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Predictions

One of the primary roles of the applied mathematician is to create mathematical models to be used in predicting the outcomes of future events. These models take the form of equations or systems of equations, inequalities or systems of inequalities, computer simulations, or any combination of these that the mathematician finds appropriate.

Mathematics can sometimes give accurate predictions about future events as with Newtonian mechanics, whereas in other cases, such as predicting public opinion and estimating animal populations, the predictions given by mathematics are probabilistic. In cases where the underlying dynamical system is chaotic, such as the weather, mathematical models can be used to describe the long-term behavior of the system, but such systems by their very nature defy specific predictions due to their sensitivity to initial conditions.

Predictive Models and their Validity

The predictive validity of a mathematical model can only be judged by how well it actually forecasts future events. Models that fail to give predictions in accordance with empirical observation are either abandoned or modified. The creation of a predictive mathematical model is part science, part mathematics, and part art.

The science and mathematics of model building are often straightforward once the conditions and assumptions about the events being modeled are set. It is in the area of making correct assumptions that a bit of art is required. On the one hand, the mathematician wants to keep the model as simple as possible, and make only the assumptions that are absolutely necessary for the process being modeled. On the other hand, leaving out some critical assumption can make the model useless in predicting future events. Thus the mathematician must balance the desire for simplicity in the model with the need to consider all relevant assumptions about the real-world processes being modeled.

In some situations, the predictive ability of a mathematical model is quite good. A classic example is Newtonian mechanics as applied to the motion of planets in the solar system. Using Newton's laws, astronomers are able to predict the exact minute a lunar or solar eclipse will occur hundreds of years in the future. Such is the predictive power of Newton's model that humans can send a spacecraft to orbit around or land on distant planets.

In other situations, such as weather forecasting, mathematical models have proved disappointing in their ability to accurately predict conditions very far in the future. In theory, one ought to be able to use Newtonian mechanics to predict the motion of particles in the atmosphere and, therefore, to predict the weather, but, in fact, when this is attempted, predictions are not so good.

The Role of Chaos. It is ironic that while it is possible to launch a spacecraft from Cape Canaveral and have it reach one of Saturn's moons at a precise time years from now, humans cannot predict with much certainty whether tomorrow's weather at the Cape will allow the launch to proceed.

The reason that the motion of celestial bodies can be predicted with a high level of accuracy, but weather conditions cannot, is related to a phenomenon known as sensitivity to initial conditions. In systems that exhibit sensitivity to initial conditions, even the smallest error in one's knowledge of the initial conditions of the system can lead to large errors in predicting the outcome of that system. Such systems are said to be "chaotic," not in the usual English language meaning of the word, but according to a precise mathematical definition. On the timescale of human existence, the solar system is not chaotic, and is therefore predictable. The weather is chaotic and hence highly unpredictable.

Statistical Models

Systems that fall somewhere between the high predictability of the solar system and the low predictability of the weather can sometimes be modeled using the mathematics of probability and statistics. In making predictions based on probabilities, it is important to understand from the outset that the predictions should not be viewed as certain. Statistical theory, though, makes it possible to quantify the degree of uncertainty present in predictions.

Polling. If an opinion poll of 1,000 randomly selected people was taken to predict the sentiments of the entire country on a certain issue, then the results of that poll will typically be reported as having a margin of error of about plus or minus 3 percentage points with 95 percent confidence. This means that, if the poll showed 65 percent of the sample in favor of a certain proposition, you could be 95 percent confident that between 62 percent and 68 percent of the entire population favored the proposition. It would be inappropriate to say that 65 percent of the population favored the proposition, because the statistical theory does not allow the making of this type of precise prediction.

Sampling and Estimation. It is often necessary for wildlife agencies to predict the number of a certain species of animal living in a particular habitat. Since it is usually impossible to do an actual count of wild animals, some method of estimation is necessary. Often a technique known as "capturerecapture" is used.

The "capture-recapture" method involves capturing a number of the animals and tagging or marking them in some way and then releasing them back into the habitat. After a period of time sufficient to allow the tagged animals to mix back into the population of untagged animals, a second capture or "recapture" is carried out, and the number of tagged animals in the recapture group is counted. Then, using a proportion, an estimate of the number of animals in the entire population can be made.

To illustrate, suppose that 50 deer are captured, tagged, and released back into the habitat. Several weeks later, 80 deer are captured, 16 of which are tagged. Then, letting N be the number of deer in the entire population, set up the proportion 16/80=50/N and solve for N to get 250 as our estimate of the entire herd.

Now, as with the opinion polling example, you cannot conclude that there are exactly 250 deer in the entire herd. In fact, because the sample size is only 80 compared with 1,000 in the opinion poll, you get an even wider margin of error. In this case, the mathematical statistics says that you can be 95 percent confident that the true number of deer in the herd lies between 213 and 303.

see also Chaos; Data Collection and Interpretation; Endangered Species, Measuring; Polls and Polling; Probability and the Law of Large Numbers; Statistical Analysis.

Stephen Robinson

Bibliography

"Exploring Surveys and Information from Samples." In The Quantitative Literacy Series. Parsipanny, NJ: Dale Seymour Publications, 1998.

Moore, David S. Statistics: Concepts and Controversies. San Francisco: W. H. Freeman and Company, 1979.

Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.