A posteriori methods

The e-constraint and weighting methods belong to a posteriori methods using the scalarization approach. These methods convert an MOO problem into a SOO problem, which can then be solved by a suitable method to find one Pareto-optimal solution. A series of such SOO problems will have to be solved to find the other Pareto-optimal solutions. See Chapter 6 for a discussion of the weighting and -constraint methods, their properties and relative merits. 

The classification in Figure 1.3 takes into account the recent developments and provides a good overview of available MOO methods. Relative merits and limitations of groups of methods are summarized in Table 1.1. A few of the MOO methods can be placed in another group. For example, weighting method in the a posteriori methods is a special case of value function methods in the a priori methods. The f-constraint method from the a posteriori methods and goal programming from... 

A Posteriori Methods Using Multi-Objective Approach (many based on evolutionary algorithms, simulated annealing, ant colony techniques etc.) These relatively recent methods have found many applications in chemical engineering. They provide many Pareto-optimal solutions and thus more information useful for decision making is available. Role of the DM is after finding optimal solutions, to review and select one of them. Many optimal solutions found will not be used for implementation, and so some may consider it as a waste of computational time. 

The weighting method can be used (as an a posteriori method) so that different weights are set to generate different Pareto optimal solutions and... 

We can say that interactive methods overcome weaknesses of a priori and a posteriori methods the DM does not need a global preference structure and only interesting Pareto optimal solutions need to be considered. The latter means both savings in computational cost, which in many computationally complicated real problems is a significant advantage, and avoids setting cognitive overload on the DM, which the comparison of many solutions typically implies. 

Dowling, A.P.G., van Veller, M.G.P., Hoberg, E.P., and Brooks, D.R. (2003) A priori and a posteriori methods in comparative evolutionary studies of host-parasite associations. Cladistics, 19 240-253. 

Whether the prediction scheme is a simple chart, a formula, or a complex numerical procedure, there are three basic elements that must be considered meteorology, source emissions, and atmospheric chemical interactions. Despite the diversity of methodologies available for relating emissions to ambient air quality, there are two basic types of models. Those based on a fundamental description of the physics and chemistry occurring in the atmosphere are classified as a priori approaches. Such methods normally incorporate a mathematical treatment of the meteorological and chemical processes and, in addition, utilize information about the distribution of source emissions. Another class of methods involves the use of a posteriori models in which empirical relationships are deduced from laboratory or atmospheric measurements. These models are usually quite simple and typically bear a close relationship to the actual data upon which they are based. The latter feature is a basic weakness. Because the models do not explicitly quantify the causal phenomena, they cannot be reliably extrapolated beyond the bounds of the data from which they were derived. As a result, a posteriori models are not ideally suited to the task of predicting the impacts of substantial changes in emissions. 

The effectiveness of the method is exhibited by Figure 2 in which the energy errors of truncated expansions are plotted versus the numbers of determinants in these expansions. For each of the four systems shown, one curve displays this relationship for the expansions generated by the just discussed a priori truncations, whereas the other curve is obtained a posteriori by starting with the full SDTQ calculation in the same orbital basis and, then, simply truncating the determinantal expansion based on the ordering established by the exact coefficients of the determinants. There is practically no difference in the number of determinants needed to achieve an accuracy of 1 mh. 

BSSE also affects the shape of the potential energy surface and the energy derivatives. There have been numerous attempts to find a general scheme to eliminate this error, and both a posteriori [2] and a priori [3] schemes are available. The counterpoise approach (CP) by Boys and Bemardi [5] and related methods are the most common a posteriori procedures. Within this method, the monomer electrons are described by the same basis functions as those used in the complex by means of the so... 

There have been many attempts to formulate a procedure to avoid it and both a posteriori and a priori schemes are available. The counterpoise approach (CP) (Boys and Bemardi, 1970) and related methods are the most conunon a posteriori procedures. Although this technique represents the most frequently employed a posteriori procedure to estimate this error, several authors have emphasised that the method introduced by Boys and Bemardi does not allow a clear and precise determination of the BSSE. The addition of the partner s functions introduces the "secondary superposition error" a spurious electrostatic contribution due to the modification of the multipole moments and polarizabilities of the monomers. This is particularly important in the case of anisotropic potentials where these errors can contribute to alter the shape of the PES and the resulting physical picture (Xantheas, 1996 and Simon et al., 1996). 

The assumption of multivariate normal distribution underlying this method gives for the conditional (a posteriori) probability density of the category g... 

The future development of chemometrics may be characterized by the balancing of opposites [GELADI, 1995] combination of hard and soft information, using a priori information when available and combining with a posteriori information. So, a holistic strategy (Fig. 1-2) with different chemometric methods in each of its steps may help to solve problems of the real world of chemistry and other disciplines . 

---