Globally Identifiable Case

In globally identifiable cases [19], the posterior/updated PDF for 0 given a large amount of data V may be approximated accurately by a Gaussian distribution, so the evidence p T Cj) can be approximated by using Laplace s method for asymptotic expansion [197]  

This represents a penalty against complicated parameterization [98,164], as demonstrated in the following discussion. 

it is attempted to show that the Ockham factor decreases exponentially with the number of uncertain parameters in the model class. For this purpose, consider an alternative 

Bayesian Methods for Structural Dynamics and Civil Engineering 

Since the prior variances will always be greater than the posterior variances if the data provide any information about the model parameters in the model class Cj, all the terms in the first summation in Equation (6.15) will be positive and so will the terms in the second summation unless the posterior most probable value 6 just happens to coincide with the prior most probable value O . Thus, one might expect that the log-Ockham factor In Oj will decrease if the number of parameters Nj for the model class Cj is increased. This expectation is confirmed by noting that the posterior variances are inversely proportional to the number of data points N in V, so the dependence of the log-Ockham factor is  

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Figure 2.19 Contours of the likelihood function (globally identifiable case)...

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 ... 

Information Entropy with Globally Identifiable Case... 

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... 

In the next section, the Bayesian model class selection method is introduced for quantification and selection of model classes. It will be discussed for the globally identifiable case and the general case. The Ockham factor is introduced and it serves as the penalty for a complicated model, which appears naturally from the evidence. Computational issues will be discussed and... 

Dp, Cj) in the integrand is the conditional likelihood function of model class Cj at the (k + )th time step. It represents the level of data fitting of the model with a given parameter vector 0,t- The second factor in the integrand is the posterior PDF of the parameter vector conditional on the previous data points Zi, Z2,. .., z. For globally identifiable cases, an asymptotic expansion of this integral can be obtained in a similar fashion as Eq. 41 ... 

A subtle concept, not readily apparent from the transfer function [Eq. (1.79)], is that model identifiability also depends on the shape of the input function, [u(t, p)] (Godfrey, Jones, and Brown, 1980). This property can be exploited, in the case of a single measurement system, to make a model identifiable by the addition of another input into the system. But care must be made in what is the shape of the second input. A second input having the same shape as the first input will not make an unidentifiable model identifiable. If, for example, a second bolus input into Compartment 2 of Model B in Fig. 1.13 were added at the same time as the bolus input into Compartment 1, the model will still remain unidentifiable. If, however, the input into Compartment 1 is a bolus and the input into Compartment 2 is an infusion the model will become globally identifiable. But, if the inputs are reversed and the input into Compartment 1 is an infusion and the input into Compartment 2 is a bolus, the model becomes globally identifiable only if there is an independent estimate of the volume of distribution of the central compartment. Hence, one way to make an unidentifiable model identifiable is to use another route of administration. 

For a given model class Ad and data V, it is useful to characterize the topology of the posterior PDF as a function of the model parameter vector by whether it has a global maximum at a single most probable parameter value, at a finite number of them, or at a continuum of most probable parameter values lying on some manifold in the parameter vector space. These three cases may be described as globally identifiable, locally identifiable, and unidentifiable model classes based on given dynamic data from the system. 

In order to accomplish with the aforementioned aim, during the first year of project, an extensive research on the different chemical additives used in six industrial sectors was conducted plastics, textiles, electronics, lubricants, leather, and paper. A list of selected chemical additives was identified for each sector and used as a study basis for the rest of the project. This is the case of the decabromo-diphenyl ether (BDE) used in electronics as a flame retardant or the triclosan used in the textile as a biocide. The results of this investigation were presented in the first volume of this book (Global Risk-Based Management of Chemical Additives I Production, Usage and Environmental Occurrence). This volume also included a section of case studies related to the selected additives in different countries (i.e., Denmark, Vietnam, Brazil, India). The main outcomes of the first part of the project are summarized below ... 

B. Empirical-inductive Global Value Chain Planning Requirements and State of the Art Analysis Identify planning requirements based on case studies and literature Analyze requirements coverage by state of the art research and further specify research gaps... 

After a suspected case of the 1918 Spanish flu vims (which, in a global pandemic during World War I, affected half the world s population and killed almost 25 million people in 18 months) was identified in 1976, Congress passed the National Swine Flu Immunization Program, releasing manufacturers from the liability, so that a flu vaccine... 

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