Bayesian Parametric Identification

Use to denote the parameter vector for identification. It includes the model parameters 9m and the parameters that determine the elements of the upper right triangular part of the prediction-error covariance matrix (symmetry defines fhe lower friangular part of this matrix). 

The dynamic data V consists of the measured time histories at N discrete time steps of the excitation and system response. Assume equal variances and stochastic independence for the prediction errors of different channels of measurements so the covariance matrix for the prediction errors is  

If Jg(0m, T, C) is known only implicitly, numerical optimization is needed to search for the optimal model parameters and this can be done by the function fminsearch in MATLAB [171]. 

The most probable value of the prediction-error variance in 0 can be obtained also by maximizing the posterior PDE  

The updated PDF in Equation (2.72) provides the complete description of the plausibility of the model parameters but its topology may be very complicated in general, especially if there are a large number of uncertain parameters so the distribution is difficult to be visualized. Use Smodi.0o, P) to denote the set of all model parameters which give the same output at the observed degrees of freedom as the model associated with 0q and the input F. 

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The Bayesian spectral density approach for parametric identification and model updating regression analysis are applied. During the monitoring period, four typhoons flitted over Macao. The structural behavior under such violent wind excitation is treated as discordance and the measurements obtained under these events are not taken into account for the analysis. By excluding these fifteen days of measurements, there are 168 pairs of identified squared fundamental frequency and measured temperature in the data set. Figure 2.28(a) shows the variation of the identified squared fundamental frequencies with their associated uncertainties represented by a confidence interval that is bounded by the plus or minus three standard derivations from the estimated values. It is noticed that this confidence interval contains 99.7% of the probability. Since the confidence intervals are narrow compared with the variation... 

Chapter 3 presented the Bayesian spectral density approach for the parametric identification of the multi-degree-of-freedom dynamical model using the measured response time history. The methodology is applicable for linear models and can also be utilized for weakly nonlinear models by obtaining the mean spectrum with equivalent linearization or strongly nonlinear models by obtaining the mean spectrum with simulations. The stationarity assumption in modal/model identification for an ambient vibration survey is common but there are many cases where the response measurements are better modeled as nonstationary, e.g., the structural response due to a series of wind gusts or seismic responses. In the literature, there are very few approaches which consider explicitly nonstationary response data, for example, [226,229]. Meanwhile, extension of the Bayesian spectral density approach for nonstationary response measurement is difficult since construction of the likelihood function is nontrivial in the frequency domain. Estimation of the time-dependent spectrum requires a number of data sets, which are associated with the same statistical time-frequency properties but this is impossible to achieve in practice. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

In Chapter 2, Section 2.4, parametric identification was introduced for linear and nonlinear regression problems. In this section, the Bayesian model class selection is applied to these problems. In order to smooth the presentation, some of the equations from Section 2.4 are repeated in this section. 

The Bayesian spectral density approach in Chapter 3 is used for parametric identification. The spectral density estimator is utilized up to 8 Hz to include all the peaks so No, = 480. Table 6.2 shows the optimal modal frequencies for model classes with different number of modes. There is in general no difficulty in identifying the first five modes but it is not the... 

In section Structural Parametric Identification by Extended Kalman Filter, online structural parametric identification using the EKF will be briefly reviewed. In section Online Identification of Noise Parameters, an online identification algorithm for the noise parameters in the EKF is introduced. Then, in section Outlier-Resistant Extended Kalman Filter, an online outlier detection algorithm is presented, and it is embedded into the EKF. This algorithm allows for robust structural identification in the presence of possible outliers. In section Online Bayesian Model Class Selection, a recursive Bayesian model class section method is presented for non-parametric identification problems. 

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