Bayesian Time-domain Approach

Keywords ambient vibration Bayesian inference best estimaton conditional probability correlation function modal analysis modal identification nonstationary response seismic response structural health monitoring 

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 

Bayesian Methods for Structural Dynamics and Civil Engineering Ka-Veng Yuen 2010 John Wiley Sons (Asia) Pte Ltd 

Bayesian Methods for Structural Dynamics and Civil Engineering 

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The additional measurement of y3 improves only slightly the prediction for y but the additional measurement of y4 has virtually no effect on the prediction. This example is useful to demonstrate the approximation used in the Bayesian time-domain approach. The random variable yi is predicted by the measurements of y2. y3 and 4, which are 0.5,1 and 1.5 periods apart from yi, respectively. It turns out that including the data points within one period is sufficient. Furthermore, in a usual situation, the sampling time step is much less than half of... 

Figure 4.4 shows the conditional PDF of the natural frequency and the damping ratio C with the spectral intensity and the prediction-error variance fixed at their optimal values. The conditional PDFs by the Bayesian time-domain approach and the Gaussian approximation are plotted with solid lines and dashed lines, respectively. The two groups of curves are on top on each other, indicating that the Gaussian approximation is accurate when the number of data points is sufficiently large. This can be used to represent the posterior PDF, e.g., for statistical moments computation. 

Another case is investigated with a very short period of measurement, namely T = 5 s, so it contains less than four fundamental periods of the oscillator. The Bayesian time-domain method is used for the identification and Figure 4.5 shows the conditional PDF of and f with all other parameters fixed at their optimal values. The solid lines show the conditional posterior PDF obtained by the Bayesian method and the dashed lines show the Gaussian approximation. It is clear that the posterior PDF is non-Gaussian. This confirms that the Bayesian time-domain approach is capable to offer the correct inference without assuming the type of the posterior PDF. In the case of a non-Gaussian posterior PDF, statistical moments, such as the variances of the estimates, can be computed by direct Monte Carlo simulation. The results are shown in Table 4.2 in the same fashion as Table 4.1. The computed uncertainty obtained here is reasonable by judging the normalized distance of the estimates. 

The solid lines are obtained by using the Bayesian time-domain approach without assuming the type of posterior distributions and the dashed lines are obtained by using the Gaussian approximation. It can be seen that the two sets of curves are on top of each other, implying that the Gaussian approximation is accurate. 

In this chapter, the Bayesian time-domain approach was introduced for identification of the model parameters and stochastic excitation parameters of linear multi-degree-of-freedom systems using noisy stationary or nonstationary response measurements. The direct exact formulation was presented but it turned out to be computationally prohibited for a large number of data points. Then, an approximated likelihood function expansion was proposed to resolve this obstacle. For a globally identifiable case with a large number of data points, the updated PDF... 

Chapter 3 and Chapter 4 presented the Bayesian spectral density approach and Bayesian time-domain approach. The comparison can be summarized as follows ... 

The Bayesian time-domain approach utilizes the Bayes theorem repeatedly to factorize the likelihood function into the product of a joint PDF and conditional PDFs ... 

Even though the spectral density approach requires computation of the inverse and determinant of a number of matrices, the size of these matrices is only No y. No. They are significantly smaller than the NgNp x NgNp matrix Eyj j (= E22) required in the time-domain approach. Comparison of the computational efficiency between the two methods depends on the number of the elements in the frequency index set and the number of data points in a fundamental period. The ratio of the computations required by the Bayesian spectral density approach and the Bayesian time-domain approach can be approximated by ... 

The only approximation made in the Bayesian time-domain approach is that the system response at a particular time step estimated by its entire history is essentially the same as conditioning on a significantly smaller number of previous time steps. In practice, the time-domain approach provides virtually an exact solution in the sense that the Bayesian approach utilizes the complete information inherited in the measurement. Therefore, the Bayesian time-domain approach provides more accurate statistical inference of the model parameters with the information in the data. 

Yuen, K.-V. and Katafygiotis, L. S. Bayesian time-domain approach for modal updating using ambient data. Probabilistic Engineering Mechanics 16(3) (2001), 219-231. 

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