4.
Process Modeling
4.4. Data Analysis for Process Modeling 4.4.2. How do I select a function to describe my process?


Choose Functions Whose Properties Match the Process  Incorporating scientific knowledge into selection of the function used in a process model is clearly critical to the success of the model. When a scientific theory describing the mechanics of a physical system can provide a complete functional form for the process, then that type of function makes an ideal starting point for model development. There are many cases, however, for which there is incomplete scientific information available. In these cases it is considerably less clear how to specify a functional form to initiate the modeling process. A practical approach is to choose the simplest possible functions that have properties ascribed to the process.  
Example: Concrete Strength Versus Curing Time  For example, if you are modeling concrete strength as a function of curing time, scientific knowledge of the process indicates that the strength will increase rapidly at first, but then level off as the hydration reaction progresses and the reactants are converted to their new physical form. The leveling off of the strength occurs because the speed of the reaction slows down as the reactants are converted and unreacted materials are less likely to be in proximity all of the time. In theory, the reaction will actually stop altogether when the reactants are fully hydrated and are completely consumed. However, a full stop of the reaction is unlikely in reality because there is always some unreacted material remaining that reacts increasingly slowly. As a result, the process will approach an asymptote at its final strength.  
Polynomial Models for Concrete Strength Deficient  Considering this general scientific information, modeling this process using a straight line would not reflect the physical aspects of this process very well. For example, using the straightline model, the concrete strength would be predicted to continue increasing at the same rate over its entire lifetime, though we know that is not how it behaves. The fact that the response variable in a straightline model is unbounded as the predictor variable becomes extreme is another indication that the straightline model is not realistic for concrete strength. In fact, this relationship between the response and predictor as the predictor becomes extreme is common to all polynomial models, so even a higherdegree polynomial would probably not make a good model for describing concrete strength. A higherdegree polynomial might be able to curve toward the data as the strength leveled off, but it would eventually have to diverge from the data because of its mathematical properties.  
Rational Function Accommodates Scientific Knowledge about Concrete Strength  A more reasonable function for modeling this process might be a rational function. A rational function, which is a ratio of two polynomials of the same predictor variable, approaches an asymptote if the degrees of the polynomials in the numerator and denominator are the same. It is still a very simple model, although it is nonlinear in the unknown parameters. Even if a rational function does not ultimately prove to fit the data well, it makes a good starting point for the modeling process because it incorporates the general scientific knowledge we have of the process, without being overly complicated. Within the family of rational functions, the simplest model is the "linear over linear" rational function $$ y = \frac{\beta_0 + \beta_1 x}{1 + \beta_2 x} + \varepsilon $$ so this would probably be the best model with which to start. If the linearoverlinear model is not adequate, then the initial fit can be followed up using a higherdegree rational function, or some other type of model that also has a horizontal asymptote.  
Focus on the Region of Interest  Although the concrete strength example makes a good case for incorporating scientific knowledge into the model, it is not necessarily a good idea to force a process model to follow all of the physical properties that the process must follow. At first glance it seems like incorporating physical properties into a process model could only improve it; however, incorporating properties that occur outside the region of interest for a particular application can actually sacrifice the accuracy of the model "where it counts" for increased accuracy where it isn't important. As a result, physical properties should only be incorporated into process models when they directly affect the process in the range of the data used to fit the model or in the region in which the model will be used.  
Information on Function Shapes  In order to translate general process properties into mathematical functions whose forms may be useful for model development, it is necessary to know the different shapes that various mathematical functions can assume. Unfortunately there is no easy, systematic way to obtain this information. Families of mathematical functions, like polynomials or rational functions, can assume quite different shapes that depend on the parameter values that distinguish one member of the family from another. Because of the wide range of potential shapes these functions may have, even determining and listing the general properties of relatively simple families of functions can be complicated. Section 8 of this chapter gives some of the properties of a short list of simple functions that are often useful for process modeling. Another reference that may be useful is the Handbook of Mathematical Functions by Abramowitz and Stegun [1964]. The Digital Library of Mathematical Functions, an electronic successor to the Handbook of Mathematical Functions that is under development at NIST, may also be helpful. 