General linear model approach

Engel, J. and Huele, A. F. (1996). A generalized linear modeling approach to robust design. 

Depending on study goals, a general linear modeling approach might be applied to these kinds of data. If there are more metals in the mixture, the independent joint action model can be expanded to Equation (1.5). 

Using a general linear model (GLM) approach to repeated measures data with time treated as a categorical variable is limited in two respects. First, such a model... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

This general approach for solving linear pharmacokinetic problems is referred to as the y-method. It is a generalization of the approach by means of the Laplace transform, which has been applied in the previous Section 39.1.6 to the case of a two-compartment model. 

We will use the constraint that the sum of squares of the residuals be minimal. The following is a brief development of the matrix approach to the least squares fitting of linear models to data. The approach is entirely general for all linear models. 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

From a conceptual point of view, nothing new is needed to further extend the above approach. For instance, the one-box model with two variables shown in Fig. 21.7 can be combined with the two-box (epilimnion/ hypolimnion) model. This results in four coupled differential equations. Even if the equations are linear, it is fairly complicated to solve them analytically. Computers can deal more efficiently with such problems, thus we refrain from adding another example. But we should always remember that independently from how many equations we couple, the solutions of linear models always consist of the sum of a number of exponential terms which have exactly one steady-state, although it may be at infinity. In Section 21.4 we will discuss the general structure of linear differential equations. 

This type of modeling approach can also be generalized to K points, and hence (K - 1) linear segments and results in ... 

Summary. Two principal methods for removal of low frequency noise transients are currently available. The model-based separation approach has shown more flexibility and generality, but is computationally rather intensive. It is felt that future work in the area should consider the problem from a realistic physical modelling perspective, which takes into account linear and non-linear characteristics of gramophone and film sound playback systems, in order to detect and correct these artifacts more effectively. Such an approach could involve both experimental work with playback systems and sophisticated non-linear modelling techniques. Statistical approaches related to those outlined in the click removal work (section 4.3.4) may be applicable to this latter task. 

An alternate approach is to impose continuity constraints prior to estimation, thus utilizing the sine waves that would be eliminated aposteriori due to ill-conditioning. For example, a linear model for each frequency trajectory can be shown to lead to a generalization of Equation (9.77). Such an approach may lead to more robust separation with the presence of closely spaced frequencies. 

The most general approach to model-based nonlinear control is the so-called Feedback Linearization (FL) [35], In fact, FL control approaches use the model of the plant to achieve a global linearization of the closed-loop systems, so as well-established linear controllers can be adopted for the globally linearized model. In... 

---