A Defined Domain of Applicability

One key aspect of model applicability is the definition of the chemical space and the way in which chemical similarity is measured, as chemical similarity is a relative concept. The similarity or distance values depend on both the type of molecular representation or the distance measure used. Due to this lack of 

For the aforementioned reasons there is, so far, no generally accepted or even standardized approach for defining the chemical space of QSAR models, and there are no reasons to expect that one method is the absolute best. However, given this uncertainty, AD is used to make the decision as to whether or not a QSAR prediction should be more or less reliable. This is a crucial and hot topic, and was dealt with at a JRC workshop, where several different approaches for linear and non-linear models were proposed in relation to different model typologies it is the topic of various publications.  

For an analysis of the AD of regression models, the author has always used the Williams plot, which is now widely applied by other authors and commercial software. The Williams plot is the plot of standardized cross-validated residuals (R) versus leverage (Hat diagonal) values (h from the HAT matrix). It allows an immediate and simple graphical detection of both the response outliers i.e., compounds with cross-validated standardized residuals greater than 2-3 standard deviation units) and structurally anomalous chemicals in a model (h h, the critical value being h = 3p /n, where p is the number of model variables plus one, and n is the number of the objects used to calculate the model).40,62,66 

Additional consideration of other components of the QSAR AD, such as the physico-chemical domain, descriptor domain, mechanistic domain and metabolic domain (when possible) would allow even more improved confidence levels in predictive model applications. 

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The most important one is that the model should be appropriately validated to confirm the reliability of its predictions. First rules of the validation were worked out in March 2002 at an international workshop held in Setubal, Portugal ( Setubal Rules ). In November 2004, the rules were discussed and modified by the OECD Work Program on QSAR they are now known as the OECD Principles. According to these principles, each QSAR model should be associated with (a) a well-defined endpoint (b) an unambiguous algorithm (c) a defined domain of applicability (d) appropriate measures of goodness-of-fit, robustness and predic-tivity and (v) a mechanistic interpretation, if possible [15, 16]. 

A defined domain of applicability It is realized that (Q)SARs are reductionist models that inevitably have limitations in terms of the types of chemical structures that can be predicted in other words, define the applicability of the model based on the domain. 

A defined domain of applicability. It is a necessary to estimate an applicability domain for each developed model ... 

Principle 3 a defined domain of applicability for example, the most preeminent ecotoxicities of the species, along mutagenicity and carcinogenicity by non-congeneric molecular series towards HTV inhibition by congeneric molecules ... 

In developing madiematical modeling in chemistry and physics various branches of mathematics have been widely employed, frequently each with a well defined domain of applicability This sometimes has not been sufficiently emphasized or recognized, particularly when less familiar or less frequently used mathematical disciplines are involved, such as combinatorics, topology, graph theory, set theory or category theory The lack of appreciation of the specifics of each such discipline... 

An extension of the application of these maps to the systematic description of certain groups of ternary alloys has been presented also by Pettifor (1988a, b). Composition averaged Mendeleev numbers can be used, for instance, in the description of pseudo-binary, ternary or quaternary alloys. All these maps show well-defined domains of structural stability for a given stoichiometry, thus making the search easier for new ternary or quaternary alloys with a particular structure type (and which, as a consequence, may have the potential of interesting properties and applications (Pettifor 1988a, b)). 

It needs to be emphasized that no matter how robust, significant, and validated a QSAR may be, it cannot be expected to reliably predict the modeled property for the entire universe of chemicals. Therefore, before a QSAR model is put into use for screening chemicals, its domain of application must be defined and predictions for only those chemicals that fall in this domain should be considered reliable. Some approaches that aid in defining the applicability domain are described below. 

The domain of applicability of the QSAR should be explicitly defined. The QSAR should be associated with a description of the chemical classes for which it is applicable (inclusion rules) or inapplicable (exclusion rules). For QSARs, there should be an indication of the range of... 

The domain of applicability of the QSAR was well defined by the model developer. The QSAR was stated to be applicable to chemicals having log K,v values in the range from -1.24 to 5.13, and operating by a non-polar narcosis mechanism of action. Such chemicals can be identified on a structural basis (Verhaar et al., 1992), or from physicochemical descriptors (Boxall et al., 1997). 

As seen previously, the chemical reactions studied most often are the exchange ones. Those requiring several potential energy surfaces of excited states (diabatic reactions) are worth special mention, since they most certainly define a domain of application with a future for classical trajectories. An electron jump from one surface to another requires either to be given a statistical probability of occurence by the Landau Zener formula (or one of its improved versions " ) or to be described by means of complex-valued classical trajectories as a direct and gradual passage in the complex-valued extension of the potential surfaces (generalization of the classical S-matrix ). 

In their present stage of development, these new tools are appropriate for a very wide range of systems. The applications presented here, in the references, and in the publications that will appear in the coming months should reasonably define their current domain of application. While they are very powerful even now, further understanding and development is... 

In summary, the use of implicit solvation models in molecular simulations offers considerable rewards, both at conceptual and practical levels. However, compared to the more established explicit solvent approach, less is known about the domain of applicability of these models, and so extra care must be taken when using them in practice. Drawing on the analogy with the development of the empirical explicit solvent force-fields over the past 30 years, it is likely that improvements in the implicit solvent framework accompanied by accumulation of practical experience will eventually make the framework a standard approach within its reasonably well-defined domain. 

The essential principles of model quality assessment are mature and well documented and aim to provide confidence in the model s ability to predict the future. The principles are not always well-applied. The complexity of the machine learning techniques and the easy availability of large numbers of descriptors can seduce or confuse model builders and users into an over-optimistic assessment of a model s quality. Describing the relationship between prediction accuracy and precision, and the distance of future compounds from the model domain of applicability defines the central uncertainty. Many different descriptions of the domain of applicability have been used and a number have shown utility in relating the distance to the error in prediction with some datasets. Further work in this area is warranted and indeed recommended by the OECD guidelines in QSAR model validation. 

Application software refers to software that is written to perform a particular function of general utility. While the software is of course run on a computer, the function it performs is not particularly computer related. For example, a word processor is a typical example of application software. It helps users to create text documents. It deals with concepts from a domain of interest to users in the document processing domain such as documents, pages, paragraphs, words, justification, hyphenation, and so on. The requirements for such software may be defined entirely in terms of these concepts independently of the execution platform. Of course, the execution platform does exert some influence on the software in terms of what the software can reasonably achieve. For example, the software has access to different amounts of resources if it runs on a powerful desktop computer or a small hand-held device. But the functionality requirements may be stated in terms of the application domain. 

Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

For a tool of either category to be employed, there should be a precise definition of the tool s functionality. For a software development tool, the domain of applicability should be precisely known, and for a software verification tool, the analyses or checks it performs should be well defined. 

For each model that abstracts from the real system, there is the concern whether the model s behavior is a reasonable approximation of the real behavior (Huselius et al., 2006). This concern is addressed by model validation, which can be defined as the "substantiation that a computerized model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model" (Schlesinger et al., 1979). Software simulation models can be validated by comparing trace data of the real system versus the model (cf. Figure 4). There are many possible approaches, including statistical and subjective validation techniques (Balci, 1990). Kraft describes a five-step validation process that combines both subjective and statistical comparisons of tracing data (Kraft, 2010, chapter 8). 

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