0.1 Models  (Page 2/2)

It may seem that the idea of a model and its use is being made so general that it is useless. Our purpose inbeing so general is to search for what is common in these diverse examples, and to extract it for study and more efficientuse.

Mathematical model

Rather than follow further examples of mental or verbal models, we will turn to another form of model: themathematical model. The incredible advances of the physical sciences and engineering disciplines have resulted from thedevelopment and use of mathematical models. When one describes the relation between the force applied to an object and its mass andacceleration by the familiar formula $F=M\cdot A$ , one is using a mathematical model of a physical phenomena. Here, mathematicalfunctions are used to represent physical qualities, so that the interrelationship can be described by equations. If these equationsare fundamental and if their solutions accurately simulate the actual phenomena, then they are given the special status of "laws".Consider the cases of Newton's laws, Kirchoff's laws, Faraday's law, Boyle's law, and in other fields, Fechner's law(psychophysics), Paneto's law (economics), etc. Indeed, model formulation and verification is the basis of the so-calledscientific method.

An important feature of the various types of models we have discussed is how one can move from one to another.Consider the following hypothetical account of how a physical "law" was developed.

At some point in time it was noticed that if a heavy object was dropped, it fell down. As further experience wasaccumulated, it was noticed that the object always fell in a straight line toward the earth. Next, after closer observation, itwas discovered that the object's speed increased as it fell. The next step was a major one. It required curiosity, mathematical ability, and a real quantum jump to move from the verbal model to amathematical one where it was conjectured that the velocity was a linear function of time after being dropped, V = Kt . This provedto be incredibly accurate, and thus, a "law" was discovered.

The use of mathematical models has been so successful in many areas that the concept of a model was sometimesforgotten. Indeed, some models are so accurate that users can forget that they are dealing with models and not the actualphenomenon. When we work in areas where accurate models are not available, a good understanding of the modeling process becomesessential.

The first step in choosing a model is deciding what the purpose of the model will be and what questionsare being asked. It is obviously an advantage to use the simplest model possible to serve a particular purpose but the danger thatover-simplification will destroy the validity of the model always exists.

The second step is the actual construction of the model. Here, the various theories, laws, relations, etc. thatapply must be used, and after that, the model requires that new relations be established. In other words, while building the model,one often discovers what data should be collected and what experiments must be performed, as well as what data is irrelevantor misleading. At this point, alterations are often substantial in the model.

The third step is verification or validation of the model. This usually involves some comparisons of the modelwith the phenomenon it models. One must be very careful at this point to test all of the characteristics the model should have,while remaining within the original goals and purposes set, not violating the assumptions that were made. A common mistake is to use models outside thearea for which they were intended.

Verification often involves applying the model to data that was not used in its construction to see if itcan explain the observations. If internal relations were used to derive some data, these can be compared with observations. On theother hand, if the model were built by forcing agreement with the observations, then the resulting implied internal relations can beexamined for their validity.

All of these steps are done in a rather circular fashion with the attempted use of, and verification of,the model suggesting modification, restructuring and reverification, or in some cases, discarding the whole approach.Some reflection will perhaps show that these are common ideas in modeling, and we try to systematically apply them to the veryinteresting but very difficult problem of modeling large groups of people.