Posteriori identifiability

With regards to the analysis of the quality of the various parts of the model, one may use the same methods as are used for practical identifiability analysis. Since the same methods are used, albeit with different objectives, one sometimes refers to this model quality analysis as a posteriori identifiability (and the previous analysis as a priori identifiability). Now, however, one is also interested in how the parametric uncertainty translates to an uncertainty in the various model predictions. For instance, it might be so that even though two individual parameters have a high uncertainty, they are correlated in such a manner that their effect on a specific (non-measured) model output is always the same. Such a translation may be obtained by simulations of the model using parameters within the determined confidence ellipsoids. A global alternative to this is to consider the outputs for all parameters that correspond to a cost function that is below a certain threshold, for example 2% above the found minimum. 

The next question is whether the parameters characterizing a modei can be estimated from a set of pharmacokinetic data. The answer to this question has two parts. The first is caUed a priori identifiability. This answers the question given a particufar modei structure and experimentai design if the data are perfect/ can the modei parameters be estimated " The second part is a posteriori identifiability. This answers the ques-tioip "given a particuiar modei structure and a set of pharmacokinetic data can the modei parameters be estimated within a reasonabie degree of statisticaf precision "... 

A posteriori identifiability is linked to the theory of optimization in mathematics because one normally uses a software package that has an optimization (data-fitting) capability in order to estimate parameter values for a multicompartmental model from a set of pharmacokinetic data. One obtains an estimate for the parameter values, an estimate for their errors, and a value for the correlation (or covariance) matrix. The details of optimization and how to deal with the output from an optimization routine are beyond the scope of this chapter, and the interested reader is referred to Cobelli et al. (12). The point to be made here is that the output from these routines is crucial in assessing the goodness-of-fit — that is, how well the model performs when compared to the data — since inferences about a drug s pharmacokinetics will be made from these parameter values. 

To use a GA for Multiobjective Optimization (MO) entails comparing two solutions with respect to the multiple objectives considered [Carlos et al., 2007], [Toshinsky et al., 2000]. In the case of a singleobjective, the comparison is trivial a vector solution X is better than another one, say y, if the corresponding objective function (fitness) value f(x) is greater than f(y).A multiobjective optimization problem, instead, deals with Nf objective functions i = 1,2,..., Nf this requires that two solutions x and y are compared in terms of dominance of one solution over the other with respect to atUV)- objectives [Sawaragi et al., 1985]. The multiobjective optimality search process, converges on a Pareto-optimal set of nondominated solutions, which provides a spectrum of possible choices for the decision-maker to a posteriori identify his or her preferred solution. 

Ad a. To establish the a priori distribution one has to take into account the actual system. It turns out that many systems have a level of description where a simple guess for the probability distribution can be made. In most cases this amounts to identifying units with equal probability. When throwing two dice one computes the a posteriori distribution of the total number of points from the assumed a priori distribution made up by equal probabilities for the 36 elementary events. There are good reasons for this assumption, but as always in physics it has to be verified by experiment no amount of mathematics can show that a die is not loaded. 

Unlike biopolymer dispersions where the intrinsic viscosity is known and the polymer concentration can be chosen a priori, often for fluid foods the concentration of soluble (e.g., pectins in fruit juices) and insoluble solids can be determined only posteriori, and the determination of their zero-shear viscosities is also difficult due to instrument limitation and due to the existence of yield stress. However, in many foods, it may be possible to identify the components, called key components, that play an important role in the rheological properties. 

A rule of logic is that first principles cannot be tested by deductions that themselves rely on the same first principles. An important issue is, therefore, the question of whether the probability that a character is a homology can be estimated using a cladistic analysis, that is, if it is possible to identify homologies a posteriori (Patterson s test of congruence). The answer is complex (Wagele, 2000). 

A Posteriori Modei Discrimination. Several discrimination criteria can be used in the case no additional experimental information can be acquired. A first criterion has already been discussed in Section 3.1.2, that is, investigating the total pressure effect on the initial reaction rate potentially allows identifying the RDS in a reaction mechanism. Similar tests can be defined in which the rate equation is rearranged into a particular form to give a linear relationship between dependent and independent variables. 

Currently, potential problems are identified a posteriori thanks to feedback analysis (Karagiannis et al. 2010). A key issue is to identify potential problems of these plans before their deployment. It is necessary to know if the designed plan will... 

In the Fisher approach, parameter estimates can be obtained by nonlinear least squares or maximum likelihood together with their precision, such as, a measure of a posteriori or numerical identifiability. Details and references on parameter estimation of physiological system models can be found in Carson et al. (1983) and Landaw and DiStefano (1984). Weighted nonlinear least squares is mostly used, in which an estimate 0 of the model parameter vector 0 is determined as... 

Roussel P, Moll MC, Guez P (2008) Etape 3 Identifier les risques a posteriori. Risques QuaUte en mRieu de soins V-l 46-58... 

The classification of adsorption hysteresis loops has been always stated In terms of the appearance of these curves, e.g. their shape or extension. Among the most Important classifications, that of de Boer (ref. I) is based on a combination between the steep or sloping character of the adsorption and desorption branches, while Everett s classification (ref. 2) emphasizes the extent of the region of relative pressures at which hysteresis occurs. A classification adopted by the lUPAC (ref. 3) considers four types of loops, which are Identified according to the slope of the boundary curves. It lias been Intended, a posteriori, to relate these shapes of hysteresis loops to some processes of filling by capillary condensate or evaporation of the liquid held in a pore, and in order to justify the existence of these mechanisms, several models of the pore geometry have been considered. 

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