### Abstract

Spectral parameters for a stationary Gaussian process are most often estimated by Fourier transformation
of a realization followed by some smoothing procedure. This smoothing is often a weighted least
square fitting of some prespecified parametric form of the spectrum. In this paper it is shown that maximum
likelihood estimation is a rational alternative to an arbitrary weighting for least square fitting. The derived likelihood
function gets singularities if the spectrum is prescribed with zero values at some frequencies. This is often
the case for models of technically relevant processes. The numerical problem caused by these singularities is
easily overcome by adding simulated low intensity white noise to the realization. Without changing its parameters
the spectrum is hereby lifted above zero by an amount equal to the white noise intensity. The knowledge
of an explicit likelihood function, even though it is of complicated mathematical form, allows an approximate
Bayesian updating and control of the time development of the parameters. Some of these parameters can be
structural parameters that by too much change reveal progressing damage or other malfunctioning. Thus current
process monitoring and updating, for example administered in a Bayesian network system, can be a useful aid
for the operation of a complicated technical system (large important structure, ship, wind power engine, etc.).
Keywords: Parametric spectral estimation, likelihood of spectral parameters, response monitoring

Original language | English |
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Title of host publication | Proc. of ICASP9 |

Place of Publication | Rotterdam |

Publisher | Millpress |

Publication date | 2003 |

Pages | 443-450 |

Publication status | Published - 2003 |

Event | 9th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP9) - San Francisco, CA, United States Duration: 6 Jul 2003 → 9 Jul 2003 Conference number: 9 |

### Conference

Conference | 9th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP9) |
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Number | 9 |

Country | United States |

City | San Francisco, CA |

Period | 06/07/2003 → 09/07/2003 |

## Cite this

Friis-Hansen, P., & Ditlevsen, O. D. (2003). Likelihood updating of random process load and resistance parameters by monitoring. In

*Proc. of ICASP9*(pp. 443-450). Millpress.