Identifiability global

Having understood the problem, a more formal definition of identifiability can be made. A compartmen-tal model is structurally identifiable (or sometimes called globally identifiable) if every estimable parameter in the... 

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. 

Audoly, S., D Angio, L., Saccomani, M.P., and Cobelli, C. Global identifiability of linear compartmental models— A computer algebra algorithm. IEEE Transactions on Biomedical Engineering 1998 45 36-47. 

If a parameter is locally identifiable but the observation function determines exactly one solution in the entire parameter space, that parameter is globally identifiable for that experiment. Thus, global identifiability is a subcategory of local identifiability. The term unique identifiability is equivalent to global identifiability. 

A property of a parameter is structural if it holds for almost all values of the parameter, i.e., almost everywhere in parameter space. The qualification, almost everywhere means that the property may not hold on a special subset of measure zero. Thus a parameter could be globally identifiable almost everywhere but only locally identifiable for a few special values. Structural global or local identifiability are generic properties that are not dependent on the values of the parameters, in the almost everywhere sense (Walter, 1982). 

If all of the parameters of a model are globally identifiable, the model is globally identifiable for that experiment. If all of the parameters are identifiable but at least one is not globally identifiable, the model is only locally identifiable. 

Consider a system for which K has been subjected to a similarity transformation to give a system with a coefficient matrix P KP, where P is nonsingular. Recall that under a similarity transformation, the eigenvalues do not change. Impose on P KP all the structural constraints on K and require that the response function of the system with matrix P" KP be the same as that of the system with matrix K. If the only P that satisfies those requirements is the identity matrix, all parameters are globally identifiable. If a P I satisfies the requirements, one can work out which parameters are not identifiable and which are. 

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... 

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... 

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... 

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] ... 

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. 

A priori identifiability thus examines whether, given the ideal noise-free data y. Equation 9.13, and the error-free compartmental model structure. Equation 9.5 or Equation 9.6, it is possible to make unique estimates of aU the unknown model parameters. A model can be uniquely (globally) identifiable — that is all its parameters have one solution — or nonuniquely (locally) identifiable — that is, one or more of its parameters has more than one but a finite number of possible values — or nonidentifiable — that is, one or more of its parameters has an infinite number of solutions. For instance, the model of Figure 9.1 is uniquely identifiable, while that of Figure 9.3 is nonidentifiable. 

Ljung, L. and Glad, T. 1994. On global identifiability for arbitrary model parametrizations. Automatica,... 

Priori global identifiability, 9-7 Prochazka, A., 70-3 Prone CPR, 18-10-18-11 Propagation, 19-9-19-12 numerical model for, 19-9 velocity of, 19-10-19-11 Propranolol, 10-12 Prostheses for limb salvage, 45-19-45-20... 

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 ... 

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