Mathematical modeling empirical-statistical models

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

While computers are a substantial aid in statistical analysis, it is also true that statistical methods have helped in certain computer applications. In Section V the subject of mathematical models will be discussed. These are in many cases based on empirical correlations. When these have been obtained by regression methods, not only is the significance of the results better understood, but also the correlation is expressed directly in a mathematical form suitable for programming. 

Response Surface Methodology (RSM) is a statistical method which uses quantitative data from appropriately designed experiments to determine and simultaneously solve multi-variate equations (3). In this technique regression analysis is performed on the data to provide an equation or mathematical model. Mathematical models are empirically derived equations which best express the changes in measured response to the planned systematic... 

A model is a representation or a description of the physical phenomenon to be modelled. The physical model (empirical by laboratory experiments) or conceptual model (assembly of theoretical mathematical equations) can be used to describe the physical phenomenon. Here the word model refers to a mathematical model. A (mathematical) model as a representation or as a description of a phenomenon (in the physical space) is a systematic collection of empirical and theoretical equations. In a model (at least in a good model) both approaches explain and predict the phenomenon. The phenomena can be predicted either mechanistically (theoretically) or statistically (empirically). 

Different results may be observed under conditions that are ostensibly the same. To keep track of this variation, we must maintain records or statistics. There are two general strategies that we may employ. First, we may simply store the results. That is, if we have a thousand observations, we can maintain access to all the individual values. The record may then be employed as an empirical distribution function, in which particular percentiles may be identified on demand. Second, we may use a mathematical model to summarize the distribution. There are two very different reasons for doing this. First, a statistical model may be used to provide a concise summary. The facility with which an analyst can store and retrieve data makes this motivation less compelling than it once was. Second, when a sparse data set is not considered representative of a large population, a model may also be used to infer or predict values that are not represented in the data set. 

Through this step and based on experimental evidence we try to develop the appropriate model to describe the test chamber kinetics. As was anticipated in the introduction of this Chapter, from a conceptual point of view, two broad categories of models can be developed empirical-statistical and physical-based mass transfer models. It should be emphasized that, in several cases, even the fundamentally based mass transfer models are indistinguishable from the empirical ones. This happens because the mass transfer models are generally very complex in both the physical concept involved and the mathematical treatment required. This often leads the modelers to introduce approximations, making the mass transfer models not completely distinguishable from some empirical models in terms of both functional formulations and descriptive capabilities. Considering the current status of models which have been developed to describe VOC emissions (and/or sink processes), we could define the mass transfer models as hybrid-empirical models. 

Statistical ideas are often used to derive empirical equations from sets of data, and these equations form the basis for mathematical models used by biological engineers (see Section 1.4). 

Numerous mathematical models exist the ones to be considered, however, are only those which are not of the empirical or semiempirical type the assumptions of which appear correct from a thermodynamical and statistical-mechanical point of view and the numerical solution of which does not present an unrealistically large computational task even in the case of thick adsorption layers. One of these is the so-called lattice model. Aversion of this model suitable for the description of the structure and thermodynamical properties of multimolecular adsorption layers was used in our calculations. 

Scientific models can take many shapes and forms, but they all seek to characterize response variables through relationships with appropriate factors. Traditional models can be divided into two main categories mathematical or theoretical models and statistical or empirical models. Mathematical models have the common characteristic that the response and predictor variables are assumed to be free of specification error and measurement uncertainty. Statistical models, on the other hand, are derived from data that are subject to various types of specification, observation, experimental, and/or measurement errors. In general terms, mathematical models can guide investigations, and statistical models are used to represent the results of these investigations. 

The problem of how to fit a random process model to a given physical situation, i.e., what values to assign to the time averages, is not a purely mathematical problem, but one involving a skillful combination of both empirical and theoretical results, as well as a great deal of judgement based on practical experience. Because of their involved nature, we shall not consider such problems (called problems in statistics to distinguish them from the purely mathematical problems of the theory of random processes) in detail here, but instead, refer the reader to the literature. ... 

In this section, methods are described for obtaining a quantitative mathematical representation of the entire reaction-rate surface. In many cases these models will be entirely empirical, bearing no direct relationship to the underlying physical phenomena generating the data. An excellent empirical representation of the data will be obtained, however, since the data are statistically sound. In other cases, these empirical models will describe the characteristic shape of the kinetic surface and thus will provide suggestions about the nature of the reaction mechanism. For example, the empirical model may require a given reaction order or a maximum in the rate surface, each of which can eliminate broad classes of reaction mechanisms. 

Section 1.6.2 discussed some theoretical distributions which are defined by more or less complicated mathematical formulae they aim at modeling real empirical data distributions or are used in statistical tests. There are some reasons to believe that phenomena observed in nature indeed follow such distributions. The normal distribution is the most widely used distribution in statistics, and it is fully determined by the mean value p. and the standard deviation a. For practical data these two parameters have to be estimated using the data at hand. This section discusses some possibilities to estimate the mean or central value, and the next section mentions different estimators for the standard deviation or spread the described criteria are fisted in Table 1.2. The choice of the estimator depends mainly on the data quality. Do the data really follow the underlying hypothetical distribution Or are there outliers or extreme values that could influence classical estimators and call for robust counterparts ... 

Any QSAR method can be generally defined as an application of mathematical and statistical methods to the problem of finding empirical relationships (QSAR models) of the form ,- = k(D, D2,..., D ), where ,- are biological activities (or other properties of interest) of molecules, D, P>2,- ,Dn are calculated (or, sometimes, experimentally measured) structural properties (molecular descriptors) of compounds, and k is some empirically established mathematical transformation that should be applied to descriptors to calculate the property values for all molecules (Fig. 6.1). The goal of QSAR modeling is to establish a trend in the descriptor values, which parallels the trend in biological activity. In essence, all QSAR approaches imply, directly or indi-... 

G. E. P. Box and N. R. Draper, Empirical Model Building and Response Surfaces, Wiley Series on Probability and Mathematical Statistics, Wiley, New York, 1987. 

The model or set of models to be used in the exposure assessment to relate the presence of a substance to human exposure/absorbed dose should be stated. The model s general description should provide enough detail so that the user or reviewer understands the input variables, underlying mathematical algorithms and data transformations and output/results, such that the model can be easily compared to other alternatives. The basis for each model, whether deterministic, empirical or statistical, should be described. The statement of the model should include which variables are measured and which are assumed. A description should be provided of how uncertainties in the parameters and the model itself are to be evaluated and treated. 

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