Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion
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 a...
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| Vydané v: | Reliability engineering & system safety Ročník 106; s. 179 - 190 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Oxford
Elsevier Ltd
01.10.2012
Elsevier |
| Predmet: | |
| ISSN: | 0951-8320, 1879-0836 |
| On-line prístup: | Získať plný text |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Oladyshkin, S. Nowak, W. |
| Author_xml | – sequence: 1 givenname: S. surname: Oladyshkin fullname: Oladyshkin, S. email: sergey.oladyshkin@iws.uni-stuttgart.de – sequence: 2 givenname: W. surname: Nowak fullname: Nowak, W. |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=26300826$$DView record in Pascal Francis |
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| Keywords | Orthonormal basis Uncertainty Random Polynomial chaos Data-driven Stochastic Arbitrary distribution Chaos Histogram Statistical distribution Probability distribution Data driven modelling Stochastic process Modeling Uncertain system Dependence Probability density function Convergence rate Existence condition Statistical analysis Statistical moment Orthogonal polynomial Statistical parameter Statistical model Reliability |
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| SubjectTerms | Applied sciences Arbitrary distribution Chaos theory Computer simulation Convergence Data-driven Demand Density Exact sciences and technology Mathematical analysis Mathematical models Mathematics Operational research and scientific management Operational research. Management science Orthonormal basis Polynomial chaos Probability and statistics Probability theory and stochastic processes Random Raw Reliability theory. Replacement problems Sciences and techniques of general use Stochastic Stochastic processes Uncertainty |
| Title | Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion |
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