Model | Research & Encyclopedia Articles

Model

A mathematical model is an equation or system of equations and/or inequalities used to describe data from real world situations and to make predictions from that data. The model may be as simple as a linear function or as complicated as a system of hundreds or even thousands of higher degree equations programmed into a powerful computer. Whatever the complexity of the model, the goal of the modeler is to create a mathematical system which comes as close as possible to capturing both the quantitative and qualitative properties of the real world system being modeled. Thus if an economist collects data on price and demand and observes that her data seems to be quite linear, she will undoubtedly attempt to find a linear function that is a good fit for the collected data. Her desire is to be able to make predictions about future demand based on future prices or vice-versa. If her assumption that this relationship will continue to remain linear is correct, then her simple linear model should do a good job at predicting future events. On the other hand, she might be interested in modeling the entire economy of a country. In this case, she would undoubtedly need to construct a far more sophisticated model consisting of many equations with many variables to account for the vast array of forces that influence the workings of a national economy. Such a complicated model would undoubtedly be programmed into a computer and many different simulations of the economy could be run on the computer based upon changing various critical assumptions about different components of the economy.

Some mathematical models give predictions that are consistently so accurate that scientists use them repeatedly with near certain confidence. One of the most famous examples of a successful mathematical model is the system of differential equations developed by Sir Isaac Newton (1642-1727) to describe the motion of the planets about the sun. Newtons laws of planetary motion have been used not only to accurately predict motion in the solar system, but to provide modern day scientists and engineers a basis for sending satellites into space and controlling their motion. Perhaps the greatest engineering achievement of the 20th century was sending humans to the moon and returning them safely to earth. Newtons mathematical model was the foundation for this tremendous feat. On the other hand some real world systems have more or less defied attempts to model them with such a high degree of certainty as Newtons model.

A notorious example is the weather. When meteorologists try to predict where a hurricane will strike land, they are able to state their predictions only within a very broad range of probabilities. The problem is that there are so many potential variables involved in weather systems that it has been impossible to construct a sophisticated enough mathematical model to take all these variables into account. Some mathematicians and meteorologists have suggested that the weather will always be impossible to predict with a high degree of accuracy because weather systems fall into a class of dynamical systems which are called chaotic. A chaotic system is one which exhibits extreme sensitivity to its initial conditions, meaning that any small error in an initial value put into a mathematical model will lead to very great errors in the final predictions of the model. Nevertheless, with computing power increasing exponentially every year, scientists continue to build more and more sophisticated mathematical models to predict the outcome of such varied systems as the weather, the economy, the stock market, and more.

It is important to understand that with very complicated systems, the best model is not always the first one proposed. Often many models are suggested, tested, and rejected before a correct model is adopted. Even then, the modelers are open to the possibility that new and unexpected conditions may arise that will necessitate modifications to the model. In fact, the modeling of sophisticated systems is very often a trial and error process, an attempt to build the best model by successive approximations to reality. One might think that a mathematician trying to capture the essence of a very complicated system would, form the beginning, attempt to account for every variable that she could possibly imagine would have an impact on the system. This is not typically the case. Usually one begins with a very simple model and only increases its complexity as the evidence warrants. It is a golden rule among mathematicians that, when it comes to models, simpler is better as long as the simpler model is able to account for the essential behavior of the system being modeled. This is not only an esthetic desire, but a practical one as well. A simpler model means a simpler mathematical analysis will be required to explain the system.