Updated PDF

Canadian HIV/AIDS Legal Network (2003). Amendment to Canada s Patent Act to Authorize Export of Generic Pharmaceuticals—Update. Available http //www. aidslaw.ca/Maincontent/issues/cts/patent-amend/Patent ActAmendment Update.pdf. Accessed August 2005. 

This form is applicable to the identification of continuous-valued uncertain parameters with observation of discrete events and p(0 A) is regarded as the updated PDF or posterior PDF of the parameter vector 0. 

In order to establish the updated PDF for Pi, a triangular distribution is used for the prior PDF ... 

Then, consider a larger number of draws (N = 1000) and 1 appeared 149 times. By using the Bayes theorem, the updated PDF for Pi is shown in Figure 2.2. The most probable value, mean, and standard deviation of the estimate are 0.150, 0.151 and 0.011, respectively. This distribution concentrates in a narrower range due to the extra information gained from the additional samples. With the updated PDF, different confidence intervals can be constructed for the uncertain parameter Pi. 

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. 

In the probabilistic approach, the solution is not simply the optimal parameters but also the probability density function that describes the complete picture of the uncertainty. It is a challenging task to demonstrate the representation of the updated PDF since it has... 

The actual value of ao is taken to be 1.0. In order to have a fair ground for comparison with the least-squares method, an improper prior is taken so the updated PDF for the uncertain... 

In evaluating the type of updated robust integral in Equation (2.126) by MCS, it requires the parameter samples distributed according to the updated PDF p 0 T>, C). Therefore, generation... 

Let p. .., be a chosen sequence of PDFs converging to p 0 T>, C) (= p ° ) so that their region of significant probability content gradually diminishes to that of p 0 V, C). For example, p - ) may be chosen as the updated PDF from Bayes Theorem based on an increasing amount of data ... 

The success of the adaptive strategy relies on effective application of the MH algorithm at each simulation level s, which requires that (which is approximated by -p< -i>) varies with a similar length scale to for s = 1, 2,..., sq. The choice of the sequence is thus important to the success of the adaptive method. If the updated PDF with data V is of the form [18,268] ... 

The prior PDF p(0 C) is taken to be independent log-normal PDFs with means of 0.9 and 1.2 and unit variance. Using the modal data V, the updated PDF for the stiffness parameter vector 0 is formulated as ... 

In this study, a non-informative prior PDF for the uncertain parameters is used and the prior PDF is absorbed into the normalizing constant. Then, the updated PDF is proportional to the likelihood function ... 

The optimal parameter vector 0 can be found by minimizing the objective function J. Furthermore, the updated PDF allows one to quantify the uncertainty of the estimation, e.g., to calculate the standard deviation or the contours with equal probability density. 

Using the Bayes theorem, the posterior/updated PDF of the model parameter vector 9, given the spectral set, Sk is ... 

This optimization problem can be solved by the MATLAB function fminsearch [171]. It has been shown numerically for the globally identifiable case with a large number of data points that the updated PDF can be well approximated by a Gaussian distribution 0(9 9, H(9 ) ) with mean 9 and covariance matrix H(9 )- -, where U(9 ) denotes the Hessian oiJ(9) calculated ate = 9 ... 

Here the frequency index set K. represents a range over which Equations (3.56) and (3.58) give a satisfactory approximation and this will be further discussed in Section 3.3.3. Using the Bayes theorem, the updated PDF of the model parameter vector 0, given the averaged spectral set [Pg.114]

In the case where a non-informative prior is used, the first term can be simply neglected. Furthermore, the updated PDF of the parameter vector 0 can be well approximated by a Gaussian distribution G(0 0, Ti(0 ) ) with mean 0 and covariance matrix H(0 ), where H 0 ) denotes the Hessian matrix of the objective function J calculated at = ... 

Another case is investigated with a very short duration of measurement, namely T = 60 s, so it contains roughly 38 fundamental periods of the oscillator. The Bayesian spectral density approach is used for its identification with the frequency index set /C = 1,2,..., 45. Figure 3.13 shows the conditional updated PDFs of and with all other parameters fixed at their optimal values. It is obvious that the conditional PDFs are non-Gaussian so the Gaussian... 

The updated PDF p 0 S jf, A, C) in Equation (3.67) is given by Equation (3.60) with Equation (3.61) for the general case of uncertain excitation. The formulation presented here is based on the spectral density estimators obtained from the measured data V and it depends on the class of structural, excitation and prediction-error models chosen to describe the system. The updated parameter vector is obtained by minimizing the objective function J 0) = — In p 0 C)p S jf 0, A, C) with the likelihood function p S 0, A, C) given by Equation (3.61). Eurthermore, the updated PDF of the model parameter vector 0 can be... 

This case is, again, unidentifiable. The updated PDF is plotted together with the previous one in Figure 3.16 and the trajectories of the peaks in the (A i, K3) plane have different slopes. By Equation (3.79), the equivaient linear system has a stiffness Ki + 3a K3 so different Duffing oscillators with K - - = K (a constant) are associated with the same equivalent linear... 

Figure 3.16 suggests that if two dynamic data sets and are utilized simultaneously, the information from is complimentary to Dfb to provide an extra mathematical constraint for the uncertain parameters, especially for Ki and K3. The updated PDF using both sets of data is given by the product of the individuals. As a result, the identification problem wfil become globally identifiable. Table 3.4 shows the updated values... 

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