### Abstract

This paper proposes a Sum of Squares (SOS) optimization technique for using multivariate data to estimate the probability density function of a non-Gaussian generating process. The class of distributions over which we optimize, result from using a polynomial map to lift the data into a higher-dimensional space, solving for an optimal Gaussian fit in this space, and then projecting a polynomial slice of the resulting joint density into physical space. The resulting distribution, to be called Sliced Normal, is able to characterize multimodal responses and strong parameter dependencies. We investigate several formulations of the problem, first maximizing a log-likelihood function, then a worst-case log-likelihood function, and finally using a heuristic to increase sparsity within the maximum log-likelihood formulation - thereby identifying independent subsets of the random variables. Using the optimal density functions in each scenario, we then propose semi-algebraic sets representing confidence regions, or 'safe sets,' for future data. Finally, we show numerically that these 'safe sets' are reliable and, hence, can be used for system identification, fault detection, robustness analysis, and robust control design.

Original language | English (US) |
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Title of host publication | 2019 American Control Conference, ACC 2019 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 5378-5384 |

Number of pages | 7 |

ISBN (Electronic) | 9781538679265 |

State | Published - Jul 1 2019 |

Event | 2019 American Control Conference, ACC 2019 - Philadelphia, United States Duration: Jul 10 2019 → Jul 12 2019 |

### Publication series

Name | Proceedings of the American Control Conference |
---|---|

Volume | 2019-July |

ISSN (Print) | 0743-1619 |

### Conference

Conference | 2019 American Control Conference, ACC 2019 |
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Country | United States |

City | Philadelphia |

Period | 7/10/19 → 7/12/19 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*2019 American Control Conference, ACC 2019*(pp. 5378-5384). [8814931] (Proceedings of the American Control Conference; Vol. 2019-July). Institute of Electrical and Electronics Engineers Inc..

**A sum of squares optimization approach to uncertainty quantification.** / Colbert, Brendon K.; Crespo, Luis G.; Peet, Matthew.

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

*2019 American Control Conference, ACC 2019.*, 8814931, Proceedings of the American Control Conference, vol. 2019-July, Institute of Electrical and Electronics Engineers Inc., pp. 5378-5384, 2019 American Control Conference, ACC 2019, Philadelphia, United States, 7/10/19.

}

TY - GEN

T1 - A sum of squares optimization approach to uncertainty quantification

AU - Colbert, Brendon K.

AU - Crespo, Luis G.

AU - Peet, Matthew

PY - 2019/7/1

Y1 - 2019/7/1

N2 - This paper proposes a Sum of Squares (SOS) optimization technique for using multivariate data to estimate the probability density function of a non-Gaussian generating process. The class of distributions over which we optimize, result from using a polynomial map to lift the data into a higher-dimensional space, solving for an optimal Gaussian fit in this space, and then projecting a polynomial slice of the resulting joint density into physical space. The resulting distribution, to be called Sliced Normal, is able to characterize multimodal responses and strong parameter dependencies. We investigate several formulations of the problem, first maximizing a log-likelihood function, then a worst-case log-likelihood function, and finally using a heuristic to increase sparsity within the maximum log-likelihood formulation - thereby identifying independent subsets of the random variables. Using the optimal density functions in each scenario, we then propose semi-algebraic sets representing confidence regions, or 'safe sets,' for future data. Finally, we show numerically that these 'safe sets' are reliable and, hence, can be used for system identification, fault detection, robustness analysis, and robust control design.

AB - This paper proposes a Sum of Squares (SOS) optimization technique for using multivariate data to estimate the probability density function of a non-Gaussian generating process. The class of distributions over which we optimize, result from using a polynomial map to lift the data into a higher-dimensional space, solving for an optimal Gaussian fit in this space, and then projecting a polynomial slice of the resulting joint density into physical space. The resulting distribution, to be called Sliced Normal, is able to characterize multimodal responses and strong parameter dependencies. We investigate several formulations of the problem, first maximizing a log-likelihood function, then a worst-case log-likelihood function, and finally using a heuristic to increase sparsity within the maximum log-likelihood formulation - thereby identifying independent subsets of the random variables. Using the optimal density functions in each scenario, we then propose semi-algebraic sets representing confidence regions, or 'safe sets,' for future data. Finally, we show numerically that these 'safe sets' are reliable and, hence, can be used for system identification, fault detection, robustness analysis, and robust control design.

UR - http://www.scopus.com/inward/record.url?scp=85072285272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072285272&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85072285272

T3 - Proceedings of the American Control Conference

SP - 5378

EP - 5384

BT - 2019 American Control Conference, ACC 2019

PB - Institute of Electrical and Electronics Engineers Inc.

ER -