Modeling | Research & Encyclopedia Articles

Modeling

Mathematical modeling is the process of constructing a mathematical system designed to describe, analyze, and predict future outcomes of real world phenomena. Mathematical models are commonly used in physics, chemistry, biology, business, economics, finance, sociology, anthropology, and many other areas as well. Practitioners in these disciplines believe that their areas of study will be most seriously regarded if they have a solid mathematical foundation under them. A mathematical model may be as simple as a linear function or as complicated as a system of thousands of equations in thousands of variables which requires a powerful computer to run simulations based upon the model. However simple or complex the model may be, the steps in the modeling process generally follow a process similar to the following steps. First, a scientist makes observations and collects data from some real world system that she wishes to understand. Second, the scientist focuses on what she believes to be the essential elements of the real world system and strips away from the modeling process anything that seems to be extraneous to the workings of the system. Third, based upon an initial analysis of the data, she hypothesizes a mathematical model - an equation or inequality or some system of these. Fourth, she tests the model by doing computations to see if the model generates values which are reasonably close to the data collected from the real world system. If so, she may either accept the model or, more likely, attempt to make some adjustments which will give even closer correspondence to the real world system. If the original model gives results which are far away from the data values, then more dramatic adjustments will need to be made, possibly even bringing into the model terms or equations which account for some of the processes originally believed to be extraneous. In either case, the modified model is tested again to see how well it simulates the real world data. This process of modifying and testing is repeated until the scientist is satisfied that she has a model system which generates results close enough to the real system data that it can be successfully used for making predictions of future events or of events in the far past before any data had been collected.

Consider an example of the modeling process described in the preceding paragraph.

Suppose an economist wishes to study the dynamics of price, supply, and demand for a certain commodity in an economy. First he would collect data either from his own or someone else's research. Ideally, this data would be based on a fairly large sample of several years duration and would consist of recorded values of price, supply, and demand for the commodity. Suppose that after looking at some graphical representations of the data, the economist decides to use a system of linear equations to model this data. He tests the model and finds that it is not so good at simulating the dynamics of the real world system. He makes modifications, perhaps changing one or more of the equations from linear to non-linear, with the hope that these modifications will allow the model to give results that are in closer correspondence with the data. He continues to make modifications in successive stages of the model until his simulations based on the model are satisfactory. The process of developing mathematical models in this way is both creative and inventive, involving both science and art in addition to mathematics. Very often a mathematical model can be viewed as a "work in progress" as the modeler continues to tweak it to get better and better correspondence with data actually collected from the real world.

With the ascent of increasingly more powerful computers, mathematicians have been able to create remarkably sophisticated models, but some of the most important breakthroughs in mathematical modeling came long before computers were available and involved relatively simple models. Sir Isaac Newton instituted a revolution in science with his laws of gravitation and planetary motion based upon a simple but powerful mathematical model involving just two differential equations and their ramifications. Newton's remarkable model formed the basis of 20th Century space exploration including the first landing of a man on the moon in 1969 and the numerous communications satellites currently orbiting the earth above us. In fact, Newton's model was the accepted explanation for all motion in the universe until the beginning of the 20th century when Albert Einstein proposed a more sophisticated model, called relativity theory, to account for certain inadequacies found in Newton's model when studying motion at near light-speed. Other famous mathematical models include James Clerk Maxwell's differential equations of electromagnetism, the equations of quantum mechanics, the laws of population growth and radioactive decay, Mandelbrot's fractal geometry, and the so called "grand unification theories" of modern nuclear physics and cosmology. Several recent Nobel prizes in economics have been awarded to economists who constructed complex mathematical models of some portion of the national economy. Wherever there is a natural system to be studied and understood, one is likely to find a mathematical modeler on the scene constructing a model which, she or he hopes, will capture the essential features of that system.