non‐random seismic data. I suggest that we need, and use, a definition of whiteness in geophysics which comes from optics. Finally I show that none of the methods I am proposing for the determination of the source signature relies on the earth being loss‐free.

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Efficient importance sampling for large sums of independent and identically distributed random variables

Statistics and Computing ◽

10.1007/s11222-021-10055-1 ◽

2021 ◽

Vol 31 (6) ◽

Author(s):

Nadhir Ben Rached ◽

Abdul-Lateef Haji-Ali ◽

Gerardo Rubino ◽

Raúl Tempone

Keyword(s):

Importance Sampling ◽

Rare Event ◽

Random Variables ◽

Moment Generating Function ◽

Accurate Estimate ◽

Probability Density Functions ◽

Density Functions ◽

The Moment ◽

Log Normal ◽

Independent And Identically Distributed

AbstractWe discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = 1 N X i ≤ γ ) , via importance sampling (IS). We are particularly interested in the rare event regime when N is large and/or $$\gamma $$ γ is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: (i) It assumes the knowledge of the moment-generating function of $$X_i$$ X i and (ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as $$x \rightarrow 0$$ x → 0 , to $$bx^{p}$$ b x p , for $$p>-1$$ p > - 1 and $$b>0$$ b > 0 . For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of $$\gamma $$ γ and/or large values of N, the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large N and/or small $$\gamma $$ γ .

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Time-Dependent System Reliability Analysis for Bivariate Responses

ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg ◽

10.1115/1.4038318 ◽

2017 ◽

Vol 4 (3) ◽

Cited By ~ 2

Author(s):

Zhen Hu ◽

Zhifu Zhu ◽

Xiaoping Du

Keyword(s):

Stochastic Processes ◽

Reliability Analysis ◽

System Reliability ◽

Random Variables ◽

General System ◽

Time Dependent ◽

Time Duration ◽

Reliability Method ◽

Bivariate Responses ◽

Time Dependent System

Time-dependent system reliability is computed as the probability that the responses of a system do not exceed prescribed failure thresholds over a time duration of interest. In this work, an efficient time-dependent reliability analysis method is proposed for systems with bivariate responses which are general functions of random variables and stochastic processes. Analytical expressions are derived first for the single and joint upcrossing rates based on the first-order reliability method (FORM). Time-dependent system failure probability is then estimated with the computed single and joint upcrossing rates. The method can efficiently and accurately estimate different types of upcrossing rates for the systems with bivariate responses when FORM is applicable. In addition, the developed method is applicable to general problems with random variables, stationary, and nonstationary stochastic processes. As the general system reliability can be approximated with the results from reliability analyses for individual responses and bivariate responses, the proposed method can be extended to reliability analysis of general systems with more than two responses. Three examples, including a parallel system, a series system, and a hydrokinetic turbine blade application, are used to demonstrate the effectiveness of the proposed method.

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Importance Sampling for Time-Variant Reliability Analysis

IEEE Access ◽

10.1109/access.2021.3054470 ◽

2021 ◽

Vol 9 ◽

pp. 20933-20941

Author(s):

Jian Wang ◽

Runan Cao ◽

Zhili Sun

Keyword(s):

Reliability Analysis ◽

Importance Sampling

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Error and Reliability in Stochastic Processes and Psychological Measurement

Psychological Reports ◽

10.2466/pr0.1972.31.1.131 ◽

1972 ◽

Vol 31 (1) ◽

pp. 131-140 ◽

Cited By ~ 2

Author(s):

Donald W. Zimmerman

Keyword(s):

Stochastic Processes ◽

Random Error ◽

Random Sampling ◽

Probability Model ◽

Random Variables ◽

Formal Structure ◽

Intermediate Structure ◽

Psychological Measurement ◽

Joint Distributions ◽

True Values

The concepts of random error and reliability of measurements that are familiar in traditional theories based on the notions of “true values” and “errors” can be represented by a probability model having a simpler formal structure and fewer special assumptions about random sampling and independence of measurements. In this model formulas that relate observable events are derived from probability axioms and from primitive terms that refer to observable events, without an intermediate structure containing variances and correlations of “true” and “error” components of scores. While more economical in language and formalism, the model at the same time is more general than classical theories and applies to stochastic processes in which joint distributions of many dependent random variables are of interest. In addition, it clarifies some long-standing problems concerning “experimental independence” of measurements and the relation of sampling of individuals to sampling of measurements.

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An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties

Algorithms ◽

10.3390/a14080229 ◽

2021 ◽

Vol 14 (8) ◽

pp. 229

Author(s):

Fangyi Li ◽

Yufei Yan ◽

Jianhua Rong ◽

Houyao Zhu

Keyword(s):

Reliability Analysis ◽

Limit State ◽

Random Variables ◽

Equivalent Model ◽

Independent Random Variables ◽

Problem Analysis ◽

Analysis Method ◽

Calculation Algorithm ◽

Reliability Problem ◽

Interval Variables

In practical engineering, due to the lack of information, it is impossible to accurately determine the distribution of all variables. Therefore, time-variant reliability problems with both random and interval variables may be encountered. However, this kind of problem usually involves a complex multilevel nested optimization problem, which leads to a substantial computational burden, and it is difficult to meet the requirements of complex engineering problem analysis. This study proposes a decoupling strategy to efficiently analyze the time-variant reliability based on the mixed uncertainty model. The interval variables are treated with independent random variables that are uniformly distributed in their respective intervals. Then the time-variant reliability-equivalent model, containing only random variables, is established, to avoid multi-layer nesting optimization. The stochastic process is first discretized to obtain several static limit state functions at different times. The time-variant reliability problem is changed into the conventional time-invariant system reliability problem. First order reliability analysis method (FORM) is used to analyze the reliability of each time. Thus, an efficient and robust convergence hybrid time-variant reliability calculation algorithm is proposed based on the equivalent model. Finally, numerical examples shows the effectiveness of the proposed method.

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