Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion
Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion
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DOI：
10.1016/j.ress.2012.05.002
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发表时间：
2012-10-01
影响因子：
8.1
通讯作者：
Nowak, W.
中科院分区：
文献类型：
--
作者：
Oladyshkin, S.;Nowak, W.
我们讨论的任意多项式混沌（aPC），这一直是在最近的一些理论论文的研究课题。与所有多项式混沌展开技术一样，aPC通过在正交多项式基上展开来近似仿真模型输出对模型参数的依赖性。APC将混沌扩展技术推广到具有任意概率测度的任意分布，这些概率测度可以是离散的、连续的或离散化连续的，并且可以在分析上（作为概率密度/累积分布函数）、在数值上作为直方图或作为原始数据集来指定。我们表明，在有限的扩展顺序的aPC只需要存在一个有限数量的时刻，并不需要完整的知识，甚至存在的概率密度函数。这就避免了分配有限的可用数据不能充分支持的参数概率分布的必要性。或者，它允许建模者自由选择技术约束的形状，他们的统计假设。我们的主要思想是将分析的复杂性水平和顺序与输入参数的统计信息的可靠性和详细程度相一致。我们提供了存在的条件，并澄清的aPC模型参数的统计矩的关系。我们用不同的统计分布和原始数据测试aPC的性能。在这些示例性的测试情况下，我们说明了收敛与增加的扩展顺序，并首次，与增加的可靠性水平的统计输入信息。我们的结果表明，aPC表现出指数收敛速度，收敛速度比经典的多项式混沌展开技术。(C)2012爱思唯尔有限公司保留所有权利。
We discuss the arbitrary polynomial chaos (aPC), which has been subject of research in a few recent theoretical papers. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. We show that the aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Our key idea is to align the complexity level and order of analysis with the reliability and detail level of statistical information on the input parameters. We provide conditions for existence and clarify the relation of the aPC to statistical moments of model parameters. We test the performance of the aPC with diverse statistical distributions and with raw data. In these exemplary test cases, we illustrate the convergence with increasing expansion order and, for the first time, with increasing reliability level of statistical input information. Our results indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques. (C) 2012 Elsevier Ltd. All rights reserved.