4. BAYESIAN ESTIMATION OF PARAMETERS Bayesian techniques for determining probabilistic model parameters have been available for quite a while, however, most of the problem formulations have been analytically unsolvable. Only very special classes of problems could be solved. With the advent of the digital computer, a new surge of interest emerged and has grown into a very well developed technique. The use of Markov Chain Monte Carlo techniques has made Bayesian analysis accessible. A more complete background is provided in [3]. In parameter estimation for structural dynamics, the work presented here follows most closely the work performed in [5] and [6]. In [5] and [6], the authors used a Bayesian updating technique to determine a probabilistic model of the optimal parameters for a finite element model. In the present work, Bayes theorem is used to estimate a probabilistic model for the elastic parameters Young’s modulus and Poisson’s ratio. The resulting models will be compared to those estimated using optimization theory from the previous section. The calibration parameters are given by ^ ` T Q V 10 , ,log E (6) where E is the Young’s modulus, Q is the Poisson’s ratio, and σ is the standard error yet to be defined. The updated or posterior probably density function is given by TH H T T p f p p , (7) where the likelihood is defined as ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § 9 1 2 9 9 9 2 7 7 7 2 2 1 exp , i calc calc i calc calc i f f f f f f p f V H T , (8) for the two frequency case and the three prior distributions are defined by ߠ ଵ~ (ݑ 1݁ 5,1.3݁ 5) ߠ ଶ~ (ݑ 0,0.45) ߠ ଷ~ (ݑ −5,5) so that the distribution function p(θ) defined by 1 5,1.3 5 0,0.45 5,5 u p u e e u T , (9) and u a b, is the uniform probability density function defined between a and b. The frequencies are normalized in Eq. (7) to equally weight the three distinct modes; otherwise, the larger magnitude mode would get the most weight. To evaluate Eq. (7), the Markov Chain Monte Carlo (MCMC) technique is used. When the MCMC technique reaches a stationary condition, the sequence of random numbers satisfy the PDF that is being evaluated. A detailed discussion on MCMC can be found in [2]. For this work, the slice sampler ([7]) as implemented in Matlab ([8]) is used to solve Eq. (7). A large (800,000) number of samples were generated using the MCMC technique. This large number of samples are only possible due to the fast running surrogate model defined by Eq. (1). The marginal probability density functions for the estimates of each parameter are given in Figure 4. In addition to plotting the margin probability density functions, the assumed prior distribution from Eq. (9) is plotted to show the range and to compare the differences between the prior and the posterior PDF. The posterior distribution was estimated using the KDE estimator, [1]. 224
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