Model ranking

This paper discusses (1) soil and groundwater and (2) aquatic equilibrium and ranking models. The second category deals with the chemical speciation in soil and groundwater, and with the environmental rating of waste sites, in cases where detailed modeling is not desirable. 

This paper presents a review discussion of soil, groundwater, aquatic equilibrium and ranking modeling concepts including selected documented models. Watershed models are not discussed, since the work of Knisel (1) is one of the most representative watershed computerized packages. 

Ranking Models Easy to use with available data Simplistic approach, output reflects user s intuition Employed by the EPA, U.S. Army, Air Fore and Navy... 

MITRE (1981). Site ranking model for determining remedial action priorities among uncontrolled hazardous substances facilities. The MITRE Co., McLean, VA 22102. 

FIGURE 8 Schematic of FDA s pilot risk-ranking model for calculation of site risk potential. 

U.S. Department of Health and Human Services (2004), Risk-based method for prioritizing cGMP inspections of pharmaceutical manufacturing sites—A pilot risk ranking model, Food and Drug Administration, Rockville, MD. 

Pavan, M., Consonni, V. and Todeschini, R. (2005) Partial ranking models by genetic algorithms variable subset selection (GA-VSS) approach for environmental priority settings. MATCH Commun. Math. Comput. Chem., 54, 583-609. 

Depending on the quality of data and the method selected, constraints on the parameters to be estimated may be required in order to get a chemically meaningful solution. In the case of multivariate curve resolution (MCR) (see Section 3.2) performed on one 2D NMR spectrum, application of constraints is mandatory. If constraints are not applied, it can be shown that there is an infinity of equally well-fitting solutions and hence the true underlying parameters (spectra, concentrations) cannot be estimated directly. This is known as the rotational ambiguity of two-way low-rank models. 

New QSAR Modelling Approach Based on Ranking Models by Genetic Algorithms - Variable Subset Selection (GA-VSS)... 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

A ranking model is a relationship between a set of dependent attributes, experimentally investigated, and a set of independent attributes, i.e. model variables. As in regression and classification models the variable selection is one of the main step to find predictive models. In the present work, the Genetic Algorithm (GA-VSS) approach is proposed as the variable selection method to search for the best ranking models within a wide set of predictor variables. The ranking based on the selected subsets of variables is... 

A ranking model is defined as a relationship between one or more dependent attributes, investigated experimentally, and a set of independent attributes, also called model attributes, which are usually theoretical calculated variables such as molecular descriptors ... 

Thus, the ranking model is given by the chosen ranking function and the ordered training set. 

It consists in the evolution of a population of models, i.e. a set of ranked models according to some objective function, based on the crossover and mutation processes, which are alternatively repeated until a stop condition is encountered (e.g., a user-defined maximum number of iterations) or the process is ended arbitrarily. 

Variable subset selection is performed by GAs, optimising populations of models according to a defined objective function related to model quality. In partial ranking models objective function is an expression of the degree of agreement between the element ranking resulting from experimental attributes and that provided by the selected subset of model attributes. 

Thus, in a first step the ranking of the unknown element u is predicted with respect to the training set elements and, in the second step, the experimental responses are predicted. In this case, despite the regression models, the ranking model provides not a single response value but an interval. 

A numerical example for partial ranking model is here provided to better explain the prediction calculation. For the sake of simplicity, let us consider an experimental ranking developed on two experimental attributes yt and y2 Table 1 shows their numerical values. Fig. 3 shows the resulted experimental Hasse diagram together with the ranking model developed on the training set composed by 9 elements a, b, c, d, e,f g, h, /, described by an arbitrary set of independent attributes. 

The overall ranking model quality, i.e. taking into account all the R responses, can be evaluated by the following expressions ... 

The goodness of fit of the partial ranking model calculated by the similarity index is lower than that calculated by T(0,0) but higher than the one by T(0,1) and T(l,l), confirming that the similarity index S(E,M) is a reasonable compromise between the over optimistic and the over pessimistic evaluation provided by T(0,0), and T(0,1), T(l,l), respectively. 

The overall ranking model quality, i.e. taking into account all the four responses, has been evaluated from the above parameters by arithmetic mean (QT), geometric mean (0G) and by the minimum value obtained on the four responses (QM) ... 

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