### Abstract

The focus of this investigation is on the formulation and validation of a modeling strategy of the uncertainty that may exist on the specification of the power spectral density of scalar stationary processes and on the spectral matrices of vector ones. These processes may, for example, be forces on a structure originating from natural phenomena which are coarsely modeled (i.e., with epistemic uncertainty) or are specified by parameters unknown (i.e., with aleatoric uncertainty) in the application considered. The propagation of the uncertainty, e.g., to the response of the structure, may be carried out provided that a stochastic model of the uncertainty in the power spectral density/matrix is available from which admissible samples can be efficiently generated. Such a stochastic model will be developed here through an autoregressive-based parameterization of the specified baseline power spectral density/matrix and of its random samples. Autoregressive (AR) models are particularly well suited for this parametrization since their spectra are known to converge to a broad class of spectra (all nonpathological spectra) as the AR order increases. Note that the characterization of these models is not achieved directly in terms of their coefficients but rather in terms of their reflection coefficients which lie (or their eigenvalues in the vector process case) in the domain [0,1) as a necessary and sufficient condition for stability. Maximum entropy concepts are then employed to formulate the distribution of the reflection coefficients in both scalar and vector process case leading to a small set of hyperparameters of the uncertain model. Depending on the information available, these hyperparameters could either be varied in a parametric study format to assess the effects of uncertainty or could be identified, e.g., in a maximum likelihood format, from observed data. The validation and assessment of these concepts is finally achieved on the response of an uncertain 4-degree of freedom system subjected to white noise and on the response of an F-15 aircraft to random buffeting loads.

Original language | English (US) |
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Title of host publication | UNCECOMP 2015 - 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering |

Publisher | National Technical University of Athens |

Pages | 757-780 |

Number of pages | 24 |

State | Published - 2015 |

Event | 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2015 - Hersonissos, Crete, United Kingdom Duration: May 25 2015 → May 27 2015 |

### Other

Other | 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2015 |
---|---|

Country | United Kingdom |

City | Hersonissos, Crete |

Period | 5/25/15 → 5/27/15 |

### Fingerprint

### Keywords

- Maximum Entropy
- Maximum Likelihood Identification
- Random Matrices
- Stochastic Autoregressive Models
- Uncertain Power Spectral Densities

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Computer Science Applications

### Cite this

*UNCECOMP 2015 - 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering*(pp. 757-780). National Technical University of Athens.

**Modeling of uncertain spectra through stochastic autoregressive systems.** / Wang, Yiwei; Wang, X. Q.; Mignolet, Marc; Yang, Shuchi; Chen, P. C.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*UNCECOMP 2015 - 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering.*National Technical University of Athens, pp. 757-780, 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2015, Hersonissos, Crete, United Kingdom, 5/25/15.

}

TY - GEN

T1 - Modeling of uncertain spectra through stochastic autoregressive systems

AU - Wang, Yiwei

AU - Wang, X. Q.

AU - Mignolet, Marc

AU - Yang, Shuchi

AU - Chen, P. C.

PY - 2015

Y1 - 2015

N2 - The focus of this investigation is on the formulation and validation of a modeling strategy of the uncertainty that may exist on the specification of the power spectral density of scalar stationary processes and on the spectral matrices of vector ones. These processes may, for example, be forces on a structure originating from natural phenomena which are coarsely modeled (i.e., with epistemic uncertainty) or are specified by parameters unknown (i.e., with aleatoric uncertainty) in the application considered. The propagation of the uncertainty, e.g., to the response of the structure, may be carried out provided that a stochastic model of the uncertainty in the power spectral density/matrix is available from which admissible samples can be efficiently generated. Such a stochastic model will be developed here through an autoregressive-based parameterization of the specified baseline power spectral density/matrix and of its random samples. Autoregressive (AR) models are particularly well suited for this parametrization since their spectra are known to converge to a broad class of spectra (all nonpathological spectra) as the AR order increases. Note that the characterization of these models is not achieved directly in terms of their coefficients but rather in terms of their reflection coefficients which lie (or their eigenvalues in the vector process case) in the domain [0,1) as a necessary and sufficient condition for stability. Maximum entropy concepts are then employed to formulate the distribution of the reflection coefficients in both scalar and vector process case leading to a small set of hyperparameters of the uncertain model. Depending on the information available, these hyperparameters could either be varied in a parametric study format to assess the effects of uncertainty or could be identified, e.g., in a maximum likelihood format, from observed data. The validation and assessment of these concepts is finally achieved on the response of an uncertain 4-degree of freedom system subjected to white noise and on the response of an F-15 aircraft to random buffeting loads.

AB - The focus of this investigation is on the formulation and validation of a modeling strategy of the uncertainty that may exist on the specification of the power spectral density of scalar stationary processes and on the spectral matrices of vector ones. These processes may, for example, be forces on a structure originating from natural phenomena which are coarsely modeled (i.e., with epistemic uncertainty) or are specified by parameters unknown (i.e., with aleatoric uncertainty) in the application considered. The propagation of the uncertainty, e.g., to the response of the structure, may be carried out provided that a stochastic model of the uncertainty in the power spectral density/matrix is available from which admissible samples can be efficiently generated. Such a stochastic model will be developed here through an autoregressive-based parameterization of the specified baseline power spectral density/matrix and of its random samples. Autoregressive (AR) models are particularly well suited for this parametrization since their spectra are known to converge to a broad class of spectra (all nonpathological spectra) as the AR order increases. Note that the characterization of these models is not achieved directly in terms of their coefficients but rather in terms of their reflection coefficients which lie (or their eigenvalues in the vector process case) in the domain [0,1) as a necessary and sufficient condition for stability. Maximum entropy concepts are then employed to formulate the distribution of the reflection coefficients in both scalar and vector process case leading to a small set of hyperparameters of the uncertain model. Depending on the information available, these hyperparameters could either be varied in a parametric study format to assess the effects of uncertainty or could be identified, e.g., in a maximum likelihood format, from observed data. The validation and assessment of these concepts is finally achieved on the response of an uncertain 4-degree of freedom system subjected to white noise and on the response of an F-15 aircraft to random buffeting loads.

KW - Maximum Entropy

KW - Maximum Likelihood Identification

KW - Random Matrices

KW - Stochastic Autoregressive Models

KW - Uncertain Power Spectral Densities

UR - http://www.scopus.com/inward/record.url?scp=84942895549&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84942895549&partnerID=8YFLogxK

M3 - Conference contribution

SP - 757

EP - 780

BT - UNCECOMP 2015 - 1st ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering

PB - National Technical University of Athens

ER -