Model, mathematical polynomial

Regression analysis is one of the main tools in generating mathematical models by fitting a model equation to experimental data. In general, the regression analysis is based on the application of the least squares method for the estimation of unknown coefficients in the model equation. This method minimizes the sum of squares of the differences between the experimental values of the dependent variable, and those estimated by the model, y . Polynomials of various degrees are often used to describe complex non-linear relationships between the dependent and independent variables because the model equation is linear with respect to the unknown coefficients, and therefore the procedure for the calculation of the coefficients reduces to the solution of a system of linear simultaneous equations. 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

Binary solutions have been extensively studied in the last century and a whole range of different analytical models for the molar Gibbs energy of mixing have evolved in the literature. Some of these expressions are based on statistical mechanics, as we will show in Chapter 9. However, in situations where the intention is to find mathematical expressions that are easy to handle, that reproduce experimental data and that are easily incorporated in computations, polynomial expressions obviously have an advantage. 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series... 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series from which the trend is to be identified is assumed to be generated by a nonstationary process where the nonstationarity results from a deterministic trend. A classical model is the regression or error model (Anderson, 1971) where the observed series is treated as the sum of a systematic part or trend and a random part or irregular. This model can be written as... 

The tools we created in Chapter 3, Physical/Chemical Models, form the core of the fitting algorithms of this chapter. The model defines a mathematical function, either explicitly (e.g. first order kinetics) or implicitly (e.g. complex equilibria), which in turn is quantitatively described by one or several parameters. In many instances the function is based on such a physical model, e.g. the law of mass action. In other instances an empirical function is chosen because it is convenient (e.g. polynomials of any degree) or because it is a reasonable approximation (e.g. Gaussian functions and their linear combinations are used to represent spectral peaks). 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

The form of the response function to be fitted depends on the goal of modeling, and the amount of available theoretical and experimental information. If we simply want to avoid interpolation in extensive tables or to store and use less numerical data, the model may be a convenient class of functions such as polynomials. In many applications, however, the model is based an theoretical relationships that govern the system, and its parameters have some well defined physical meaning. A model coming from the underlying theory is, however, not necessarily the best response function in parameter estimation, since the limited amount of data may be insufficient to find the parameters with any reasonable accuracy. In such cases simplified models may be preferable, and with the problem of simplifying a nonlinear model we leave the relatively safe waters of mathematical statistics at once. 

The so-called S-FIT method and the sum-of-broadened-rectangles (SOBR) method both use more sophisticated mathematical assumptions to model the shape of the S-phase region of the histogram. A polynomial equation (S-FIT) or a series of broadened Gaussian dis-... 

Lack of fit of the obtained model has to be statistically checked, so that, if needed, the polynomial degree may be augmented. Knowing the mathematical model of the research subject for several responses is a prerequisite in solving optimization with multiple responses. The computation of this is solved geometrically or by use of computers and the method of linear algebra. 

When several mathematical models fulfill the requirements, then, in principle, a simpler model is accepted. For us, it is the polynomial model that in the case of two factors may have these forms ... 

Apart from optimization, a problem is often set for mathematical modeling or interpolation. The optimum does not interest us in that case but the model that adequately describes the obtained results in the experimental field. A subdomain is not chosen in that case, but the polynomial order is moved up until an adequate model is obtained. When a linear or incomplete square model (with no members with a square factors) is adequate it means that the research objective corresponds to the optimization objective. 

Naturally, the number of initial experiments required to start the optimization procedure will increase if either the number of parameters considered or the complexity of the model equations increases. As far as the number of parameters is concerned, we have seen this to be true with any optimization procedure, and hence the number of parameters should be carefully selected. In order to avoid a large number of initial experiments, the complexity of the model equations may be increased once more data become available during the course of the procedure. For example, retention in RPLC may be assumed to vary linearly with the mixing ratio of two iso-eluotropic binary mixtures at first. When more experimental data points become available, the model may be expanded to include quadratic terms. However, complex mathematical equations, which bear no relation to chromatographic theory (e.g. higher order polynomials [537,579]) are dangerous, because they may describe a retention surface that is much more complicated than it actually is in practice. In other words, the complexity of the model may be dictated by experimental... 

---